Stability of spherically symmetric solutions in modified theories of gravity
氢原子量子力学模型
氢原子量子力学模型英文回答：The quantum mechanical model of the hydrogen atom is a fundamental concept in physics that describes the behavior of a single hydrogen atom. This model is based on the principles of quantum mechanics, which is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels.In the quantum mechanical model, the hydrogen atom is treated as a system consisting of a single electron orbiting a nucleus. The electron is described by a wave function, which is a mathematical function that determines the probability of finding the electron at a particular position in space. The wave function is governed by the Schrödinger equation, which is a differential equation that describes the behavior of quantum systems.The wave function of the hydrogen atom can be solvedanalytically, resulting in a set of wave functions called the hydrogen atom orbitals. These orbitals describe the different energy levels and spatial distributions of the electron in the hydrogen atom. The lowest energy level is called the ground state, while higher energy levels are called excited states.Each orbital is characterized by a set of quantum numbers, which specify the energy, shape, and orientation of the orbital. The principal quantum number (n) determines the energy level of the orbital, with larger values of n corresponding to higher energy levels. The azimuthal quantum number (l) determines the shape of the orbital, with different values of l corresponding to different shapes such as s, p, d, and f orbitals. The magnetic quantum number (m) determines the orientation of theorbital in space.For example, the 1s orbital is the ground state orbital of the hydrogen atom, with n=1, l=0, and m=0. This orbital is spherically symmetric and has the lowest energy level. The 2s and 2p orbitals are examples of excited stateorbitals, with n=2. The 2s orbital is spherically symmetric like the 1s orbital, while the 2p orbitals have different shapes and orientations.The quantum mechanical model of the hydrogen atom provides a detailed understanding of the behavior of electrons in atoms. It explains phenomena such as the quantization of energy levels, the stability of atoms, and the formation of chemical bonds. This model has been successful in predicting and explaining a wide range of experimental observations in atomic physics.中文回答：氢原子的量子力学模型是物理学中的一个基本概念，描述了单个氢原子的行为。
The field of heterogeneous catalysis中文翻译
Rational Design of Low-Temperature Hydrogenation Catalysts:Theoretical Predictions and Experimental Verification低温加氢催化剂的设计: 理论与实践The field of heterogeneous catalysis, specifically catalysis on bimetallic alloys, has seen many advances over the past few decades. One of the main goals of the catalysis industry is to develop new materials that have novel catalytic properties. Bimetallic catalysts, which often show electronic and chemical properties that are distinctly different from those of the parent metals, offer the opportunity to design new catalytic materials with enhanced activity, selectivity, and stability [1-2] . Currently bimetallic catalysts are widely utilized in many heterogeneous catalysis [3] and electro-catalysis [4] applications.In order to understand the origins of the novel catalytic properties, bimetallic surfaces have been the subject of many experimental and theoretical studies, as summarized in several reviews [5-7] . It is now well established that bimetallic surfaces often show novel properties that are not present on either of the parent metal surfaces [5-16] . The modification effect is especially important when the admetal coverage is in the sub monolayer to monolayer regime. However, it is difficult to know a priori how the electronic and chemical properties of a particular bimetallic surface will be modified relative to the parent metals. For this reason, the study of bimetallic surfaces in the field of catalysis has gained considerable interest. There are two critical factors that contribute to the modification of the electronic and chemical properties of a metal in a bimetallic surface. First, the formation of the hetero-atom bonds changes the electronic environment of the metal surface, giving rise to modifications of its electronic structure through the ligand effect. Second, the geometry of the bimetallic structure is typically different from that of the parent metals, e.g. the average metal-metal bond lengths change. This lattice mismatch leads to the strain effect that is known to modify the electronic structure of the metal through changes in orbital overlap [16] . While studies on model bimetallic surfaces provide fundamental insights into the novel properties, in an industrially relevant supported catalyst the active metal will be present in the form of nanoparticles. As shown in Fig.1, research efforts in our group involve three parallel approaches, with the goals being to bridge the“materials gap”and“pressure gap”between fundamental surface science studies and real world catalysis. In the current review we will utilize hydrogenation reactions as examples to demonstrate how the utilization of these three parallel approaches can lead to the rational design of bimetallic catalysts with novel low-temperature hydrogenation activities.Catalytic hydrogenations are among the most commonly practiced catalytic processes, ranging from common steps in organic synthesis, to batch processes in pharmaceutical production, to stabilization of edible oils, and to petroleum upgrading processes. Because hydrogenation reactions are typically exothermic, it is advantageous to carry out these reactions at low temperatures. In the current review we will first use the hydrogenation of cyclohexene to demonstrate the feasibility of increasing the low-temperature hydrogenation activity by reducing the binding energies of atomic hydrogen and cyclohexene, which can be achieved by designing bimetallic surfaces with specific surface structures. We will then discuss several other types of hydrogenation reactions to further illustrate the advantages of bimetallic catalysts in terms of both hydrogenation activity and selectivity.1 Structures of bimetallic surfacesIn the current review we will focus mainly on bimetallic surfaces by depositing one monolayer ofa 3d transition metal on either a Pt(111) single crystal or a polycrystalline Pt substrate. As shown in Fig.2, monolayer bimetallic surfaces can have three ideal configurations: a surface 3d-Pt-Pt(111) configuration, where the 3d monolayer grows epitaxially on the surface of the Pt substrate; an intermixed configuration, where the 3d atoms reside in the first two Pt layers to some varying degree; and the unique subsurface Pt-3d-Pt(111) configuration, where the first layer is comprised of Pt atoms and the second layer is occupied with the 3d metals.Procedures for the preparation of bimetallic surface structures under ultra-high vacuum (UHV) conditions have been described in detail previously [5] . For example, the Ni/Pt(111) bimetallic surfaces have been characterized using a wide range of experimental techniques and DFT modeling [17] . When Ni is deposited with the Pt(111) surface held at 300 K, Ni atoms stay on the top-most layer to produce the Ni-Pt-Pt(111) surface configuration. If this surface is subsequently heated to 600 K, or if the monolayer deposition of Ni occurs with the Pt(111) substrate held at 600 K, most of the Ni atoms diffuse into the subsurface region to produce the Pt-Ni-Pt(111) subsurface structure. Similar surface and subsurface structures have been obtained for several other 3d metals on the Pt(111) substrate [5,18] .The ab initio calculations in the current review were performed using the Vienna Ab initio Simulation Package (V ASP) version 4.6 [19-20] . The monolayer bimetallic systems were modeled on the closed-packed Pt(111) substrate. The PW91 functional was used within the generalized gradient approximation with an energy cutoff on the basis set of 396 eV. The bimetallic systems were modeled using a periodic 2*2 or 3*3 unit cell with four metal layers, with the slabs being separated by 6 equivalent layers of vacuum in the epitaxial direction. The top two layers were allowed to relax to the lowest energy configuration while the third and fourth layers were frozen at the bulk Pt-Pt distance. More details about the DFT modeling procedures on monolayer bimetallic surfaces can be found in a recent review [5] .2 Low-temperature hydrogenation of cyclohexeneCyclohexene is used as a probe molecule to study the hydrogenation because cyclic hydrocarbons are important reaction intermediates in many refinery and petrochemical processes, in addition to serving as building blocks for many chemicals produced in the chemical industry. Furthermore, cyclohexene has several competitive reaction pathways, including decomposition, dehydrogenation, disproportionation (self-hydrogenation), and hydrogenation. Comparative studies of these reaction pathways provide an opportunity to determine how the hydrogenation activity and selectivity are affected by the formation of bimetallic surfaces.2.1 DFT and experimental studies on single crystal surfacesOne hypothesis for promoting the low-temperature hydrogenation of alkene is that both reactants, atomic hydrogen and alkene, should bond relatively weakly on the catalyst surface to facilitate the hydrogenation steps. DFT calculations were performed to estimate the values of hydrogen binding energy (HBE) on several 3d-Pt-Pt(111) and Pt-3d-Pt(111) surfaces, as shown in Fig.3A [18] . Fig.3A reveals that HBE is related to the position of the surface d-band center with respect to the Fermi level, in agreement with the trend observed in previous studies for other surfaces [5] . In general, the addition of a 3d metal surface layer on Pt(111)moves the d-band center closer to the Fermi level as compared to the bulk 3d metals. This is primarily due to the tensile strain induced by the Pt lattice as the ligand effect is the weakest between late transition metal over layers and the Pt(111) substrate [17] . Conversely, subsurface 3d metals shift the surface d-band center of Pt away from the Fermi level as compared to that of Pt(111), mainly due to the electronic interactionof Pt and the subsurface 3d atoms [17] . The comparison in Fig.3A demonstrates that HBE typically follows the trend of 3d-Pt-Pt(111)>Pt(111)> Pt-3d-Pt(111). In addition, the nearly linear correlation between HBE and the surface d-band center should enable one to predict HBE on other bimetallic surfaces based on the extensive database of d-band center values for many bimetallic surfaces [5] .In addition to the trend in the correlation of HBE with surface d-band center, the binding energies of unsaturated hydrocarbons, such as cyclohexene, follows the same trend as HBE. As shown in Fig.3B, DFT calculations reveal that the Pt-3d-Pt(111) subsurface structures bond to cyclohexene more weakly than Pt(111) and the corresponding 3d-Pt-Pt(111) surface structures [18] . For example, DFT results indicate that both cyclohexene and atomic hydrogen are more weakly bonded on Pt-Ni-Pt(111) than on Ni-Pt-Pt(111), Pt(111) and Ni(111), suggesting that the subsurface Pt-Ni-Pt(111) structure should be more effective in the hydrogenation of cyclohexene than the surface structure. This has been confirmed experimentally by comparing the hydrogenation activity of cyclohexene using temperature programmed desorption (TPD), as shown in Fig.4 [18] . As illustrated in the TPD peak area of the cyclohexane product, the subsurface Pt-Ni-Pt (111) structure shows the highest hydrogenation yield, with the desorption peak centered at a very low temperature of 203 K. Similar bimetallic surface structure can also be produced by depositing one monolayer of Pt on a Ni(111) substrate, which also possesses the novel low-temperature pathway for cyclohexene hydrogenation [21] .The trend in the DFT calculations in Fig.3B also shows that the binding energy of cyclohexene on Pt-Co-Pt(111) and Pt-Fe-Pt(111) is even weaker than that on Pt-Ni-Pt(111). Although this might suggest that the former two surfaces would be more active toward the hydrogenation than Pt-Ni-Pt(111), one should keep in mind that the adsorption of cyclohexene needs to be strong enough for the hydrogenation to take place. One would therefore expect to observe a volcano relationship for the hydrogenation activity as the d-band center moves further away from the Fermi level, i.e., when the adsorption of cyclohexene becomes too weak for the hydrogenation to occur. This is verified experimentally in the results shown in Fig.5. The hydrogenation yield from TPD measurements is the highest on Pt-Ni-Pt(111), but starts to decrease on the Pt-Co-Pt(111) and Pt-Fe-Pt(111) surfaces, where the binding of cyclohexene becomes too weak for hydrogenation to occur. On the other side of the volcano curve, the binding energies of cyclohexene on the 3d-Pt-Pt(111) surfaces are too strong, preventing the effective hydrogenation of cyclohexene [18] 2.2 Polycrystalline bimetallic surfacesIndustrial catalysts are often supported nanoparticles of varying shape and size. Polycrystalline bimetallic films provide a potential way to bridge the “materials gap”between single crystal surfaces and supported catalysts. As illustrated in Fig.6, it is possible to assume that the surface chemistry of the nanoparticle should be dominated primarily by the first few atomic layers. It is also reasonable to assume that the chemistry of the individual crystal facets on the nanoparticle (primarily (111) and (100) for an FCC nanoparticle) can be approximated by their respective single crystal extension [22] .With these assumptions we have investigated the chemical properties of 3d-Pt bimetallic structures prepared on a polycrystalline Pt film that contained mainly the (111) and (100) facets. Similar to Pt(111), monolayer Ni was deposit on a Pt foil at room temperature to produce the Ni-Pt-Pt surface structure, followed by annealing to higher temperatures to obtain the Pt-Ni-Pt subsurface structure [22] . The TPD results of the hydrogenation of cyclohexene are shown inFig.7. Similar to the corresponding single crystal surfaces, the subsurface Pt-Ni-Pt polycrystalline structure shows significantly higher hydrogenation activity than that from the polycrystalline Pt and Ni surfaces. These results confirm the assumption that the trend observed on single crystal bimetallic surfaces can be extended to the polycrystalline counterparts.2.3 Thermodynamic stability of bimetallic surfaces under hydrogenation conditionsBefore extending the surface science results to supported catalysts, it is important to verify that the desirable Pt-Ni-Pt subsurface structure is the thermodynamically preferred configuration under hydrogenation conditions. As demonstrated in several recent studies, including single crystal surfaces [23-24] , polycrystalline films [22,25] and supported catalysts [26] , the thermodynamically preferred Pt-Ni bimetallic structure is directly related to the chemical environment present on the surface. Fig.8 shows the DFT predicted potential for segregation for a 3d metal atom to segregate from the subsurface to the surface of Pt(111). These values were calculated for the environments of vacuum, and with 0.5 monolayer (ML) atomic hydrogen and 0.5 ML atomic oxygen, using procedures described previously [24] . The thermodynamic potential for segregation is defined as follows:where ΔE seg is the thermodynamic potential for segregation per Pt-3d pair, E A/3d-Pt-Pt is the total energy for the surface configuration with adsorbate A, E A/Pt-3d-Pt is the total energy for the subsurface configuration with adsorbate A, and M is the total number of Pt-3d pairs per unit cell. As defined in a previous publication [22] , a positive ΔE seg value indicates that the subsurface Pt-3d-Pt is more stable. The DFT results in Fig.8 predict that for the reducing environment of vacuum and 0.5 ML atomic hydrogen, the subsurface configuration is thermodynamically preferred, whereas in 0.5 ML atomic oxygen the surface configuration is preferred. There is a nearly linear trend between ΔE seg and the difference in d-band, ΔƐd , which leads to a generalized equation in predicting the thermodynamic stability of a wide range of bimetallic surfaces [24] . Because the environment of hydrogenation reactions is similar to that of the reducing environment, with the bimetallic surface being partially covered by hydrogen, the results in Fig.8 suggest that the desirable subsurface Pt-Ni-Pt configuration should be thermodynamically stable, making it possible to extend model surfaces to supported catalysts for hydrogenation reactions.2.4 Synthesis and evaluation of supported catalystsSupported monometallic Pt and bimetallic Ni-Pt and Co-Pt catalysts were synthesized on γ-Al 2 O 3 using the incipient wetness method [27-28] . The catalysts were characterized using a variety of techniques, including extended X-ray absorption fine structure (EXAFS). The utilization of EXAFS is critical in these studies because it provides direct information on the extent of bimetallic bond formation based on the coordination numbers of Ni-Pt and Co-Pt under in-situ reaction conditions. As summarized in Table 1, the detection of the Ni-Pt and Co-Pt nearest neighbors confirms that bimetallic bonds are indeed produced on the supported catalysts [28] . The supported catalysts were evaluated using both batch and flow reactors to determine the reaction kinetics of the hydrogenation of cyclohexene at a low temperature of 303 K [28] . Fig.9 shows the production of cyclohexane from cyclohexene on Pt/γ-Al 2 O 3 , Co-Pt/γ-Al 2 O 3 , and Ni-Pt/γ-Al 2 O 3 , using a batch reactor equipped with Fourier transform infrared (FTIR)spectroscopy. The solid lines are fittings using the Langmuir-Hinshelwood model, resulting in a rate constant of 1.7, 21 and 24 min -1 for supported Pt, Co-Pt, and Ni-Pt, respectively [28] . The trend observed in the rate constant of cyclohexene hydrogenation is consistent with that from the single crystal surfaces for the same reaction, Ni-Pt>Co-Pt>Pt, as shown earlier in the volcano curve in Fig.5. The observation of the similar trend between model surfaces and supported catalysts provides an important demonstration of the rational design of bimetallic catalysts from combined theoretical and experimental approaches.3 Research opportunities in bimetallic catalysis3.1 Low-temperature hydrogenation reactionsWe have applied similar combined approaches for the design of bimetallic catalysts for the low-temperature hydrogenation of several types of hydrocarbon molecules. Below we will provide several examples of hydrogenation reactions that are of both fundamental and practical importance.Hydrogenation of acrolein. Studies of the selective hydrogenation of unsaturated aldehydes, such as αβ-unsaturated aldehydes, have been of growing interest for the production of fine chemical sand pharmaceutical precursors [29] .The hydrogenation of the C=C and/or C=O bonds in unsaturated aldehydes offers the possibility to improve both the hydrogenation activity and selectivity through the formation of bimetallic surfaces. Using the hydrogenation of acrolein as a probe reaction, we demonstrated that the selective hydrogenation of the C=O bond can be achieved through the formation of the subsurface Pt-Ni-Pt(111) and Pt-Co-Pt(111) bimetallic structures [30-31] .Hydrogenation of benzene.The hydrogenation of benzene to cyclohexane is of significant importance in the petroleum industry and for environmental protection. The process of benzene hydrogenation has been utilized commercially for the production of cyclohexane, which is one of the key intermediates in the synthesis of Nylon-6 and Nylon-66 [32] . We have identified Co-Pt bimetallic catalysts as promising materials for the hydrogenation of benzene at a relatively low temperature of 343 K [28,33] . For example, Table 2 summarizes the batch and flow reactor results of benzene hydrogenation on several Co-based bimetallic catalysts. The Co-Pt catalyst shows the highest rate constant and lowest activation barrier for the hydrogenation of benzene, which is consistent with the relatively weak binding energies of atomic hydrogen and benzene from DFT calculations [33] . In addition, the catalyst support also plays a role in controlling the hydrogenation activity of Co-Pt catalysts [34]Selective hydrogenation of acetylene in ethylene. The selective hydrogenation of acetylene in the presence of ethylene is an important reaction because acetylene poisons the catalysts in ethylene polymerization reactions [35-36] . By supporting Pd-Ag bimetallic catalysts on ion-exchanged β-zeolites, we observed a synergistic effect that led to a higher selectivity for acetylene hydrogenation in the presence of excess ethylene [37] .The increase in the hydrogenation selectivity is attributed to a combination of an enhanced π-cation interaction between acetylene and zeolite at low temperatures and the ability of the Pd-Ag bimetallic catalysts to perform hydrogenation at such low temperatures [37] .3.2 Reducing bulk Pt in bimetallic catalysts with metal carbidesAs demonstrated in Figs.3-5, the subsurface Pt-Ni-Pt structure is desirable to enhance the activity and selectivity of the hydrogenation of unsaturated hydrocarbons. However, if elevated temperatures are required for reactions, the subsurface Ni atoms start to diffuse into bulk Pt,leaving a monometallic Pt surface and therefore the disappearance of the enhanced bimetallic hydrogenation activity [17,22] . In addition, as shown in Fig.8, adsorbates such as oxygen can cause the subsurface Ni atoms to segregate to the surface, forming the Ni-Pt-Pt surface that is not active for hydrogenation reactions. One idea to overcome such inherent instability of Pt-Ni-Pt is to replace the bulk Pt with an alternative substrate, such as transition metal carbides that often show catalytic properties similar to Pt [38-44] . We have explored the utilization of tungsten monocarbide (WC) to produce the Pt-Ni-WC structure [45] . As WC has been shown to be an effective diffusion barrier layer [46] , thermal deactivation due to Ni diffusion will be alleviated. Furthermore, it is also possible that the WC substrate would anchor Ni by the formation of W—Ni or C—Ni bonds to prevent its segregation to the surface in an oxygen-rich environment. Fig.10 shows a comparison of the hydrogenation of cyclohexene from Pt-Ni-Pt and Pt-Ni-WC surfaces [45] . The Pi-Ni-WC surface shows higher hydrogenation activity, which is consistent with parallel DFT calculations [45] . The promising results in Fig.10 suggest the possibility to synthesize a more active and stable Pt-Ni-WC hydrogenation catalyst with much lower loading of Pt than that in Pt-Ni-Pt.3.3 Production of hydrogen using bimetallic catalystsAll examples presented above are for hydrogenation reactions, which can be classified as hydrogen-consuming reactions and require catalysts to bond to atomic hydrogen and adsorbates relatively weakly. Using the surface d-band center argument in Fig.3, these reactions are preferred on bimetallic catalysts with surface d-band center away from the Fermi level, such as the Pt-3d-Pt subsurface structures. In contrast, for hydrogen-producing reactions, the desirable catalysts should be those that bond to hydrogen and adsorbates more strongly, i.e., with d-band center closer to the Fermi level, such as the 3d-Pt-Pt surface structures. This hypothesis has been confirmed experimentally in our recent studies for the production of H 2 from the reforming of biomass derived molecules, including ethanol, ethylene glycol and glycerol on the Ni-Pt-Pt(111) surface [47] . Similarly, the Ni-Pt-Pt(111) surface also shows a very high activity for H 2 production from the decomposition of ammonia [48] .We have also demonstrated that coking of the catalyst surfaces, a common deactivation mechanism in dehydrogenation reactions, can be reduced by the formation of bimetallic surfaces [49] . These results further demonstrate the possibility of designing bimetallic catalysts from combined theoretical predictions and experimental verification.4 ConclusionsThe field of catalysis is undergoing a revolution in the selection process of catalytic materials, from the traditional“trial-And-error”method to the“rational design”approach, with the latter requiring atomic level understanding of the catalyst structures and reaction mechanisms. In the current review we utilized several hydrogenation reactions to demonstrate the importance of combining theoretical and experimental approaches for designing bimetallic structures with desirable catalytic properties. In addition, the examples also illustrated the possibility to bridge the “materials gap”and “pressure gap”between fundamental studies on single crystal surfaces and catalytic evaluation of supported catalysts. Similar approaches can be adopted for the rational design of bimetallic catalysts beyond hydrogenation reactions.多相催化领域，特别是在催化作用双金属合金，在过去的几十年里看到了许多进展。
有机化学结构与功能第一章
The structure of the molecule is related to the reactions that it can undergo
What is organic synthesis? Organic synthesis is the making of new organic molecules
有 机化
H
C Br
Cl F
1
学
Chapter 1 Structure and Bonding In Organic Molecules
1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 The Scope of Organic Chemistry: An Overview Coulomb Forces: A Simplified View of Bonding Ionic and Covalent Bonds: The Octet Rule Electron-Dot Model of Bonding: Lewis Structures Resonance Forms Atomic Orbitals: A Quantum Mechanical Description Molecular Orbitals and Covalent Bonding Hybrid Orbitals: Bonding in Complex Molecules Structures and Formulas of Organic Molecules
14
Elements tend to form molecules in such a way
物理专业 词汇S2
specific conductance 导电率
specific electronic charge 电子的比电荷
specific gravity 比重
specific gravity bottle 比重瓶
specific heat 比热
spectroscopic parallax 分光视差
spectroscopic photography 分光摄影术
spectroscopy 光谱学
spectrum 光谱
spectrum locus 光谱轨迹
spectrum selector 光谱选挥器
spectrum variable 光谱变星
source of light 光源
source of sound 声源
south pole 南极
space 空间
space astronomy 空间天文学
space charge 空间电荷
space charge density 空间电荷密度
space charge effect 空间电荷效应
spectrohelioscope 太阳光谱观测镜
spectrometer 光谱仪
spectrometry 光谱测定法
spectrophotofluorometer 荧光分光光度计
spectrophotography 光谱摄影学
spectrophotometer 分光光度计
spectrophotometry 光谱测定法
spark chamber 火花室
spark counter 火花计数器
英语作文分析椭圆
IntroductionThe ellipse, a fundamental geometric figure that transcends the boundaries of mathematics and permeates various scientific disciplines, is a captivating subject for in-depth analysis. This elegant shape, defined as the locus of points such that the sum of the distances from two fixed points (the foci) is constant, embodies a rich tapestry of properties, applications, and theoretical underpinnings. This essay embarks on a comprehensive, multifaceted exploration of ellipses, delving into their mathematical definition, properties, conic section origins, practical applications, and connections to advanced mathematical concepts.I. Mathematical Definition and PropertiesAn ellipse is a two-dimensional curve characterized by its eccentricity, major axis, minor axis, and foci. Its defining property, as mentioned earlier, is that for any point P on the ellipse, the sum of the distances from P to the two foci, F1 and F2, remains constant:PF1 + PF2 = 2a,where 'a' is the semi-major axis, or half the length of the major axis, which is the longest chord passing through the center of the ellipse. The semi-minor axis, 'b', is the corresponding half-length of the minor axis, perpendicular to the major axis.The eccentricity, denoted by 'e', quantifies the "flatness" of the ellipse, ranging from 0 (a circle) to less than 1 (a non-circular ellipse). It is given by the ratio of the distance between the foci and the length of the major axis:e = c/a = √(a² - b²)/a,where 'c' is the distance between the center and either focus. The area of an ellipse is A = πab, while its perimeter, or circumference, cannot be expressed in a simple closed-form expression but can be approximated using various numerical methods.Ellipses exhibit several notable properties, including:1. Reflective Property: Light emanating from one focus reflects off theellipse and passes through the other focus.2. Conjugate Diameters: Any two diameters of an ellipse that intersect at right angles are called conjugate diameters. Their midpoints lie on a circle called the director circle.3. Harmonic Property: If a tangent and a secant of an ellipse intersect ata point outside the ellipse, then the product of the lengths of the segments of the secant is equal to the square of the length of the tangent segment.II. Origins in Conic SectionsEllipses find their roots in the study of conic sections, dating back to the work of ancient Greek mathematicians such as Apollonius of Perga. A conic section is formed when a plane intersects a double-napped cone at various angles. When the plane is neither parallel nor perpendicular to the cone's axis and intersects both nappes, it generates an ellipse. This perspective provides a geometric intuition for the ellipse's defining property, as the sum of distances from any point on the cutting plane to the cone's apexes is constant.III. Practical ApplicationsEllipses have myriad practical applications across various fields, exemplifying their versatility and significance:A. Astronomy: The orbits of planets and other celestial bodies around the sun or any central mass are elliptical due to the laws of gravitational attraction, as postulated by Johannes Kepler and later incorporated into Isaac Newton's laws of universal gravitation. The Earth's orbit, for instance, is an ellipse with the Sun at one focus.B. Architecture and Engineering: Elliptical shapes are frequently employed in architectural designs for aesthetic appeal and structural efficiency. Examples include elliptical domes, bridges, and arches, such as the famous St. Louis Gateway Arch. In engineering, ellipses are used in the design of reflectors and lenses for optical systems, ensuring optimal light distribution or focusing.C. Image Processing and Computer Graphics: In digital imaging, the Hough transform is a technique used to detect lines, circles, and ellipses in images.Ellipses also feature prominently in computer graphics, particularly in modeling and rendering curved surfaces, texture mapping, and in the design of user interfaces.D. Geographical Information Systems (GIS): Ellipsoidal approximations are used to model the Earth's surface in GIS applications, accounting for the planet's slightly oblate spheroidal shape rather than treating it as a perfect sphere. This is crucial for accurate geodetic calculations, cartography, and satellite navigation systems like GPS.IV. Connections to Advanced Mathematical ConceptsEllipses serve as a gateway to deeper mathematical concepts and theories:A. Elliptic Integrals and Functions: Problems involving arc lengths, areas, and volumes of solids of revolution derived from ellipses lead to the study of elliptic integrals, which are a class of special functions with numerous applications in physics, engineering, and number theory. Elliptic functions, which generalize trigonometric functions, arise as inverses of elliptic integrals and play a vital role in the theory of complex analysis.B. Elliptic Curves: In modern algebraic geometry, elliptic curves are a particular type of smooth, projective curve defined by a cubic equation in two variables. Although they are named due to their superficial resemblance to ellipses, their intrinsic properties and applications, such as in cryptography and number theory (specifically the Birch and Swinnerton-Dyer conjecture), are vastly different.C. General Relativity: In Einstein's general theory of relativity, the path followed by a particle in a gravitational field is described by a geodesic, which in the case of a spherically symmetric mass distribution is an ellipse in the weak-field limit. This provides a direct link between the ellipse's fundamental nature and our understanding of gravity and spacetime curvature.ConclusionThe ellipse, a seemingly simple geometric figure, reveals itself as a profound and multifaceted construct upon closer examination. From its elegantmathematical definition and intriguing properties to its origins in conic sections, practical applications across diverse fields, and connections to advanced mathematical concepts, the ellipse stands as a testament to the beauty and power of mathematics in describing and understanding the world around us. This in-depth analysis underscores the ellipse's enduring significance and invites further exploration of its intricate relationships with various branches of science and mathematics.。
On spherically symmetric metric satisfying the positive kinetic energy coordinate condition
22
E W Ω , g 01 = g 10 = 2 2 , g 11 = 2 2 , 2 2 2 W Ω +E W Ω +E W Ω + E2
2
2
2
=
1 R2
,g
33
=
1 R 2 sin 2θ
(1-9)
; g
µν
= 0, others;
g ij = ΩR 2 sin θ and the positive kinetic energy coordinate condition (0-5) becomes
σ σ σ ρ ρ ρ
& = ∂K (t , r ) , K ′ = ∂K (t , r ) , etc; we have where K ∂t ∂r λ 0 2 2 2 g ij g = sin θ K Q R (rρ ′ tσ ′ − tρ ′ rσ ′) Ω3 g 00 ,λ
LGNK 2 = 3g 0λ g ij g g 00 , ,λ
2
(0-1)
(0-2)
where g = g µν < 0 , g ij is the determinant of the 3- dimensional metric g ij . For the Schwarzschild metric indicated by the line element rs 2 1 ds 2 = − dr 2 + r 2 dθ 2 + sin 2 θdϕ 2 , (0-3) dt + 1 − r rs 1− r ∂ g ij ∂ r 4 sin 2 θ = = 0 in the area of rs < r , hence, according where rs = 2m , we see that rs ∂t ∂t 1− r to (0-2), in the area of rs < r the negative kinetic energy term of gravitation field vanishes. But (0-3) cannot be continued into the area of 0 < r < rs . For continuing (0-3) into the area of 0 < r < rs , one has used the method of coordinate transformation and obtained some metrics,
Chap2 第二章 微透镜(2)
2.2微球透镜2.2.1 均匀介质球透镜一．球透镜的性质仅有两个参数D：ball lens diametern：index of refraction通常商用透镜直径在0.3—10mm。
有效焦距EFL(effective focus length)后焦距 BFL （back focus length ）2D F BFL −= ()14E −=n nD FL 显然焦点的位置与输入直径d 有关。
反映了球透镜固有的球差特性。
数值孔经()nD n d 12NA −＝数值孔经随增大而增大，。
见图2D d /二．在光纤耦合应用用于光纤耦合的球透镜，应考虑光纤的NA 和激光束的直径。
因为光束直径与球透镜的NA 有关。
球透镜的NA 要略小于光纤的数值孔经。
图3中光纤直接贴着球透镜放置。
三．例子LD 光束直径 1.7mm ，光纤NA=0.14（λ=1310nm ）,用BK7玻璃（n=1.50,λ=1310nm ）,可计算出理想透镜的直径为8.09mm ，可选择直径8mm 的元件。
即()mm nNAn d D nD n d 09.814.05.1)15.1(7.12)1(2;12NA =×−××=−=−＝图4为用两球透镜实现光纤对光纤的耦合，其中光纤紧贴球透镜放置。
2.2.2 变折射率球透镜In optical design we often aspire to two objectives; excellent image fidelity and high flux concentration at high radiative efficiency. But maximum flux concentration requires large convergence angles, at which it becomes exceedingly difficult to eliminate all geometric aberrations. Can an imaging system be free of all geometric aberrations and still attain the thermodynamic limit to flux concentration? There is a conjecture that such an ideal optical system can not be constructed from any finite number of mirrors or homogeneous lenses. Although the hypotheses has remained unproved, no exceptions to it have been found.However, with the extra degree of freedom of a refractive-index gradient, there are spherically symmetric solutions that produce perfect imagery at maximum concentration.( Whether spherical symmetry is necessary to produce perfect imaging at the thermodynamic limit for concentration also remains a unresolved question.).All the geometrical optics ,i.e. the law of refraction and reflection, can be derived from Fermat’s principle .Fermat’s principle states that a physically possible ray path is one for which the optical path length along it from A to B is an extremum as compared to neighboring paths, where “extremum” mean having zero derivatives in space, that usually means “minimum,” as Fermat’s original statement. This leads to the result that the geometrical wave front are orthogonal to the ray(the theorem of Malus and Dupin) ,i.e. ,the ray is normal to the wave front. This in turn means that if there is no aberration ,i.e. all ray meet at one point, then the wave fronts must be portions of spheres , and all the refractive and reflecting surface must be concentric portions of spheres. Thus ,also,if there is no aberration , the optical path length from the object point to image point is the same along all rays. Aberrations can then be considered in term of the departure of wave fronts from the ideal spherical shape. This concept is helpful to understand such few “perfect” optical system such as Maxwall’s fisheye and the Lunemburg lens.1．Maxwell fish eye lensRef:Herzberger, M.(1958) “Modern Geometrical Optics” p426.,Weley (Interscience) New York. Born,M.,and E. Wolf(1970). “Principles of Optics,” 4th ed. Pergamon, Oxford.In the first published analysis of gradient-index lenses, Maxwell formulated the problem of finding the refractive index profile n(r) such that a point on a sphere of unit radius in an environment of refractive index n env is imaged to a diametrically opposite point on the same sphere. He demonstrated that the profile2env r 12)(+=n r n (1)is a solution. Maxwell referred to the device as a fish-eye lens(even though it does not actually describe the fish eye). The ray trajectories within the lens are circular arcs. A hemisphere with the refractive-index profile of (1) can sever as a focusing lens: A collimated beam impingingupon the circular cross section is focused to a point with exit angle over 2/π±rad. A principal contribution of Maxwell’s proof was demonstrating a perfect imaging system, i.e., free of geometric aberration in all orders, if a continuously varying refractive index is admitted.Generally only points on the surface and within the lens are sharply imaged. Furthermore , the images of extended objects surfer from severe aberration.2. Luneburg lensLuneburg ( or Luneberg) formulated the problem of calculating n(r ) for a spherically symmetric gradient-index lens capable of perfect imaging The source and the focus are located outside (or on)the lens and are collinear with the center of the sphere. Luneburg derived a solution for n(r ) for the special case of practical interest with a far-field source and the focus lying on the surface of a sphere of unit radius:222)(r r n −= (2)The ray trajectories are elliptical arcs . Luneburg’s analysis is trivially extended to the case in which the environment has a refractive index n env greater than unity:)2)(222r n r n env −=（(3)Application:1. As a multi-beam system, it can be used for optical interconnectionswithin electronic system;2. In a variety of optical signal processing application ,it may possible toimplement a varity of 2-D Fourier transformation simultaneously. 3. In wireless communications system ,as an antenna.satellite-basedmobile communication systems.2.3 固体浸液透镜（solid immersion lens）SIL1990年由S.M.Mansfild和G.S.Kino〔APL57(24)2615〕提出一种与浸液显微镜类似的显微透镜，但不同的是全固体的透镜，其聚焦点在固体内部表面，由此由高折射率材料制成的透镜，其内部波长相对空气缩短了1/n，由此分辨率提高了n倍。
三维Stokes近似系统的爆破准则
三维Stokes近似系统的爆破准则郭蒙;郭真华【摘要】Aim To study the initial boundary value problem of the three-dimensional Stokes approximation equations in a bounded smooth domain. Methods Contradiction methods. Results A blow-up criterion for the local strong solutions in terms of the gradient of the velocity is established. Conclusion If the velocity satisfies certain condition, a local strong solutions can be continued globally in time.%目的研究三维Stokes 近似系统在一个有界区域上的初始边界值问题.方法利用反证法.结果建立了仅关于速度梯度的局部强解爆破准则.结论如果速度满足一定条件,那么局部强解将关于时间是全局连续的.【期刊名称】《西北大学学报（自然科学版）》【年(卷),期】2011(041)003【总页数】4页(P391-394)【关键词】三维Stokes近似系统；爆破准则；全局解【作者】郭蒙;郭真华【作者单位】西北大学数学系/非线性中心,陕西西安,710127;西北大学数学系/非线性中心,陕西西安,710127【正文语种】中文【中图分类】O175.2可压Navier-Stoke方程可以表示为其中t≥0表示时间，x=(x1，…，xd)∈Rd，ρ(x，t)，u(x，t)=(u1(x，t)，…，ud(x，t))，P=aργ(a＞0，γ≥1)分别表示流体密度，速度和压力，这里黏性系数μ，λ满足μ＞0，λd+2μ≥0。
主要学术贡献
主要学术贡献主要从事非线性偏微分方程的研究，着重探讨流体动力学等领域中的数学理论及其应用。
多次应邀到美国、香港和新加坡等地学术访问或演讲。
主持或参加了多个科研项目，在国内外学术刊物发表论文60余篇。
主讲高等数学，数学物理方程，偏微分方程泛函方法，流体动力学的数学理论，非线性偏微分方程的某些理论。
指导博士研究生11人，其中9人已经毕业并获得博士学位。
指导硕士研究生26人，其中20人已经毕业并获得硕士学位。
合作编写教材《数学物理方程》一部，已由高等教育出版社出版。
获奖励情况(1) 1999年获得教育部科学技术进步一等奖；(2) 2000年获得教育部首届青年教师奖；(3) 2002年获得吉林省第七届青年科技奖；(4) 2006年获得吉林省长春市政府特殊津贴。
科研项目1) “相变和图像处理等领域中的某些非线性扩散方程”，教育部优秀青年教师教学与科研奖励基金项目，项目负责人,2000—2004；2) “图像处理中的非线性扩散模型”，国家自然科学基金项目青年基金项目，项目负责人,2001—2003；3) “流体动力学等领域中的具有退化性或奇异性的某些数学模型”，国家自然科学基金面上项目，项目负责人,2006—2008；4）“数学与其它领域交叉的若干专题”，国家重点基础研究发展计划973计划，参加者，2006—2011；5）“带有奇异性的某些流体动力学模型”,国家自然科学基金面上项目，项目负责人，2010年～2012年。
发表论文目录1.Yuan, Hongjun; Xu, Xiaojing Existence and uniqueness of solutions for a classof non-Newtonian fluids with singularity and vacuum. J. Differential Equations 245 (2008), no. 10, 2871--2916.2.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie; Xu, Xiaojing On theCauchy problem for the evolution $p$-Laplacian equations with gradient term and source. J. Differential Equations 235 (2007), no. 2, 544--585.3.Lei, Yutian; Wu, Zhuoqun; Yuan, Hongjun Radial minimizers of a Ginzburg-Landaufunctional. Electron. J. Differential Equations1999, No. 30, 21 pp.4.Wu, Zhuoqun; Yuan, Hongjun; Yin, Jingxue Some properties of solutions for asystem of dynamics of biological groups. Comm. Partial Differential Equations22 (1997), no. 9-10, 1389--1403.5.Yuan, Hong Jun, Hölder continuity of interfaces for the porous medium equationwith absorption. Comm. Partial Differential Equations 18 (1993), no. 5-6, 965--976.6.Yuan, Hongjun; Wang, Changjia Unique solvability for a class of fullnon-Newtonian fluids of one dimension with vacuum. Z. Angew. Math. Phys. 60 (2009), no. 5, 868--898. 35Q357.Yin, Li; Xu, Xiaojing; Yuan, Hongjun Global existence and uniqueness ofsolution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum. Z. Angew. Math. Phys. 59 (2008), no. 3, 457--474.8.Xu, Xiaojing; Yuan, Hongjun Existence of the unique strong solution for a classof non-Newtonian fluids with vacuum. Quart. Appl. Math. 66 (2008), no. 2, 249--279.9.Wang, Changjia; Yuan, Hongjun Global strong solutions for a class ofheat-conducting non-Newtonian fluids with vacuum. Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3680–3703,10.Lining, Tong; Hongjun, Yuan Classical solutions to Navier-Stokes equationsfor nonhomogeneous incompressible fluids with non-negative densities. J. Math.Anal. Appl. 362 (2010), no. 2, 476–504.11.Lian, Songzhe; Gao, Wenjie; Cao, Chunling; Yuan, Hongjun Study of thesolutions to a model porous medium equation with variable exponent ofnonlinearity. J. Math. Anal. Appl. 342 (2008), no. 1, 27--38.12.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie The limiting problemof the drift-diffusion-Poisson model with discontinuous $p$-$n$-junctions.J. Math. Anal. Appl. 347 (2008), no. 1, 157--168.13.Yuan, Hongjun; Chen, Mingtao Positive solutions for a class of $p$-Laplaceproblems involving quasi-linear and semi-linear terms. J. Math. Anal. Appl.330 (2007), no. 2, 1179--1193.14.Xin, Zhouping; Yuan, Hongjun Vacuum state for spherically symmetric solutionsof the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ. 3 (2006), no. 3, 403--442.15.Yuan, Hongjun; Tong, Lining; Xu, Xiaojing BV solutions for the Cauchy problemof a quasilinear hyperbolic equation with $\sigma$-finite Borel measure and nonlinear source. J. Math. Anal. Appl. 311 (2005), no. 2, 715--735.16.Yuan, Hongjun; Xu, Xiaojing; Gao, Wenjie; Lian, Songzhe; Cao, ChunlingExtinction and positivity for the evolution $p$-Laplacian equation with $L^1$ initial value. J. Math. Anal. Appl. 310 (2005), no. 1, 328--337.17.Hongjun, Yuan; Songzhe, Lian; Wenjie, Gao; Xiaojing, Xu; Chunling, CaoExtinction and positivity for the evolution $p$-Laplacian equation in $R^n$.Nonlinear Anal. 60 (2005), no. 6, 1085--1091.18.Hongjun, Yuan; Xiaoyu, Zheng Existence and uniqueness for a quasilinearhyperbolic equation with $\sigma$-finite Borel measures as initial conditions.J. Math. Anal. Appl. 277 (2003), no. 1, 27--50.19.Yuan, Hongjun The Cauchy problem for a singular conservation law with measuresas initial conditions. J. Math. Anal. Appl. 225 (1998), no. 2, 427--439.20.Hongjun, Yuan Source-type solutions of a singular conservation law withabsorption. Nonlinear Anal. 32 (1998), no. 4, 467--492.21.Yuan, Hong Jun Extinction and positivity for the evolution $p$-Laplacianequation. J. Math. Anal. Appl. 196 (1995), no. 2, 754--763.22.Yuan, Hong Jun The Cauchy problem for a quasilinear degenerate parabolicsystem. Nonlinear Anal. 23 (1994), no. 2, 155--164.23.Yuan, Hong Jun Finite velocity of the propagation of perturbations for generalporous medium equations with strong degeneracy. Nonlinear Anal. 23 (1994), no. 6, 721--729.24.Yuan, Hongjun Continuity of weak solutions for quasilinear parabolicequations with strong degeneracy. Chin. Ann. Math. Ser. B 28 (2007), no. 4, 475--498.25.Yuan, Hong Jun; Lian, Song Zhe; Cao, Chun Ling; Gao, Wen Jie; Xu, Xiao JingExtinction and positivity for a doubly nonlinear degenerate parabolic equation.Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1751--1756.26.Yuan, Hong Jun; Tong, Li Ning BV solutions for a quasilinear hyperbolicequation with nonlinear source and finite Radon measures as initial conditions.(Chinese) Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 1, 54–70,27.Yuan, Hongjun; Wang, Shu The zero-Mach limit of a class of combustion flow.J. Partial Differ. Equ. 22 (2009), no. 4, 362–375,28.Ren, Chang Yu; Guan, Jin Rui; Yuan, Hong Jun A class of general-form parabolicMonge-Ampère equations. (Chinese) Chinese Ann. Math. Ser. A 30 (2009), no.3, 421--432. 35K9629.Yuan, Hongjun; Yan, Han Existence and uniqueness of BV solutions for a classof degenerate Boltzmann equations with measures as initial conditions. J.Partial Differ. Equ. 22 (2009), no. 2, 127--152. 35F2530.Yuan, Hongjun; Xu, Xiaojing Some entropy inequalities for a quasilineardegenerate hyperbolic equation. J. Partial Differential Equations18 (2005), no. 4, 289--303.31.Yuan, Hong Jun; Wu, Gang Quasilinear degenerate parabolic equation with Diracmeasure. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 4, 515--526;translation in Chinese J. Contemp. Math.26 (2005), no. 3, 291--30232.Yuan, Hongjun; Jin, Yang Existence and uniqueness of BV solutions for theporous medium equation with Dirac measure as sources. J. Partial Differential Equations 18 (2005), no. 1, 35--58.33.Yuan, Hong Jun; Xu, Xiao Jing Existence and uniqueness of BV solutions fora quasilinear degenerate hyperbolic equation with local finite measures asinitial conditions. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 1, 39--48; translation in Chinese J. Contemp. Math.26 (2005), no. 1, 43--5434.Yuan, Hongjun Instantaneous shrinking and localization of functions in$\roman Y_\lambda(m,p,q,N)$ and their applications. Chinese Ann. Math. Ser.B 22 (2001), no. 3, 361--380.35.Yuan, Hongjun Cauchy's problem for degenerate quasilinear hyperbolicequations with measures as initial values. J. Partial Differential Equations12 (1999), no. 2, 149--178.36.Yuan, Hongjun Localization condition for a nonlinear diffusion equation.Chinese J. Contemp. Math. 17 (1996), no. 1, 45--58.37.Yuan, Hongjun Existence and nonexistence of interfaces of weak solutions fornonlinear degenerate parabolic systems. J. Partial Differential Equations 9 (1996), no. 2, 177--185.38.Yuan, Hongjun Extinction and positivity for the non-Newtonian polytropicfiltration equation. J. Partial Differential Equations 9 (1996), no. 2, 169--176.39.Yuan, Hong Jun A localization condition for a class of nonlinear diffusionequations. (Chinese) Chinese Ann. Math. Ser. A 17 (1996), no. 1, 47--58.40.Zhao, Junning; Yuan, Hongjun The Cauchy problem of some nonlinear doublydegenerate parabolic equations. Chinese J. Contemp. Math. 16 (1995), no. 2, 173--192.41. Zhao, Jun Ning; Yuan, Hong Jun The Cauchy problem for a class of nonlinear doubly degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A 16 (1995), no. 2, 181--196.42. Wu, Zhuo Qun; Yuan, Hong Jun Uniqueness of generalized solutions for a quasilinear degenerate parabolic system. J. Partial Differential Equations 8 (1995), no. 1, 89--96.43.Yuan, Hong Jun Regularity of free boundary for certain degenerate parabolic equations. Chinese J. Contemp. Math. 15 (1994), no. 1, 77--86.44.Zhao, Jun Ning; Yuan, Hong Jun Uniqueness of the solutions of $u_t=\Delta u^m$ and $u_t=\Delta u^m-u^p$ with initial datum a measures: the fast diffusion case. J. Partial Differential Equations 7 (1994), no. 2, 143--159.45.Yuan, Hong Jun Regularity of the free boundary for a class of degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A15 (1994), no. 1, 89--97.。
symmetrical
symmetricalSymmetricalIntroductionSymmetry is a fundamental concept in various fields of science, art, mathematics, and nature itself. It refers to a balanced arrangement of parts on either side of a central axis, resulting in a mirror image. The concept of symmetry has intrigued humans for centuries and has been used to create aesthetically pleasing designs, to study patterns in nature, and to understand fundamental principles in various disciplines. In this document, we will delve into the concept of symmetry, explore its different forms, discuss its significance, and look at some real-world examples.Understanding SymmetrySymmetry can be defined as a balanced and harmonious arrangement of parts. It can be observed in different aspects of our daily lives, ranging from the human body to plants, animals, and even inanimate objects. The concept ofsymmetry can be categorized into different types based on the nature of the objects being observed.Types of Symmetry1. Reflectional Symmetry: Also known as mirror symmetry, reflectional symmetry occurs when an object can be divided into two equal halves along a line or a plane. The two halves are mirror images of each other. Examples of objects exhibiting reflectional symmetry include butterflies, human faces, and buildings that have symmetrical facades.2. Rotational Symmetry: Rotational symmetry occurs when an object can be rotated around a central point and still maintains its original appearance. The object may have multiple axes of rotational symmetry. Examples of objects with rotational symmetry include wheels, flowers like sunflowers, and snowflakes.3. Translational Symmetry: Translational symmetry occurs when an object can be shifted along a specific direction and still maintains its original appearance. This type of symmetry is commonly observed in patterns such as wallpaper designs, textiles, and some architectural elements.4. Rotational-Reflectional Symmetry: This is a combination of both rotational and reflectional symmetries. An object with rotational-reflectional symmetry can be both rotated and reflected to create a perfect match. Examples include snowflakes and many intricate geometric designs.The Significance of SymmetrySymmetry holds great importance in various disciplines, from mathematics and physics to biology and art. Here are some ways symmetry influences these fields:1. Mathematics: Symmetry is an essential concept in the field of mathematics, exhibiting deep connections with geometry, group theory, and topology. Symmetry is used to study patterns, shapes, and fundamental principles in these areas.2. Physics: Symmetry plays a crucial role in physics, particularly in quantum mechanics and particle physics. The laws of physics are often described by symmetrical equations and principles. Symmetry has aided in predicting the existence of new particles and understanding the behavior of fundamental forces.3. Biology: Symmetry is evident in the natural world, from the bilateral symmetry of animals like insects and humans to the radial symmetry of plants like flowers. It is believed that symmetrical features in organisms are associated with better health and genetic fitness.4. Art and Design: Symmetry has been utilized by artists and designers throughout history to create aesthetically pleasing compositions. It is seen in various art forms, architectures, and even in the human perception of beauty. Symmetry provides a sense of balance and harmony in visual compositions.Real-World Examples of Symmetry1. Taj Mahal: The iconic Taj Mahal in India is known for its symmetrical design. The mausoleum is perfectly mirrored along a central axis, creating a stunning reflection in the surrounding water.2. Butterfly Wings: The intricate patterns on butterfly wings often exhibit reflectional symmetry. The wings are divided into two equal halves, each mirroring the other.3. Snowflakes: Snowflakes are renowned for their symmetrical and intricate structures. Each snowflake exhibits both rotational and reflectional symmetries, making them unique and beautiful.4. The Sunflower: The spiral arrangement of seeds in a sunflower exhibits rotational symmetry. The seeds form concentric circles, giving the flower a harmonious and balanced appearance.ConclusionWhether found in nature, mathematics, or art, symmetry is an intriguing concept that captivates our attention. The different types of symmetry, including reflectional, rotational, and translational, are present in various aspects of our lives. Symmetry plays a crucial role in understanding the world around us, from predicting the behavior of subatomic particles to appreciating the beauty of a natural flower. As we continue to explore the wonders of symmetry, we gain deeper insights into the principles that govern our universe and the underlying harmony that exists in our surroundings.。
大学物理英语词汇
Chapter 1 Introduction（引言）§1.1 Space and Time（空间与时间）universe宇宙object物体measurement 测量kinematics运动学motion of objects 物体的运动mass point/particle质点center of mass 质心space and time 时空rotation 旋转subject研究的对象phenomena 现象intergalactic星系间的submicroscopic 亚微观的dimension尺度uniform均匀的isotropic各向同性的continuous连续的direction方向graininess 颗粒性location位置frame of reference 参考系specify确定、规定simultaneously 同时地inconsistent with与…不一致define/definition 定义platinum-iridium铂铱合金atomic standard 原子标准transition 跃迁meridian子午线general conference on weights and measures国际计量大会vacuum真空former standard of length米原器atomic energy level原子能级isotope cesium 铯同位素krypton 氪angstrom埃§1.2 Coordinate Systems and Frames of Reference（坐标系与参考系）frame of reference 参考系coordinate system坐标系rectangular Cartesian coordinates直角笛卡儿坐标系axis / axes (pl.)（坐标）轴origin坐标原点at rest静止dimension维mutually perpendicular 互相垂直intersection 交点§1.3 Idealized Models（理想模型）idealized model 理想模型simplified version简化方式neglect忽略particle质点air resistance 空气阻力vacuum真空in terms of 利用rigid body刚体insulator绝缘体§1.4 Vectors（矢量）vector矢量scalar标量magnitude大小velocity速度acceleration 加速度momentum动量proportional to正比于parallel平行position vector位置矢量§1.5 Properties of Vectors（矢量的特点）resultant/net vectoradditionsubtractionequivalenttranslatehead-to-tail methodparallelogram methoddiagonalcommutative lawscalar productdot productdistributive lawmultiplicationcross productvector productarearight-hand ruleparallelmultiplyfunctionsome variable§1.6 Components of a Vector（矢量的分量）component分量absolute value绝对值projection投影perpendicular 垂线rectangular component正交分量§1.7 Unit Vectors（单位矢量）unit vector单位矢量dimensionless 无量纲的unit magnitude单位大小respectively分别地Chapter 2 Kinematics: Motion in Two and Three Dimensions （运动学：二维与三维运动）§2.1 Kinematical Function of a Point（质点的运动函数）position vector位置矢量trigonometry 三角学§2.2 Displacement and Velocity（位移与速度）trajectory轨迹displacement vector位移矢量velocity速度ratio比值，比率straight line直线approach趋近、接近limit极限average velocity 平均速度instantaneous velocity瞬时速度slope斜率chord弦limiting process 求极限过程curved path弯曲路径derivative导数magnitude and direction大小和方向speed速率scalar components标量分量limiting value极限值limiting process 求极限过程tangent相切、切线change增量、改变量differential n.微分differentiate v. 微分、求导integrate v.积分integration n.积分coefficient系数module (矢量的)模successively 连续地square root 平方根§2.3 Acceleration（加速度）acceleration 加速度average acceleration 平均加速度instantaneous acceleration 瞬时加速度second derivative二阶导数positive正的negative负的respectively 分别地one-dimensional motion一维运动uniform circular motion匀速圆周运动projectile motion抛体运动§2.4 Motion with Constant Acceleration（匀加速运动）无§2.5 Linear Motion with Constant Acceleration（匀加速直线运动）linear线性的one-dimensional一维的corresponding对应的eliminate消去freely falling bodies自由落体air resistance 空气阻力acceleration due to gravity 重力加速度altitude高度vertical direction 竖直方向negative sign 负号latitude经度regardless of与.无关maximum value最大值minimum value最小值§2.6 Projectile Motion （抛体运动）projectile抛体trajectory轨迹assumption 假设negligible可忽略的rotation 转动air friction 空气摩擦parabola抛物线parabolic trajectory 抛物线轨迹initial初始的horizontal水平的independent 独立的superposition叠加flight time飞行时间horizontal range射程maximum height最大高度horizontal surface水平面a body projected horizontally平抛物体vertical竖直的firing angle抛射角§2.7 Circular Motion（圆周运动）circular motion 圆周运动uniform circular motion匀速圆周运动circular motion with varying speed变速圆周运动centripetal向心的arc length 弧长angular displacement 角位移instantaneous angular velocity（瞬时）角速度radian(s) 弧度dimensional有量纲的counterclockwise 逆时针clockwise顺时针circle圆center of a circle圆心vectorially矢量地angular acceleration 角加速度tangential acceleration 切向加速度center-seeking 向心resolve （矢量）分解centripetal acceleration 向心加速度normal acceleration 法向加速度perpendicular to垂直于radial径向的radius半径§2.8 Relative Motion（相对运动）relative velocity相对速度relative acceleration 相对加速度observer观察者outcome结果measurement 测量stationary 静止的differentiate求微分Galilean transformation equation伽利略变换valid有效的special theory of relativity狭义相对论as it turns out结果是relative to相对于heading due north头朝北right triangle直角三角形upstream逆流hypotenuse直角三角形的斜边Chapter 3 Newton’s Laws of Motion（牛顿运动定律）§3.1 Newton’s First Law（牛顿第一定律）at rest静止net external force/ resultant force合外力inertial frame of reference 惯性参考系inertia惯性act on = exert（力）作用于approximation近似inertial mass 惯性质量interact (n. interaction)相互作用resultant external force合外力momentum动量unless stated otherwise 除非另有说明§3.2 Newton’s Second Law（牛顿第二定律）nonzero非零的mass质量momentum动量rate of change变化率directly proportional to正比于inversely proportional to反比于§3.3 Newton’s Third Law（牛顿第三定律）interact相互作用opposite相反、相对isolated 孤立的action force 作用力reaction force反作用力§3.4 Applications of Newton’s Laws（牛顿运动定律的应用）tension 张力diagram示意图isolate 隔离free-body diagram受力图unknown未知量Atwood’s Machine阿特伍德机light string轻绳vertically 竖直地frictionless 无摩擦的incline斜面pulley滑轮balanced平衡的block 木块、滑块wedge楔、斜铁plane 平面horizontal surface水平面§3.5 International Units and Dimensions（国际单位制与量纲）physical quantity物理量fundamental unit基本单位universally普遍scientific community科学界luminous intensity光强度abbreviation缩写lowercase小写的uppercase大写的rectangle矩形§3.6 Introduction to Some Common Forces（几种常见力）electromagnetic电磁的lean against 倚靠compress 压mattress spring 床垫弹簧normal force 法向力、支持力stiffness倔强性stretch 拉伸frictional force / force of friction 摩擦力viscous medium粘滞媒质（介质）resistance 阻力force of static friction 静摩擦力maximum force of static friction最大静摩擦力is proportional to正比于proportionality constant比例常数coefficient of static friction 静摩擦系数coefficient of kinetic friction 滑动摩擦系数variation变化§3.7 The Four Fundamental Forces（四种基本力）gravitational force 引力universal gravitational constant万有引力常数electromagnetic force电磁力bind约束Coulomb’s law库仑定律charged particle带电粒子strong nuclear force 强力hydrogen氢nucleus (pl. nuclei or nucleuses)原子核neutron 中子proton质子counteract抵抗repulsive排斥的strength强度weak nuclear force弱力short-range force 短程力radioactivity放射性radioactive decay 放射性衰变nucleons核子massless 无质量的action at a distance远程作用hypothesis 假设field场Chapter 4Linear Momentum and Angular Momentum （动量与角动量）§4.1 Linear Momentum and Impulse（动量与冲量）(linear) momentum动量impulse 冲量impulse-momentum theorem动量定理time-average force 平均冲力§4.2 Impulse-momentum Theorem for Particles System（质点系的动量定理）particles system 质点系internal forces 内力external forces 外力§4.3 Conservation of Linear Momentum（动量守恒定律）momenta（pl.）动量§4.4 Center of Mass（质心）vector notation矢量表示continuous object连续物体element of mass 质元§4.5 Motion of the Center of Mass（质心的运动）conserved 守恒的isolated system 孤立系统§4.6 Angular Momentum of a Particle（质点的角动量）conserved 守恒的isolated system 孤立系统§4.7 Conservation Law of Angular Momentum（角动量守恒定律）Kepler 开普勒ellipse椭圆Chapter 6 Rotation of a Rigid Body about a Fixed Axis （刚体的定轴转动）§6.1 Motion of a Rigid Body（刚体的运动）rigid body刚体parallelogram rule 平行四边形法则translation 平动an extended body 空间实体rotation 转动nondeformable 不变形的resultant motion 合运动parallel平行fixed axis 固定轴counterclockwise motion 逆时针运动angular acceleration 角加速度clockwise motion顺时针运动separation 间隔translation 平动angular velocity 角速度trajectory 轨迹§6.2 Law of Rotation of a Rigid Body about a Fixed Axis（刚体定轴转动定律）moment of inertia 转动惯量rotation axis 旋转轴torque 力矩proportionality constant比例常数element of mass 质元line of action of force 力的作用线analogue 类似；相似perpendicular distance垂直距离distribution of mass 质量分布pivot about 围绕…旋转；以…为轴旋转moment arm 力臂is proportional to与…成正比§6.3 Calculation of Moments of Inertia for Rigid Bodies（转动惯量的计算）an extended body 延续实体hoop圆环spherical shell薄球壳solid sphere实心球spherical cavity球腔linear density线密度§6.4 Application of Law of Rotation of a Rigid Body about a FixedAxis（刚体定轴转动定律应用）orientation 方向；方位atwood’s machine伍德机brake制动器，刹车pedal踏板sprocket链轮齿bearing轴承pulley滑轮nonslip 无滑动§6.5 Conservation of Angular Momentum with Respect to the Fixed Axis（对定轴角动量守恒）resultant external torque合外力矩isolated隔离的valid 有效；适用pin 销；轴hapter 7Electric Fields of Stationary Electric Charges（静止电荷的电场）§7.1 Charge（电荷）Electricity电学magnetism磁学accelerator 加速器interatomic原子间的amber琥珀magnetite磁铁矿electrification充电magnet磁铁charge 电荷quantized量子化的quantization量子化proton质子electrically charged带电的charged body带电体conservation守恒uncharged不带电的§7.2 Coulomb’s Law（库仑定律）Coulomb’s Law库仑定律inversely proportional to相反地separating 分开的permittivity介电常数hydrogen氢opposite sign符号相反§7.3 The Electric Field（电场）electric field 电场test charge检验电荷distribution分布X-ray X-射线lightning闪电electronic电子的intermolecular分子间的rub摩擦magnesia氧化镁electromagnetism电磁学plastic rod塑料棒repel排斥attract 吸引suspend悬挂neutron中子electron电子neutral中性的integer整数integral multiple整数倍proportional to正比于square平方product乘积repulsive排斥Coulomb constant库仑常数superposition principle叠加原理electric field (intensity) 电场强度source charge场源电荷radio waves无线电波atmosphere大气thundercloud 雷雨云§7.4 Calculation of Electric Field（电场的计算）stationarydenominatorelectric dipoleelectric dipole momentspherically symmetriccontinuous charge distributioncharge elementstrategy静止的分母电偶极子电偶极矩球对称电荷连续分布元电荷策略bisector平分线manipulation处理linear charge density电荷线密度surface charge density 电荷面密度volume charge density电荷体密度ring charge带电圆环charged disk带电圆盘infinite plate of charge无限大带电平面§7.5 Electric Field Lines and Electric Flux（电场线和电通量）electric field lines电场线electric flux电通量infinity无穷远visualize形象化strength强度penetrate穿过qualitative定量的closed surface闭合曲面align排列thread线intersection 相交cross交叉§7.6 Gauss’s Law（高斯定理）Gauss’s law 高斯定理arbitrary shape 任意形状gaussian surface 高斯面electric flux电通量principle 原理practice实际§7.7 Application of Gauss’s Law（高斯定理的应用）algebraic代数的rearrange重新整理charge distribution电荷分布spherical symmetry 球对称cylindrical symmetry 柱对称plane symmetry 平面对称symmetric对称的spherical shell球壳infinite length无限长infinite plane无限大平面Chapter 8 Electric Potential（电势）§8.1 Conservativity of Electrostatic Field（静电场的保守性）line integral线积分conservative force field 保守力场closed path闭合路径conservative保守的circuital theorem for electrostatic field静电场环路定理§8.2 Potential Difference and Electric Potential（电势差和电势）potential difference电势差electric potential电势infinity无穷远electrostatic potential energy 静电势能volt伏特voltage电压electron volt电子伏特battery电池§8.3 Calculation of Electric Potential（电势的计算）equipotential surface等势面broken lines虚线semicircular半圆的insulating绝缘的infinite无限的dashed lines虚线extend延伸solid lines实线finite有限的arbitrary任意的function函数curved surface曲面§8.4 Electric Potential Gradient（电势梯度）gradient梯度notation符号potential Gradient电势梯度maximum最大值right angle 直角sketch勾画§8.5 Electrostatic Potential Energy（静电势能）electrostatic potential energy 静电势能vicinity附近Chapter 9 Conductor in Electrostatic Field（静电场中的导体）§9.1 Conductors in Electrostatic Equilibrium（导体的静电平衡）isolated conductor 孤立导体electrostatic equilibrium静电平衡equipotential body等势体radius of curvature曲率半径electrostatic shielding静电屏蔽neutralize电中和sharp point discharge尖端放电lightning rod 避雷针cosmic rays 宇宙射线lightning stroke雷击glow discharge 辉光放电ion离子corona discharge电晕放电shrink收缩cavity 空腔grounding接地curved surface 曲面conducting wire导线collision碰撞thunderstorm雷暴induced charge 感应电荷insert插入guarantee保证contradiction 矛盾§9.2 Calculation of Electrostatic Field with Conductors Nearby（有导体存在时静电场的分析与计算）conducting slab 导电板lateral area侧面uncharged conductor不带电导体edge effect边缘效应redistribute 重新分配external外部Chapter 10 Capacitors and Dielectrics in Electrostatic Field （电容器和静电场中的电介质）§10.1 Capacitance and Capacitors（电容和电容器）Leyden jar 莱顿瓶flash 闪光灯capacitance电容coaxial同轴的capacitor电容器coaxial cable同轴电缆parallel-plate capacitor 平行平板电容器concentric同心的cylindrical capacitor圆柱形电容器parallel combination 并联spherical capacitor 球形电容器series combination串联submultiple因数farad 法拉microfarad 微法拉picofarad 皮法拉rectify 调整inductance 自感应ignition 点火sparking打火花metallic金属(性)的combination联合、组合equivalent相当的§10.2 Dielectrics and Electric Field（电介质与电场）dielectric电介质relative dielectric constant 相对介电常数voltmeter 伏特计insulating绝缘的dielectric breakdown介质击穿dielectric strength介电强度§10.3 Polarization of Dielectrics（电介质的极化）polarize极化polar molecules极性分子polarization 极化nonpolar molecules非极性分子induced dipole moments 感应电矩permanent electric dipole moments 固有电矩surface charge表面电荷align排成一线orient取向bound charge束缚电荷homogeneous 均匀的free charge 自由电荷microwave 微波oven 烤箱vibrate 振动tune 调整resonate 共振oscillate 振荡§10.4 Gauss’s Law for Electric Displacement Vector（高斯定律）electric displacement 电位移dielectric constant介电常数deliberately故意地the flux of D(r) D(r) 的通量permittivity电容率§10.5 Energy Stored in a Charged Capacitor（电容器的能量）transfer转移electrostatic potential energy 静电势能battery电池electrostatic field energy 静电场能increment 增量energy density能量密度transformation转化maximum operating voltage 最大工作电压terminal 终端deliver递送dissipate消散pathway路径Chapter 11 Magnetic Force (磁力)§11.1 Nature of Magnetic Force(磁力的本质)magnetite磁铁矿石bar magnet条形磁铁interaction 相互作用magnetic pole磁极electric current loops of molecules分子环形电流§11.2 Magnetic Field and Magnetic Field Vector(磁场和磁感应强度)magnetic field磁场magnetic field vector=magnetic induction =magnetic flux density磁感应强度magnetic force 磁场力Lorentz force 洛仑兹力B-line磁感(应)线magnetic flux磁通量tesla(T)特（斯拉）weber韦伯§11.3 Motion of a Charged Particle in a Magnetic Field(带电粒子在磁场中的运动)cyclotron period回旋周期magnetic focusing磁聚焦helix螺旋线pitch螺距magnetic lens磁镜magnetic confinement 磁约束a magnetic bottle磁瓶the mass spectrometer 质谱仪schematic drawing示意图ion离子precision 精确度proton质子deuteron 氘核bombard 轰击cyclotron 加速器dees D型盒evacuate抽成真空shield屏蔽oscillate 振动plasma等离子体nuclear fusion核聚变Van Allen belts范阿仑辐射带§11.4 The Hall Effect(霍尔效应)the Hall voltage 霍尔电压the drift velocity漂移速度§11.5 Magnetic Force on a Current-carrying Conductor(载流导体在磁场中受力—安培力)current-carrying conductor/wire载流导体/导线current loop in a uniform magnetic field匀强磁场中的载流线圈linear element 线元current element vector 电流元矢量loop 环, 回路magnetic moment of a current loop载流线圈磁矩rectangular loop矩形回路a wire segment 一段导线strip 条；带Chapter 12 Source of Magnetic Field（磁场的源）§12.1 The Magnetic Field of Moving Point Charges（运动点电荷的磁场）permeability of free space真空磁导率§12.2 The Biot-Savart Law（毕奥-萨伐尔定律）the Biot-Savart Law毕奥-萨伐尔定律permeability of free space真空磁导率Gauss’law in magnetism磁场的高斯定律magnetic monopoles磁单极solenoid螺线管turn匝current-carrying wire 载流导线encircle环绕current element电流元diverge发散converge聚合magnetic pole磁极magnet磁铁magnetic flux磁通量§12.3 Ampere Circuital Theorem （安培环路定理）penetrate穿过bounded by以…为边界finite point 有限点line integral线积分§12.4 Application of Ampere Circuital Theorem（安培环路定理的应用）current-carrying wire 载流导线circumference 周长cylindrical shell圆柱形壳toroid螺绕环inner radius 内径outer radius外径spherical conductor 球形导体§12.5 Magnetic Field due to Varying Electric Field（与变化的电场相联系的磁场）displacement current位移电流generalized Ampere’s Law广义安培环路定理conduction current传导电流magnetic monopole磁单级postulate假设total current全电流steady current恒定电流§12.6 The Magnetic Force Between Two Parallel Current-carryingWires（平行电流间的相互作用力）antiparallel反平行Chapter 13 Magnetic Media in Magnetic Field（磁场中的磁介质）§13.1 Effect on Magnetic Field Caused by Magnetic Media（磁介质对磁场的影响）magnetic medium磁介质diamagnetic medium抗磁质paramagnetic medium顺磁质ferromagnetic material 铁磁质magnetic moment 磁矩paramagnetism 顺磁性partial alignment部分取向electron spin 电子自旋magnetic dipole 磁偶极子ferromagnetism 铁磁性diamagnetism抗磁性induced magnetic moment感生磁矩permanent magnetic moment固有磁矩§13.2 Atomic Magnetic Dipole Moments（原子磁矩）magnetization磁化atomic原子的magnetic dipole moment磁矩orbital magnetic moment 轨道磁矩quantum theory量子理论intrinsic spin angular momentum内禀自旋角动量§13.3 Magnetization（磁介质的磁化）magnetization n.磁化、磁化强度magnetize . 磁化atomic current loopamperian currentcross-sectional area分子环流v安培电流横截面积induced magnetic dipole moments感生磁矩surface magnetization current/ bound current面磁化电流(面束缚电流)applied magnetic field外加磁场magnetic susceptibility磁化率relative permeability相对磁导率bismuth 铋Bohr magneton玻尔磁子superconductor超导体emf电动势§13.4 Ferromagnetic Materials（铁磁质）iron铁cobalt钴nickel镍alloy 合金ferromagnetism 铁磁性magnetic domain磁畴critical temperature临界温度Curie temperature居里温度thermal agitation热扰动end effect边界效应magnetic saturation磁饱和reversible 可逆的magnetic hysteresis磁滞效应hysteresis loop 磁滞回线magnetization curve磁化曲线initial magnetization curve起始磁化曲线remnant magnetization剩磁coercive force矫顽力memory 记忆能力magnetize磁化demagnetize去磁，退磁transformer 变压器motor 电动机secondary coil副线圈cycle循环irreversible process 不可逆过程hard ferromagnetic materials硬磁性材料soft ferromagnetic materials软磁性材料hysteresis loss磁滞损耗（铁损）Curie point居里点permanent magnet永久磁体,magnetic tape磁带，memory unit记忆元件iron cores铁芯galvanometer 电流计rr§13.5 Circuital Theorem for H （H 的环路定理）magnetic intensity磁场强度magnetization current 磁化电流free current自由电流isotropic各向同性的permeability磁导率relative permeability相对磁导率Chapter 14 Electromagnetic Induction（电磁感应）§14.1 Faraday Law of Electromagnetic Induction（法拉第电磁感应定律）electromagnetic induction 电磁感应induction current感应电流emf (electromotive force) 电动势induction emf 感生电动势weber韦伯Lenz Law楞次定律polarity极性§14.2 Motional emf（动生电动势）motional emf 动生电动势§14.3 Induced emf and Induced Electric Field（感生电动势和感生电场）nonelectrostatic force非静电力induced emf 感生电动势induced electric field感生电场vortex field涡旋场eddy currents 涡流nonconservative field 非保守场time-varying field时变场alternate变化alternative 交流电的，交变的laminated叠片（组成）的§14.4 Mutual Induction（互感现象）mutual induction互感现象mutual inductance互感系数emf by mutual induction互感电动势orientation 方位§14.5 Self-induction（自感现象）self-induction自感现象self-inductance 自感系数inductor电感self-induced emf 自感电动势is proportional to正比于§14.6 Energy of Magnetic Field（磁场的能量）magnetic energy density磁场能量密度energy due to mutual induction互感磁能Chapter 15 Maxwell’s Equations and Electromagnetic Waves （麦克斯韦方程组组与电磁波波）§15-1 Maxwell’s Equations（麦克斯韦方程组）§15-2 Electromagnetic Waves（电磁波）propagation传播in phase同相、同步transverse waves横波wavelength波长visible spectrum可见光谱infrared waves 红外波radiation 辐射ultraviolet ray紫外线Poynging vector 坡印亭矢量§15-3 The Wave Equation for Electromagnetic Waves（电磁波的方程）wave function波函数wave equation波的方程wave number 波数angular frequency 角频率plane wave平面波Chapter 16 Temperature and the Kinetic Theory of Gases（温度与气体运动论）§16.1 Thermal Equilibrium and Temperature （热平衡及温度）temperature 温度hotness热coldness冷thermometric property热力学特性thermal contact热接触the average internal molecular kinetic energy 分子内平均动能thermal equilibrium热平衡electrical conductor 导电器the zeroth law of thermodynamics热力学第零定律temperature scale温标§16.2 The Celsius and Fahrenheit Temperature Scales（摄氏温标与华氏温标）thermometer温度计temperature scale温标the ice-point temperature冰点温度freezing point冰点steam-point沸点normal boiling point标准沸点the steam-point temperature 气化点温度the Celsius temperature scale摄氏温标the Fahrenheit temperature scale华氏温标§16.3 Gas Thermometers and the Absolute Temperature Scale（气体温度计和绝对温标）calibrate 校对、校准discrepancy差异volume 体积density密度sufficiently low 足够低sulfur硫a constant-volume gas thermometer等容气体温度计triple point of water 水的三相点ideal-gas temperature scale理想气体温标absolute temperature scale绝对温标nitrogen氮hydrogen氢oxygen氧recalibrate再校准extrapolate外推，向外延长triple point 三相点coexist共存helium氦liquefy液化in terms of 利用rigid body刚体insulator绝缘体Kelvin scale 开尔文温标§16.4 The Ideal-Gas Law（理想气体定律）Boyle’s law玻意耳定律constant volume 等体Boltzmann’s constant玻耳兹曼常量mole摩尔Avogadro’s number 阿伏伽德罗常量carbon atom碳原子universal gas constant普适气体常量ideal gas理想气体equation of state状态方程state variable状态参量standard condition标准条件subscript 下标§16.5 The Kinetic Theory of Gases（气体分子运动论）macroscopic state variable宏观状态变量microscopic quantity微观量walls of a container容器壁translational kinetic energy平动动能root mean square (rms) speed方均根速率order of magnitude量级piston活塞redistribute 再分布partition 分配equipartition theorem（能）均分定理classical statistical mechanics经典统计力学degree of freedom自由度monatomic 单原子的bond键diatomic 双原子的polyatomic 多原子的vibration振动mean free path平均自由程air current 气流convection 对流diffuse扩散reciprocal倒数frequency频率§16.6 Maxwell Speed Distribution Function（麦克斯韦速率分布函数）probability概率abscissa横坐标normalization condition 归一化条件most probable distribution最概然分布Chapter 17 Heat and the First Law of Thermodynamics （热及热力学第一定律）§17.1 Heat Capacity and Specific Heat（热容与比热）atomist 原子学家thermal energy 热能manifestation 表现形式molecular motion 分子运动thermal contact热接触caloric a.热的n.热（质）internal energy 内能heat capacity热容量phase相heat conduction热传导calorie卡（路里）molar mass摩尔质量Law of conservation of energy能量守恒定律The first law of thermodynamics 热力学第一定律be proportional to和…成正比molar specific heat摩尔比热solar heating system太阳能热系统coolant冷却液§17.2 Change of Phase and Latent Heat（相变与潜热）heat capacity热容量phase change相变vaporization汽化，蒸发fusion 熔化melting融化condensation 凝聚sublimation升华carbon dioxide二氧化碳crystalline a. 结晶的、晶状的n.结晶体average translational kinetic energy平均平动动能latent heat潜热§17.3 Joule’s Experiment（焦耳实验）thermally insulated绝热的mechanical equivalence of heat热功当量§17.4 The Internal Energy of an Ideal Gas（理想气体内能）internal energy 内能real gas实际气体§17.5 Work and the PV Diagram for a Gas（功与气体PV图）quasi-static process准静态过程piston活塞isobaric等压的isothermal 等温的§17.6 The First Law of Thermodynamics（热力学第一定律）§17.7 Heat Capacities of Gases（气体的热容）infinitesimal无穷小的§17.8 The Quasi-Static Adiabatic Process for an Ideal Gas（理想气体准静态绝热过程）compression 压缩Poisson formula 泊松公式process equations 过程方程§17.9 Kinds of Thermodynamic Processes（热力学过程的种类）Chapter 18 The Second Law of Thermodynamics（热力学第二定律）§18.1 Heat Engines and the Second Law of Thermodynamics（热机与热力学第二定律）second law of thermodynamics 热力学第二定律Kelvin statement开尔文表述Clausius statement 克劳修斯表述heat engine热机working substance工质steam engine蒸汽机internal-combustion engine内燃机heat reservoir热库vaporize 汽化condenser冷凝器exhaust valve 排气阀combustion chamber 燃烧室intake stroke进气冲程ignite点燃spark plug火花塞power stroke做功冲程Otto cycle 奥拓循环efficiency of a heat engine热机的效率§18.2 Refrigerators and the Second Law of Thermodynamics（制冷机与热力学第二定律）§18.3 Equivalence of the Heat-Engine and Refrigerator Statements（开尔文表述与克劳修斯表述的等价性）§18.4 The Carnot Engine and Carnot Cycle（卡诺热机与卡诺循环）refrigeration cycle 制冷循环efficiency of positive cycle正循环的效率Carnot cycle 卡诺循环Carnot engine 卡诺热机expansion 膨胀compression压缩schematic diagram原理图、示意图friction摩擦reversible可逆的Carnot efficiency 卡诺效率irreversible 不可逆的dissipative force耗散力nonequilibrium state 非平衡态viscous force粘滞力§18.5 Degradation of Energy（能量的损失）§18.6 Irreversibility and Disorder（不可逆性与无序性）microscopic state微观状态macroscopic state宏观状态disorder无序性random motion无规则运动§18.7 Entropy（熵）Boltzmann entropy玻耳兹曼熵principle of entropy increase熵增加原理additivity of entropy熵的可加性statistical meaning 统计意义reversible process 可逆过程infinitesimal无穷小的function of state 状态函数§18.8 Entropy and Probability（熵与概率）microscopic state微观态macroscopic state宏观态spontaneously自发地evacuated 真空的Chapter 19 Oscillatory Motion（振动）§19.1 Description of Simple Harmonic Motion（简谐运动的描述）oscillation/ vibration振动oscillatory motion 振动simple harmonic motion (SHM)简谐运动Spring弹簧equilibrium position平衡位置period 周期frequency频率hertz(Hz)赫兹reciprocal倒数amplitude振幅phase相、位相、周相phase angle /(phase constant) 初相、初位相integer times 整数倍a variety of很多displacement 位移velocity速度acceleration 加速度substitute替换（代）disturb扰动trigonometric三角法的，据三角法的kinematics 运动学compensate补偿§19.2 Phasor and Phase（旋转矢量与振动的相）circular motion 圆周运动initial position初始位置angular displacement 角位移angular speed 角速度in phase同相antiphase /180°out of phase反相component分量projection投影§19.3 Dynamic Equation for Simple Harmonic Motion（简谐运动的动力学方程）force constant of a spring倔强系数restoring force 恢复力harmonic oscillator谐振子in phase同相derivative/differentiate 微商，导数substitute替换、替代damping减幅，衰减sustain 维持compensate补偿，修正stiffness坚硬，硬度pitch定调negative/minus 负，减dynamics 动力学§19.4 Examples of Simple Harmonic Motion（简谐运动的实例）simple pendulum单摆physical pendulum物理摆/复摆pendulum bob摆锤pivot转动的轴moment of inertia转动惯量component分量torque扭矩，转矩§19.5 Energy in Simple Harmonic Motion（简谐运动的能量）§19.6 Combination of Simple Harmonic Motion（简谐运动的合成）combination 合成component分量periodically 周期性地triangle relation三角关系beat拍beat frequency 拍频§19.7 Damped Oscillations, Driven Oscillations and Resonance（阻尼振动受迫振动共振）damped oscillations阻尼振动overdamping过阻尼overdamped 过阻尼的underdamped欠阻尼的critical damping 临界阻尼nonoscillatory不摆动的，不波动的molasses 糖蜜critically damped 临界阻尼的driven oscillations受迫振动resonance共振natural angular frequency固有角频率natural period固有周期Chapter 20 Waves（波动）§20.1 Traveling Waves（行波）disturbance扰动propagate v.传播propagation n.传播medium媒质、媒介mechanical waves机械波。
工程光学英文版课后练习题含答案
工程光学英文版课后练习题含答案IntroductionEngineering Optics is a branch of optics that studies the application of optical principles and devices to solve engineering problems, including optical design, imaging systems, and measurement techniques. As an important part of Engineering Optics, the homework exercises help students understand the theoretical knowledge and familiarize themselves with practical problems. In this document, we provide a set of homework exercises with answers for Engineering Optics, which are designed to help students review the knowledge they learned in class and prepare for exams.Chapter 1: Introduction1.What is the definition of light?–Light is an electromagnetic wave that travels through space and has both electric and magneticcomponents perpendicular to each other and to thedirection of propagation.2.What are the primary properties of light?–The primary properties of light include reflection, refraction, diffraction, interference,and polarization.3.What is the difference between coherent andincoherent light?–Coherent light is light that has a constant phase relationship between two or more waves, whileincoherent light is light that has a random phaserelationship between two or more waves.4.What is the difference between monochromatic andpolychromatic light?–Monochromatic light consists of a single wavelength, while polychromatic light consists ofmultiple wavelengths.5.Define dispersion.–Dispersion is the phenomenon of different wavelengths of light traveling at different speedsthrough a medium, leading to a separation of thecolors of light.Chapter 2: Geometrical Optics1.Define ray and expln how rays are used ingeometrical optics.–A ray is an idealized model of the path that light travels through space, represented as a linewith an arrow indicating the direction ofpropagation. Rays are used in geometrical optics to determine the behavior of light as it passesthrough lenses, mirrors, and other optical devices.2.Define optical axis and principal plane.–The optical axis is the imaginary line passing through the center of curvature of a sphericallysymmetric optical system. The principal plane isthe plane perpendicular to the optical axis thatpasses through the focal point of the system.3.Define focal length and expln how it relates to the curvature of a lens.–The focal length is the distance between the center of curvature of a lens and the point whereparallel rays of light converge after passingthrough the lens. The curvature of a lensdetermines its focal length.4.Define the focal plane and expln how it relates to the focal length.–The focal plane is the plane perpendicular to the optical axis that passes through the focalpoint of a lens or mirror. The distance from thelens or mirror to the focal plane is equal to thefocal length.5.Expln the concept of conjugate planes.–Conjugate planes are prs of object and image planes that are related by an optical system suchthat an object in one plane is imaged onto theother plane. The distance between the two planes isequal to the sum of the object distance and imagedistance.Chapter 3: Optical Instruments1.Define the resolving power of an optical system.–The resolving power of an optical system is its ability to distinguish two closely spacedobjects as separate entities. It is determined bythe numerical aperture and wavelength of the lightused in the system.2.Define the magnification of an optical system.–The magnification of an optical system is the ratio of the size of the image produced by thesystem to the size of the object being imaged.3.What is a camera and how does it work?–A camera is an optical instrument that uses a lens to focus an image onto a light-sensitivesurface, such as film or a digital sensor. Theimage is formed by the interaction of light withthe surface, creating a chemical or electronicpattern that can be developed into a visible image.4.What is a microscope and how does it work?–A microscope is an optical instrument that uses a lens or a series of lenses to magnify small objects that cannot be seen with the naked eye. The specimen is placed on a stage and illuminated witha light source, and the image is formed by lensesthat focus the light onto the observer’s eye or a camera sensor.5.What is a telescope and how does it work?–A telescope is an optical instrument that usesa lens or a mirror or a combination of both tocollect and focus light from distant objects, such as stars, galaxies, or planets. The image is formed by lenses that magnify the light and focus it onto the observer’s eye or a camera sensor.ConclusionIn conclusion, the homework exercises and answers provided in this document are intended to help students review key concepts and prepare for exams in Engineering Optics. By solving these problems, students can deepen their understanding of optical principles and devices and develop their problem-solving skills. We hope that this resource will be useful for students and instructors alike in the study of Engineering Optics.。
The Density Profile of Massive Galaxy Clusters from Weak Lensing
a r X i v :a s t r o -p h /0310549v 1 20 O c t 2003THE DENSITY PROFILE OF MASSIVE GALAXY CLUSTERS FROM WEAK LENSING H.DAHLE Institute of Theoretical Astrophysics,University of Oslo,P.O.Box 1029,Blindern,N-0315Oslo,Norway We use measurements of weak gravitational shear around a sample of massive galaxy clusters at z =0.3to constrain their average radial density profile.Our results are consistent with the density profiles of CDM halos in numerical simulations and inconsistent with simple models of self-interacting dark matter.Unlike some other recent studies,we are not probing the scales where the baryonic mass component becomes dynamically important,and so our results should be directly comparable to CDM N-body simulations.1IntroductionWhile the concordance flat ΛCDM model,in which the matter density is dominated by cold dark matter (CDM),provides a good fit to observed large scale-properties of the universe,there remain some possible small-scale problems for this model.Numerical simulations of structure formation in a CDM model predict that the dark matter (DM)halos of L ⋆galaxies such as the Milky Way should contain a number of subhalos that exceed the observed number of satellite dwarf galaxies by 1-2orders of magnitude (e.g.Klypin et al.1999;Moore et al.1999a).Strongly suppressed star formation in the subhalos could be a possible solution to this problem.Observations of anomalous flux ratios of strongly gravitationally lensed multiple quasar images (Kochanek &Dalal 2003)and observations of the dynamics of optically dark high-velocity gas clouds in the local group (Robishaw,Simon &Blitz 2002)appear to be qualitatively consistent with this proposed solution.In addition,the simulations predict that DM halos have cuspy inner density profiles ρ(r )∝r −α,with αsomewhere in the range between 1.0(Navarro,Frenk &White 1997;hereafter NFW)and 1.5(Moore et al.1999b).This appears to contradict the observed dynamics of DM-dominated low surface brightness galaxies which favour softer cores with α=0.2±0.2(de Blok,Bosma,&McGaugh 2003).On the scales of galaxy clusters,some studies indicate shallowerdensity profiles than those predicted from CDM simulations(Sand et al.2003),while others give αvalues that are consistent with CDM predictions(Bautz&Arabadjis2003).Attempts have been made to solve these small-scale problems of CDM by proposing DM models that modify its behavior on small scales.Some examples of these are models in which the DM is self-interacting(Spergel&Steinhardt2000),self-annihilating(Kaplinghat,Knox&Turner 2000),fluid(Peebles2000;Arbey,Lesgourgues&Salati2003),warm(e.g.,Sommer-Larsen& Dolgov2001),repulsive(Goodman2000),fuzzy(Hu,Barkana&Gruzinov2000),decaying(Cen 2001),is both self-interacting and warm(Hannestad&Scherrer2000),acts as mirror matter (Mohapatra,Nussinov&Teplitz2002)or has its gravitational interaction with baryonic matter suppressed on small scales(Piazza&Marioni2003).Of these,the self-interacting DM model of Spergel&Steinhardt is the one which has been explored in most detail.Here,we put limits on this model by using weak gravitational lensing to measure the average density profile of an ensemble of massive galaxy clusters.Details of this work are given by Dahle,Hannestad& Sommer-Larsen(2003).2Constraints on the DM halo profileOur data set is a subset of the weak gravitational lensing measurements of38X-ray luminous clusters presented by Dahle et al.(2002).This subset consists of6clusters at z=0.3for which weak gravitational shear has been measured out to a projected radius of3h−165Mpc.Wefit the average observed radial shear profile to a“generalized NFW profile”on the formρ(r)=δcρc3 1x2(cx)−α(1+cx)α−3dx −1.(2)This model has a concentration parameter c defined by c=r200/r s,whereFigure1:The contours show the68%and95%confidence intervals for the concentration c vir and inner slopeαof our average cluster halo.Also shown is the mean value and scatter in c vir for an NFW halo of similar mass, predicted by Bullock et al.(2001).The dashed lines indicate lines along which the two parameters are degenerate.See also Dahle et al.(2003).∆α∼0.3.On the other hand,Bautz&Arabadjis(2003)find1<α<2and Lewis,Buote &Stocke(2003)findα=1.19±0.04,based on Chandra observations of the X-ray luminous intracluster medium in four clusters and in one cluster,respectively.In contrast to our weak lensing study(which only probe the DM density profile at radii where the baryonic component is not dynamically dominant),these strong lensing and X-ray studies are not directly compara-ble to simulations that only contain collisionless CDM.The above results indicate that future observational studies should simultaneously take into account both the baryonic component in stars and in the X-ray luminous intracluster medium as well as the DM.Similarly,all these components must be properly modeled in numerical simulations,if the simulations are to be directly compared to cluster observations on small(≤10kpc)scales.In any case,all the recent studies indicate that the core sizes of massive clusters are too small to be consistent with any self-interacting dark matter having a cross section large enough to explain the rotation curves of dwarf galaxies.Like previous weak lensing studies(e.g.,Clowe&Schneider2001,Hoekstra et al.2002), we are not able to strongly distinguish between the outer slope of an isothermal sphere,ρ∝r−2,and the NFW slopeρ∝r−3.However,in a recent work,Kneib et al.(2003)use a combination of weak and strong gravitational lensing data based on HST imaging of the cluster CL0024+17tofind an outer slope>2.4.Their data is adequatelyfit by a NFW profile with c=22+9−5,significantly higher than typical observed concentration parameters of rich clusters (e.g.,Hoekstra et al.2002;Katgert,Biviano&Mazure2003),which are generally consistent with CDM predictions(see also Fig.1).However,Chandra X-ray data(Ota et al.2003),as well as dynamical studies based on galaxy spectroscopy(Czoske et al.2002),indicate that this is not a fully relaxed,spherically symmetric system.Weak lensing measurements of a representative sample of dynamically relaxed clusters out to even larger radii than we probe in our study should eventually settle the issue of the value of the outer slope.AcknowledgmentsI thank my collaborators Steen Hannestad and Jesper Sommer-Larsen,and acknowledge support from The Reseach Council of Norway through a post-doctoral research fellowship. References1.Arbey,A.,Lesgourgues,J.,&Salati,P.2003,Phys.Rev.D,68,235112.Bautz,M.W.&Arabadjis,J.S.2003,ArXiv Astrophysics e-prints,33133.Bullock,J.S.,Kolatt,T.S.,Sigad,Y.,Somerville,R.S.,Kravtsov,A.V.,Klypin,A.A.,Primack,J.R.,&Dekel,A.2001,MNRAS,321,5594.de Blok,W.J.G.,Bosma,A.,&McGaugh,S.2003,MNRAS,340,6575.Cen,R.2001,ApJL,546,L776.Clowe,D.&Schneider,P.2001,A&A,379,3847.Czoske,O.,Moore,B.,Kneib,J.-P.,&Soucail,G.2002,A&A,386,31.8.Dahle,H.,Kaiser,N.,Irgens,R.J.,Lilje,P.B.,&Maddox,S.J.2002,ApJS,139,3139.Dahle,H.,Hannestad,S.,&Sommer-Larsen,J.2003,ApJL,588,L7310.Dav´e,R.,Spergel,D.N.,Steinhardt,P.J.,&Wandelt,B.D.2001,ApJ,547,57411.Goodman,J.2000,New Astronomy,5,10312.Hannestad,S.,&Scherrer,R.J.2000,Phys.Rev.D,62,04352213.Hoekstra,H.,Franx,M.,Kuijken,K.,&van Dokkum,P.G.2002,MNRAS,333,91114.Hu,W.,Barkana,R.,&Gruzinov,A.2000,Phys.Rev.Lett.,85,115815.Jing,Y.P.2000,ApJ,535,3016.Kaplinghat,M.,Knox,L.,&Turner,M.S.2000,Phys.Rev.Lett.,85,333517.Katgert,P.,Biviano,A.,&Mazure,A.2003,ArXiv Astrophysics e-prints,1006018.Klypin,A.,Kravtsov,A.V.,Valenzuela,O.,&Prada,F.1999,ApJ,522,8219.Kneib,J.et al.2003,ArXiv Astrophysics e-prints,729920.Kochanek,C.S.&Dalal,N.2003,ArXiv Astrophysics e-prints,203621.Lewis,A.D.,Buote,D.A.,&Stocke,J.T.2003,ApJ,586,13522.Meneghetti,M.,Yoshida,N.,Bartelmann,M.,Moscardini,L.,Springel,V.,Tormen,G.,&White,S.D.M.2001,MNRAS,325,43523.Mohapatra,R.N.,Nussinov,S.,&Teplitz,V.L.2002,Phys.Rev.D,66,6300224.Moore,B.,Ghigna,S.,Governato,F.,Lake,G.,Quinn,T.,Stadel,J.,&Tozzi,P.1999a,ApJL,524,L1925.Moore,B.,Quinn,T.,Governato,F.,Stadel,J.,&Lake,G.1999b,MNRAS,310,114726.Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1997,ApJ,490,49327.Ota,N.,Pointecouteau,E.,Hattori,M.,&Mitsuda,K.2003,ArXiv Astrophysics e-prints,658028.Peebles,P.J.E.2000,ApJL,534,L12729.Piazza,F.&Marioni C.2003,Phys.Rev.Lett.,91,14130130.Robishaw,T.,Simon,J.D.,&Blitz,L.2002,ApJL,580,L12931.Sand,D.J.,Treu,T.,Smith,G.P.,&Ellis,R.S.2003,ArXiv Astrophysics e-prints,946532.Sommer-Larsen,J.&Dolgov,A.2001,ApJ,551,60833.Spergel,D.N.&Steinhardt P.J.2000,Phys.Rev.Lett.,84,376034.Yoshida,N.,Springel,V.,White,S.D.M.,&Tormen,G.2000,ApJL,544,L87。
AP Physics C 物理词汇
Chapter 1 Background基础知识vector/ˈvektər/矢量scalar/ˈskeɪlər/标量magnitude/ˈmæɡnɪtuːd/数值，大小polar/ˈpoʊlər/极坐标Cartesian Coordinates/kɑːrˈtiːziən/直角坐标, 笛卡尔坐标algebraic/ˌældʒɪˈbreɪɪk/代数的dot product点乘cross product叉乘unit analysis单位分析，量纲分析base units基本单位derived units/dɪ'raɪvd 'juːnɪts/导出单位qualitative/ˈkwɑːlɪteɪtɪv/定性的Chapter 2 One-DimensionalKinematics 个维运动学One-Dimensional/'wʌndɪ'mɛnʃənəl/一维的kinematics/,kɪnɪ'mætɪks/运动学Instantaneous speed/ˌɪnstənˈteɪniəs spi:d/瞬时速率velocity/vəˈlɑːsəti/速度speed/spiːd/速率acceleration/əkˌseləˈreɪʃn/加速度Uniformly accelerated motion匀加速运动displacement/dɪsˈpleɪsmənt/位移distance/ˈdɪstəns/距离derivation/ˌderɪˈveɪʃn/导数integrate/ˈɪntɪɡreɪt/积分Separating variables变量分离法parabola/pəˈræbələ/抛物线trajectory/trəˈdʒektəri/轨道, 轨迹Chapter 3 Two-DimensionalKinematics 二维运动学two-dimensional二维的synchronize/ˈsɪŋkrənaɪz/同步collide/kəˈlaɪd/碰撞collision/kəˈlɪʒn/碰撞rest/rest/静止inertial reference frame惯性参考系counterclockwise/ˌkaʊntərˈklɑːkwaɪz/反时针方向的anticlockwise clockwise/ˈklɑːkwaɪz/顺时针方向的phase/feɪz/相位，相Uniform Circular Motion (UCM)匀速圆周运动UCMcentripetal acceleration/senˈtrɪpɪtl əkˌseləˈreɪʃn/向心加速度nonuniform circular motion非匀速圆周运动tangential component/tænˈdʒenʃl kəmˈpoʊnənt/切向分量tether/ˈteðər/n. 系链；拴绳 v. （用绳或链）拴elevation/ˌelɪˈveɪʃn/海拔, 高度angle of elevation仰角angle of depression俯角simultaneously/ˌsaɪmlˈteɪniəsli/adv. 同时地phase shift angle相移角Chapter 04 Newton’s Laws 牛顿定律static equilibrium/ˈstætɪk ˌiːkwɪˈlɪbriəm/静力平衡dynamic equilibrium/daɪˈnæmɪk ˌiːkwɪˈlɪbriəm/动态平衡inertial reference frames/ɪˈnɜːrʃl ˈrefrəns freɪmz/惯性系mass/mæs/质量weight/weɪt/重量gravitational force/ˌɡrævɪˈteɪʃənl fɔːrs/万有引力normal force/ˈnɔːrml fɔːrs/正压力, 法向力frictional force/ˈfrɪkʃənə fɔːrs/摩擦力static friction/ˈstætɪk ˈfrɪkʃn/静摩擦kinetic friction/kɪˈnetɪk ˈfrɪkʃn/动摩擦incipient motion of the object/ɪnˈsɪpiənt/物体的运动趋势tension force/ˈtenʃn fɔːrs/弹力, 张力external forces/ɪkˈstɜːrnl fɔːrs/外力massless and taut/ˈmæsləs ənd tɔːt/(指在考虑绳的弹力时)无质量且张紧的incline/ɪnˈklaɪn/斜面Inclined Plane Static Friction on an Incline斜面上的静摩擦Atwood machine阿特伍德机pulley/ˈpʊli/滑轮pendulum bob/ˈpendʒələm bɑːb/摆锤banked curve problem倾斜跑道问题Chapter 05 Work, Energy; andPower 功、能量和功率work功joule/dʒuːl/焦耳dot product点积；标量积kinetic energy动能The Law of Conservation of Energy能量守恒定律Potential energy/pəˈtenʃl ˈenərdʒi/势能gravity/ˈɡrævəti/重力gravitational force/ˌɡrævɪˈteɪʃənl fɔːrs/万有引力,重力spring force弹力HOOKE’S LAW胡克定律elastic potential energy/ɪˈlæstɪk pəˈtenʃl ˈenərdʒi/弹性势能Chapter 06 Linear Momentum andCenter of Mass 线性动量和质心momentum/moʊˈmentəm/动量elastic/ɪˈlæstɪk/弹性的elastic collision/ɪˈlæstɪk kəˈlɪʒn/弹性碰撞impulse/ˈɪmpʌls/冲量center of mass质心weighted average加权平均The Ballistic Pendulum/bəˈlɪstɪk ˈpendʒələm/冲击摆，弹道摆Chapter 07 Rotation I: Kinematics,Force, Work, and Energy 转动1：运动学、力学、功和能量rotation/roʊˈteɪʃn/旋转torque/tɔːrk/n. 转矩，力矩rigid body/ˈrɪdʒɪd ˈbɑːdi/刚体radian/ˈreɪdiən/弧度revolution/ˌrevəˈluːʃn/旋转angular position/ˈæŋɡjələr pəˈzɪʃn/角位置angular velocity/ˈæŋɡjələr vəˈlɑːsəti/角速度angular acceleration/ˈæŋɡjələr əkˌseləˈreɪʃn/角加速度rotational inertia/roʊˈteɪʃənl ɪˈnɜːrʃə/转动惯量Chapter 08 Rotation Ⅱ: InertiaEquilibrium, and CombinedRotation/Translation 转动Ⅱ：惯量、平衡和转动/ 平移的结合angular momentum/ˈæŋɡjələr moʊˈmentəm/角动量top/tɑːp/陀螺Chapter 09 Simple HarmonicMotion 简谐运动simple harmonic motion/ˈsɪmpl hɑːrˈmɑːnɪk ˈmoʊʃn/简谐运动(SHM) amplitude/ˈæmplɪtuːd/振幅frequency/ˈfriːkwənsi/频率period/ˈpɪriəd/周期oscillation/ˌɑːsɪˈleɪʃn/振荡periodic motion/ˌpɪriˈɑːdɪk ˈmoʊʃn/周期运动phase/feɪz/相位phase shift相移phase shift angle相移角cycle/ˈsaɪkl/周期hertz/hɜːrts/赫兹pendulum/ˈpendʒələm/摆Chapter 10 Universal Gravitation万有引力universal gravitation/ˌjuːnɪˈvɜːrsl ˌɡrævɪˈteɪʃn/万有引力principle of superposition/ˈprɪnsəpl əv ˌsuːpərpəˈzɪʃn/叠加原理spherically symmetric mass球形对称质量分布Kepler's laws开普勒定律trajectory/trəˈdʒektəri/轨道foci/ˈfəʊsaɪ, ˈfəʊkaɪ/焦点；焦距 focus的复数asteroid/ˈæstərɔɪd/小行星slingshot/ˈslɪŋʃɑːt/弹弓Chapter 11 Coulomb’s Law andElectric Fields Due to PointCharges 库仑定律和点电荷产生的电场Coulomb’s Law库仑定律Electric fields and field lines电场和电场线permittivity/,pɝmɪ'tɪvəti/介电常数εfree space真空empty space electrostatic constant/ɪˌlektroʊˈstætɪk ˈkɑːnstənt/静电常数kcharge/tʃɑːrdʒ/电荷volt/voʊlt/伏特（电压单位）electronvolt/ɪˈlektrɑːn voʊlt/电子伏特electrostatic force/ɪˌlektroʊˈstætɪk fɔːrs/静电力conservative force/kənˈsɜːrvətɪv fɔːrs/守恒力potential/pəˈtenʃl/电势，电位equipotential line/,iːkwɪpə'tenʃ(ə)l laɪn/等势线, 等位线, 等电位线Chapter 12 Calculating ElectricFields and Potentials Due toContinuous Charge Distributions 由连续电荷分布产生的电场和电势的计算Chapter 13 Gauss’s Law 高斯定律flux/flʌks/通量electric flux电通量，电通GAUSS’S LAW高斯定律Chapter 14 Analysis of CircuitsContaining Batteries and Resistors包含电池和电阻的电路分析current/ˈkɜːrənt/电流amp/æmp/安培nuclei/'njʊklɪ,ai/原子核（nucleus的复数形式）current density/ˈkɜːrənt ˈdensəti/电流密度drift velocity/drɪft vəˈlɑːsəti/(电子)漂移速度Ohm’s law/oʊm/欧姆定律resistance/rɪˈzɪstəns/电阻resistor/rɪˈzɪstər/电阻器resistivity/ˌriːzɪˈstɪvəti/电阻率conductivity/ˌkɑːndʌkˈtɪvəti/电导率voltage/ˈvoʊltɪdʒ/电压Resistors in Parallel并联电阻Equivalent Resistance等效电阻Resistors in Series串联电阻voltmeter/ˈvoʊltmiːtər/电压表ammeter/ˈæmiːtər/电流表Kirchhoff’s laws/ˈkɜrkhɔf/基尔霍夫定律loop equations/luːp ɪˈkweɪʒn/环路方程battery/ˈbætəri/电池node/noʊd/节点internal resistance内电阻, 内阻electromotive force/ɪˌlektrəˈmoʊtɪv fɔːrs/电动势(EMF)Chapter 15 Capacitors 电容器capacitor/kəˈpæsɪtər/电容器capacitance/kəˈpæsɪtəns/电容,电容量conductor/kənˈdʌktər/导体farad/ˈfæræd/法拉(电容单位), 法microfarad/ˌmaɪkroʊˈfærəd/微法拉(电容量的实用单位) nanofarad/ˈneɪnəˈfærəd/纳法picofarad/ˈpaɪkoʊˌfærəd/皮法edge effects边缘效应vacuum/ˈvækjuːm/真空, 真空的equivalent capacitance/ɪˈkwɪvələnt kəˈpæsɪtəns/等效电容array/əˈreɪ/数组，阵列Complicated Capacitor Arrays复杂电容器阵列dielectric/ˌdaɪɪˈlektrɪk/adj. 非传导性的；诱电性的 n. 电介质；绝缘体dielectric material介电材料dielectric constant介电常数Chapter 16 RC Circuits RC电路RC circuit RC电路discharge/dɪsˈtʃɑːrdʒ/放电charge/tʃɑːrdʒ/vi. 充电Chapter 17 Magnetic Fields 磁场magnetic/mæɡˈnetɪk/磁性magnetic field磁场tesla/'teslə; 'tezlə/特斯拉（磁通量单位）intrinsic spin/ɪnˈtrɪnzɪk spɪn/固有自旋；内蕴自旋，特征自旋permanent magnet/ˈpɜːrmənənt ˈmæɡnət/永磁体, 永磁铁the right-hand rule右手定则magnetic field line/mæɡˈnetɪk fiːld laɪn/磁场线, 磁力线magnetic pole/mæɡˈnetɪk poʊl/磁极current-carrying wire载流导线Biot-Savart’s law/ˈbaɪət 'sɑ:vɑ:t lɔː/毕奥萨伐尔定律mass spectrometer/mæs spekˈtrɑːmɪtər/质谱仪point charge点电荷permeability/ˌpɜːrmiəˈbɪləti/磁导率(μ0the permeability offree space)Ampere’s law/æm'pɪr/安培定律Amperian path安培路径Amperian loop安培环路solenoid/ˈsɑːlənɔɪd,ˈsoʊlənɔɪd/螺线管toroid/ˈtoʊrɔɪd/环形线圈；圆环面；螺旋管coil/kɔɪl/线圈current density/ˈkɜːrənt ˈdensəti/电流密度Chapter 18 Faraday's and Lenz'sLaw 法拉第定律和楞次定律magnetic flux/mæɡˈnetɪk flʌks/磁通量Faraday’s law/ˈfærədeɪ/法拉第定律Faraday/ˈfærədeɪ/法拉第（电量单位）Weber/ˈveɪbər/韦伯（磁通单位）electromotive force/ɪˌlektrəˈmoʊtɪv fɔːrs/电动势(EMF)uniform magnetic field均匀磁场nonuniform magnetic field不均匀磁场Lenz’s law/lentsiz 'lɔː/楞次定律Motional EMF动生电动势Chapter 19 Inductors 电感器mutual inductance/ˈmjuːtʃuəl ɪnˈdʌktəns/互感；互感系数self-inductance/self ɪnˈdʌktəns/自感inductor/ɪnˈdʌktər/（电路、电子电路的）电感器concentric solenoid/kənˈsentrɪk ˈsɑːlənɔɪd/同轴螺线管henry亨利(电感单位)LR circuits LR电路decay/dɪˈkeɪ/衰减LC circuits LC电路coaxial cable/ko'æksɪəl ˈkeɪbl/同轴电缆Chapter 20 Maxwell’s Equations 麦克斯韦方程组magnetism/ˈmæɡnətɪzəm/磁性，磁力；磁学maxwell's law麦克斯韦定律electric monopole/ɪˈlektrɪk ˈmɑːnəˌpoʊl/电单极电偶极子dipole/ˈdaɪpoʊl/偶极；双极子bar magnet/bɑːr ˈmæɡnət/磁棒, 条形磁铁displacement current位移电流。
合肥工业大学大学物理考试试题Word版
合肥⼯业⼤学⼤学物理考试试题Word版Exercise:1. A particle moving along x axis starts from x 0 with initialvelocity v 0. Its acceleration can be expressed in a =-kv 2 wherek is a known constant. Find its velocity function v =v (x ) with the coordinate x as variable.2. A particle moves in xy plane with the motion function asj t i t t r )3sin 5()3cos 5()(+=(all in SI). Find (a) its velocity )(t v and(b) acceleration )(t a in the unit-vector notation. (c) Showthat v r ⊥.3. A bullet of mass m is shot into a sand hill along a horizontal path, assume that the drag of the sand is kv f -=, find the velocity function v(t) if 0)0(v v = and the gravitation of thebullet can be ignored.4. what work is done by a conservative force j i x f 32+= thatmoves a particle in xy plane from the initial positionj i r i 32+= to the final position j i r f 34--=. All quantities arein SI.5. The angular position of a point on the rim of a rotating wheel is given by 320.30.4t t t +-=θ, where θ is in radians and t is in seconds. Find (a) its angular velocities at t=0s and t =4.0s? (b) Calculate its angular acceleration at t =2.0s. (c) Is its angular acceleration constant?6. A uniform thin rod of mass M and length L can rotate freely about a horizontal axis passing through its top end o (231ML I =).A bullet of mass m penetrates the rod passing its center of mass when the rod is in vertical stationary. If the path of the bullet is horizontal with an initial speed v o beforepenetration and 20v after penetration . Show that (a) the angular velocity of the rod just after the penetration is MLmv 430=ω. (b) Find the maximum angular max θ the rod will swing upward after penetration.7. A 1.0g bullet is fired into a block (M=0.50kg) that is mounted on the end of a rod (L=0.60m). The rotational inertia of the rod alone about A is 206.0m kg ?. The block-rod-bullet system then rotates about a fixed axis at point A. Assume the block is small enough to treat as a particle on the end of the rod. Question: (a) What is the rotational inertia of the block-rod-bullet system about A? (b) If the angular speed of the system about A just after the bullet ’s impact is 4.5rad/s ,8. A clock moves along the x axis at a speed of 0.800c and readszero as it passes the origin. (a) Calculate the Lorentz factorγ betweenthe rest frame S and the frame S* in which the clock is rest. (b) what time does the clock read as it passes x =180m ?9. What must be the momentum of a particle with mass m so that its total energy is 3 times rest energy?10. Ideal gas within a closed chamber undergoes the cycle shownthe Fig. Calculate Q net the net energy added to the gas as heatduring one complete cycle.11. One mole of a monatomic ideal gas undergoes the cycle shown in the Fig. temperature at state A is 300K.(a). calculate the temperature of state B and C.(b). what is the change in internal energy of the gas between state A and state B? (int E ?)(c). the work done by the gas of the whole cycle .(d). the net heat added to the gas during one complete cycle.12. The motion of the electrons in metals is similar to the motion of molecules in the ideal gases. Its distribution function of speed is not Maxwell ’s curve but given by.=0)(2Av v pthe possible maximum speed v F is called Fermi speed. (a)plot the distribution curve qualitatively. (b) Express the coefficient A in terms of v F . (c) Find its average speed v avg .13. Two containers are at the same temperature. The first contains gas with pressure 1p , molecular mass 1m , and rms speed 1rms v . The second contains gas with pressure 12p , molecularmass 2m , and average speed 122rms avg v v =. Find the mass ratio21m m .14. In a quasi-static process of the ideal gas, dW =PdV and d E int =nC v dT . From the 1st law of thermodynamics show thatthe change of entropy i f v i fT T nC V V nR S ln ln +=? .Where n is the numberof moles, C v is the molar specific heat of the gas at constantvolume, R is the ideal gas constant, (V i, T i) and (V f, T f) . are the initial and final volumes and temperatures respectively.15. It is found experimentally that the electric field in a certain region of Earth ’s atmosphere is directed vertically down. At an altitude of 300m the field is 60.0 N /C ; at an altitude of 200m , the field is 100N /C . Find the net charge contained in a cube 100m on edge, with horizontal faces at altitudes of 200m and 300m . Neglect the curvature of Earth.16. An isolated sphere conductor of radius R with charge Q . (a) Find the energy U stored in the electric field in the vacuum outside the conductor. (b) If the space is filled with a uniform dielectrics of known r ε what is U * stored in the field outside the conductor then?17. Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius R. (a) show that, at a distance r from the cylinder axis (r2ερ=, where ρis the volume charge density. (b) write the expression for E when r>R .18. A non-uniform but spherically symmetric distribution of charge has a volume density given as follow:-=0)/1()(0R r r ρρ0ρ is a positive constant, r is the distance to the symmetric center O and R is the radius of the charge distribution. Within the chargedistribution (r < R ), show that (a) the charge contained in the co-center sphere of radius r is )34(31)(430r Rr r q -=πρ, (b) Find themagnitude of electric field E (r ) within the charge (r < R ). (c) Find the maximum field E max =E (r *) and the value of r *.19. In some region of space, the electric potential is the following function of x,y and z: xy x V 22+=, where the potential is measured in volts and the distance in meter . Find the electric field at the point x=2m, y=2m . (express your answer in vector form)20. The Fig. shows a cross section of an isolated spherical metal shell of inner radius R 1 and outer radius R 2. A pointcharge q is located at a distance 21R from the center of the shell. If the shell is electrically neutral, (a) what are the induced charges (Q in , Q out ) on both surfaces of the shell? (b)Find the electric potential V(0)V (∞)=0.21. Two large metal plates of equal area S are parallel and closed to each other with chargesQ A ,Q B respectively. Ignore the fringing effects, find (a) the surface charge density on each side of both plates, (b) the electric field at p1, p2 . (c) the electric potential difference between the two plates(d is the distance between palte A and B)（注：可编辑下载，若有不当之处，请指正，谢谢!）。
一种催化剂的微观形貌控制,一篇英文文献,总结200字
一种催化剂的微观形貌控制,一篇英文文献,总结200字Title: Microstructure Control of Catalysts: A ReviewMicrostructure control of catalysts is a crucial factor in enhancing their performance and efficiency in various chemical reactions. This review article aims to summarize recent advancements in manipulating the microstructure of catalysts to achieve tailored catalytic properties.The article first discusses the significance of microstructure control in catalysis. It highlights how specific properties, such as particle size, shape, composition, and surface morphology, greatly influence catalytic activity, selectivity, and stability. Various techniques for microstructure control are then presented, including synthesis methods like sol-gel, hydrothermal, and template-assisted approaches, as well as post-treatment techniques like annealing, doping, and surface modification.Moreover, the article provides examples of microstructure control in different types of catalysts, such as metal nanoparticles, metal oxides, and zeolites. It showcases how adjusting the microstructure can lead to improved catalytic performance in applications such as hydrogenation, oxidation, and dehydrogenation reactions.The review also highlights the importance of characterization techniques for evaluating catalyst microstructures, including transmission electron microscopy (TEM), X-ray diffraction (XRD), and scanning electron microscopy (SEM). These techniques facilitate understanding of the relationship between microstructure and catalytic activity.In conclusion, the article emphasizes the significance of microstructure control in catalyst design and optimization. It emphasizes the need for further research in this area to develop more effective catalysts for various industrial applications. The review provides important insights for researchers and engineers working in the field of catalysis and offers valuable information for the rational design of catalysts with desired microstructures.。
Navier-Stokes-Poisson方程的两个注记
Navier-Stokes-Poisson方程的两个注记周海军;高真圣【摘要】研究空间维数为2或3情形下Navier-Stokes-Poisson方程组中的“Poisson”项.一方面得到了该项在旋转变化下的形式不变性以及在Riesz算子作用下的有界性;另一方面利用Helmholtz分解等方法,给出未知向量函数的计算公式.【期刊名称】《贵州师范大学学报（自然科学版）》【年(卷),期】2016(034)004【总页数】4页(P54-57)【关键词】Navier-Stokes-Poisson方程;Riesz算子;傅里叶变换;Helmholtz分解【作者】周海军;高真圣【作者单位】华侨大学数学科学学院,福建泉州362021;华侨大学数学科学学院,福建泉州362021【正文语种】中文【中图分类】O175.29Navier-Stokes-Poisson方程具有很强的物理背景，主要可以描述天体星云在有粘性和有重力时的运动状态，它的一般形式为：其中Φ是牛顿重力势能(G>0是牛顿常数),应力张量D(u)=1/2(▽u+(▽u)T)。
密度ρ=ρ(x,t)和速度u=u(x,t)是时间变量t∈[0,∞)和空间变量x∈Ω的函数(Ω⊂Rn为有界区域)。
另外P=P(ρ)为压力函数，粘性系数μ和λ满足。
由于牛顿重力势能函数Φ满足一个Poisson方程，从而称其为“Poisson项”。
此外方程组中“▽”为梯度算子，“div”为散度算子，“Δ”为拉普拉斯算子，u⊗u为张量积,“T”表示矩阵的转置。
Navier-Stokes-Poisson方程是目前流体研究的热门之一，许多优秀的数学家都致力于这方面的研究，并且取得了许多结果，例如文[1]证明了二维可压该方程解的存在性问题，文[2]研究了高维全空间中可压情形下的适定性问题，文[3] 研究了球对称系统的稳定性，文[4]研究了三维以及更高维可压方程组解的全局存在性，文[5]研究了三维空间中系统的最优衰减率；文[6]研究了非等熵可压系统的最优衰减率等。
Keywords 单词表
Chapter 1 Measurement 测量Physical quantity 物理量, Standard 标准, unit单位, Precision精度significant figures 有效数字, Dimension 量纲,维度， Force 力， Velocity 速度,Speed 速率, Acceleration 加速度， Momentum 动量, Base units 基本单位，Derived units 导出单位， phenomenon 性质， Cesium atom clock 铯原子钟，Definition 定义， vibrations 振动， platinum-iridium alloy 铂铱合金，multiply乘, divide除, product积, quotient商， consistent 一致,pendulum单摆, frequency 频率Chapter 2 Motion in one dimension 一维运动Kinematics运动学, Dynamics 动力学，vector矢量， Scalar 标量constant acceleration 匀加速度， Freely falling body 自由落体， Object 物体，Particle 质点， Translation 平动， Rigid body 刚体， Rotation 转动，Reference frame 参考系， magnitude and direction 大小和方向， equal 相等， opposite 相反addition 加，subtraction减， Triangle， Parallelogram 平行四边形，Commutative law 交换律， Associative law 结合律， Component分量，rectangular coordinate system直角坐标系， Unit Vectors 单位矢量，spherical coordinate system 球坐标系， Dot product 点乘， Cross product 叉乘，Radius 半径， Average velocity平均速度，Instantaneous velocity 瞬时速度，tangent 切线，initial position初始位置，vertical垂直， positive正， Derivative 导数，Integration 积分， Calculus 微积分， Simple harmonic motion 简谐运动Chapter 3 Force and Newton’s Laws 力和牛顿定律Classical mechanics 经典力学, interaction互作用, counteraction 反作用action-reaction force 作用与反作用力, environment 环境net force 净力remain at rest 保持静止，linear motion 线性运动， reference frame 参照系, inertia 惯性, inertial frame 惯性参照系，independent 独立的，无关的, friction 摩擦力,frictionless 无摩擦的, resistance 阻力，external force 外力, internal force 内力, horizontal 水平的, calibration 校准，定标, weight 重力, weightless 失重， perception 感觉，equator 赤道，North pole 北极，vary 变化Chapter 4 Motion in two and three dimensions 二维和三维运动Projectile motion 抛体运动, Drag force 曳力,阻力, Uniform circular motion 匀速圆周运动, Equivalent 相等,等价, Parabolic trajectory 抛物型轨道, range 射程,origin of coordinate 坐标原点, Differential equation 微分方程, angular velocity 角速度, centripetal acceleration 向心加速度, Relative Motion 相对运动, due east 正东Chapter 5 Application of Newton’s Laws牛顿定律的应用non-constant force 非恒力，inertial force 惯性力，pseudo-force 赝力，viscous force 黏滞力,tensile force, tension 张力，拉力，spring force 弹力，normal force正压力，frictionalforce 摩擦力，gravitational force万有引力，centrifugal force 离心力，Non-inertial frame 非惯性系， negligible 可忽略的， Massless and non-stretch string 无质量不可伸长的绳子，Passive force 被动力，active force 主动力，pulley 滑轮， Free-body diagram 隔离图，subscript 下标， superscript 上标，Coefficient of friction 摩擦系数， static friction静摩擦力，kinetic friction动摩擦力， conical pendulum 锥摆Chapter 6 Momentum 动量momentum 动量，collision 碰撞，impulse 冲量， conservation 守恒，strike 击打，theorem 定理， internal force 内力，external force 外力，elastic 弹性的，completely inelastic 完全非弹性， Center-of-mass reference frame 质心参考系，Coefficient of restitution 恢复系数Chapter 7 Systems of Particles 质点系Translational motion 平动，Rotational motion 转动，Combination 合成，simultaneous 同时的，trajectory 轨道，center of mass 质心， geometry 几何， symmetry 对称性， eliminate 消除，snapshot 快照，distribute 分布，numerator 分子，denominator 分母， decompose 分解，corollary 推论，distinction 区别，wobble 摇摆，oscillation 振动，tedious 冗长的，infinitesimal 无穷小，argument 论证，spherical symmetry 球对称，cylindrical symmetry 柱对称，irregular 不规则的，superimpose 重叠，mirror image 镜像，variable mass 可变质量Chapter 8 Rotational Kinematics 转动运动学rigid body 刚体，axis 轴，pure rotation 纯转动，angular displacement 角位移，angular velocity 角速度， angular acceleration 角加速度， degree of freedom 自由度，orientation 方向，spin 旋转，radian 弧度，circumference 一周，revolution 圈，increase增加, decrease 减少, counterclockwise 逆时针， characteristic 特征， finite 有限的，represent 表示， right-hand rule 右手定则， curl 卷曲， tangential 切向的，radial 径向的Chapter 9 Rotational Dynamics 转动动力学torque (moment of force)力矩, analogy 类比， analogue 类似物， analogous 类似的， emerge 出现， rotational inertia (moment of inertia) 转动惯量, intrinsic 固有的， arbitrary 任意的， resolve 分解， radial 径向的， tangential 切向的， moment arm力臂, cross-section 截面, cross product (vector product) 叉乘，矢量积，align 排列，substitute 替代，proportionality 比例性， parallel-axis theorem 平行轴定理, perpendicular-axis theorem 垂直轴定理, discrete 离散的， continuous 连续的， approximate 近似， uniform 均匀的，algebraic 代数的， calculus 微积分，approach 方法，instantaneous axis 瞬时轴, corollary 推论, equilibrium 平衡, nonequilibrium 非平衡, rolling without slipping 无滑滚动， inch 缓慢移动， superposition 叠加， inclined plane 斜面Chapter 10 Angular Momentum 角动量Specify 指定， right angle 直角， with respect to the origin 关于原点，time rate of change 随时间的变化率， moment of impulse 冲量矩，resolve a vector into components 将矢量分解，gyroscope 陀螺仪，回转仪， top 陀螺， axle 轮轴， shaft 轴， bearing 轴承， wobble 摇晃，symmetric 对称的， asymmetric 非对称的，lopsided 倾向一方的， sideways 向侧面的，invert 反转， divert 转向， deflect 偏转， orientational stability 取向稳定性，configuration 组态，构型， precession 进动， nutation 章动Chapter 11 Energy 1: Work and Kinetic Energy 功和动能Work 功, Kinetic energy 动能, Power 功率, Variable force 变力, Mechanical energy 机械能, Invariant 不变的, Work-energy theorem 功能原理, Negative work 负功,Electron-volt 电子伏特, Stretch 拉伸, Compress 压缩, Restoring force 恢复力, Integrate, integral, integration 积分, Elastic/inelastic collision 弹性(非弹性碰撞)Chapter 12 Potential Energy 势能Conservative force 保守力，Potential energy 势能， Agent Body, Round trip 环程，Cycle 循环， Characteristic 特征， Configuration 构型， State quantity 状态量，Process quantity 过程量， Isolated system 孤立系， Turning points 拐点，Stable equilibrium 稳定平衡， Unstable 非稳定的，Neutral 中性的，Dissociation energy 离解能, Gradient ▽梯度， Divergence ▽· 散度，Rotation ▽×旋度，Partial derivative 偏导数Chapter 13 Conservation of Energy 能量守恒Internal energy 内能， environment 环境， transfer 传递， microscopic 微观的，Weld 接触点， Protrusion 突出物， dissipative force 耗散力， decay 衰变，Exoergic 放能的， endoergic 吸能的， calorie 卡路里Chapter 14 Gravitation 引力Origin 起源，Universal gravitation 万有引力，Attract 吸引，Superposition 叠加，Shell theorem 壳层定理，Latitude 纬度，Altitude 高度，Inverse square law 平方反比律，Equator 赤道，Spherically symmetric 球对称的，Planet 行星，Satellite 卫星，Escape speed 逃逸速度，Perturb 扰动， Mutual interaction 相互作用，Torsion 扭力，Ellipsoid 椭圆体，Elliptic 椭圆的，Parabolic 抛物线的，Hyperbolic 双曲线的， Focus 焦点，Semi-major axis 半长轴，Eccentricity 离心率，Aphelion 远日点，Perihelion 近日点，Dark matter 暗物质， Dark energy 暗能量， General theory of relativity 广义相对论。
激励器结构对三电极等离子体高能合成射流流场及其冲量特性的影响
激励器结构对三电极等离子体高能合成射流流场及其冲量特性的影响张宇;罗振兵;李海鹏;王林;夏智勋【摘要】等离子体激励器以其结构简单、响应速度快、环境适应性强等优势，已成为主动流动控制技术和流体力学研究的前沿与热点。
相比于传统两电极激励器，三电极等离子体高能合成射流激励器具有更高的能量效率，形成射流冲量更大，有望成为新型快响应直接力产生装置。
为揭示激励器结构对射流流场和冲量特性的影响规律，进而优化激励器结构参数，利用电参数测量装置、高速阴影系统及自主设计的单丝扭摆式微冲量测量系统对不同射流孔径、腔体体积和电极间距的三电极激励器放电特性、射流流场及其冲量进行了实验研究。
为对比激励器在不同工况条件下的工作特性，定义无量纲能量沉积ε和无量纲射流冲量 I *，并分析了激励器结构参数对ε和 I *的影响。
结果表明对于给定无量纲能量沉积ε，激励器存在最优射流孔径；激励器无量纲能量沉积ε和无量纲射流冲量I *随腔体体积增加而减小，随激励器电极间距增加而增加；射流强度及其流场影响区域随腔体体积增加而减小，随激励器电极间距增加而增加。
对比不同腔体体积和电极间距工况条件下 I *随ε的变化可知，为设计具有较好射流冲量水平的激励器，在相同无量纲能量沉积ε条件下，应尽量增大激励器无量纲射流冲量 I *。
当设计激励器无量纲能量沉积ε小于初始工况时，应增大初始工况激励器腔体体积使无量纲能量沉积ε降低至设计值；当设计激励器无量纲能量沉积ε大于初始工况时，应增大初始工况激励器电极间距使无量纲能量沉积ε增加至设计值，使设计激励器具有较好的射流冲量水平。
%Plasma actuators have been become the research focus in flow control and fluid dynamic fields because of the advantages of simplicity,fast response and pared with typical two-electrode actuator,three-electrode Plasma Synthetic Jet Actuator (TE-PSJA) possesses the advantages of higher energy efficiency and bigger jet impulse,and has potential to be adopted as a fast-response direct force generation device.In order to reveal the effects of geometric parameters on the flow field and impulse,discharge characteristic,flow field and impulse of TE-PSJA with different orifices,volumes and electrode gaps were experimentally studied using electric parameter measurement device,high-speed shadowgraphy and single line torsion pendulum system.In order to compared the working characteristics of the actuator in different conditions,dimensionless energy depositionε and dimensionless jet impulse I * were defi ned,and the effects of geometric parameters onεand I * were analyzed.Results showed that the best jet orifice diameter is existed,ε and I * decreases as volumes increase,but increases as electrode gaps increase,strength and affected area of the jet decrease as volumes increase,while increase as electrode gaps increase.A similar flow structure which contains a mushroom-shaped jet and a spherically symmetric precursor shock above the jet front was pared with the variation of I * with ε on the different volumes and electrode gaps conditions,it can be concluded that in order to design actuators which possessed better jet impulse level, the dimensionless jet impulse I * should be increased as possible with the sameε.The cavity volume should bein creased so as the dimensionless energy depositionεis to be the design value when the dimensionless energy deposition is less than the initial case.On the contrary,the electrode gaps should be increased so as thedimensionless energy depositionε be the desi gn value when the dimensionless energy deposition is larger than the initial case.【期刊名称】《空气动力学学报》【年(卷),期】2016(034)006【总页数】7页(P783-789)【关键词】等离子体合成射流;高速阴影;单丝扭摆;结构参数;主动流动控制;能量沉积;射流冲量;射流流场【作者】张宇;罗振兵;李海鹏;王林;夏智勋【作者单位】国防科学技术大学航天科学与工程学院，湖南长沙 410073;国防科学技术大学航天科学与工程学院，湖南长沙 410073;中国国防科技信息中心，北京 100142;国防科学技术大学航天科学与工程学院，湖南长沙 410073;国防科学技术大学航天科学与工程学院，湖南长沙 410073【正文语种】中文【中图分类】V211.1等离子体激励器作为一种新型的流动控制方式,以其结构简单、响应迅速、工作频带宽、适应多工况等优点正受到越来越多的关注,极有可能成为主动流动控制技术的新突破[1-4]。
12个字母的英文单词
12个字母的英文单词你知道的12个字母的单词有哪些吗?一小编为大家整理了一些，起来学习一下吧!1.handkerchief 手帕12个字母的英语单词例句：1. She balled the handkerchief up and threw it at his feet. 她把手绢揉成一团，丢在了他的脚边。
2. He pulled a handkerchief from his pocket and gave it to him. 他从口袋里掏出手绢递给他。
3. He took out a handkerchief and blew his nose. 他掏出一块手帕，擤了擤鼻子。
4. Blake held his handkerchief over the mouthpiece to muffle his voice. 布莱克把手绢包在话筒上，以使自己的声音听起来模糊不清。
5. Tom tried to staunch the blood with his handkerchief. 汤姆试图用手帕来止血。
6. She lowered the handkerchief which she had kept dabbing at her eyes. 她放下了那块一直用来擦眼睛的手帕。
7. He took a handkerchief from his pocket and lightly wiped his mouth. 他从口袋里掏出手帕，轻轻擦了擦嘴。
8. "Is this what you were looking for?" Bradley produced the handkerchief. “你刚才找的是不是这个?”布拉德利掏出那块手绢。
9. Her sister broke down, sobbing into her handkerchief. 她姐姐再也控制不住自己的情绪，捂着手绢呜咽起来。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
a r X i v :g r -q c /0703060v 4 8 A u g 2007Stability of spherically symmetric solutions in modified theories of gravityMichael D.SeifertDept.of Physics,University of Chicago,5640S.Ellis Ave.,Chicago,IL,60637∗In recent years,a number of alternative theories of gravity have been proposed as possible res-olutions of certain cosmological problems or as toy models for possible but heretofore unobserved effects.However,the implications of such theories for the stability structures such as stars have not been fully investigated.We use our “generalized variational principle”,described in a previous work [1],to analyze the stability of static spherically symmetric solutions to spherically symmetric perturbations in three such alternative theories:Carroll et al.’s f (R )gravity,Jacobson &Mat-tingly’s “Einstein-æther theory”,and Bekenstein’s TeVeS.We find that in the presence of matter,f (R )gravity is highly unstable;that the stability conditions for spherically symmetric curved vac-uum Einstein-æther backgrounds are the same as those for linearized stability about flat spacetime,with one exceptional case;and that the “kinetic terms”of vacuum TeVeS are indefinite in a curved background,leading to an instability.PACS numbers:04.20.Fy,04.40.Dg,04.50.+hI.INTRODUCTIONThe idea of using a variational principle to bound the spectrum of an operator is familiar to anyone who has taken an undergraduate course in quantum mechanics:given a Hamiltonian H whose spectrum is bounded be-low by E 0,we must have for any normalized state |ψ in our Hilbert space E 0≤ ψ|H |ψ .We can thus obtain an upper bound on the ground state energy of the sys-tem by plugging in various “test functions”|ψ ,allowing these to vary,and finding out how low we can make the expectation value of H .This technique is not particular to quantum mechan-ics;rather,it is a statement about the properties of self-adjoint operators on a Hilbert space.In particular,we can make a similar statement in the context of linear field theories.Suppose we have a second-order linear field the-ory,dependent on some background fields and some dy-namical fields ψα,whose equations of motion can be put into the form−∂2(ψ,ψ).(3)2by the work of Carroll et al.[5];Einstein-æther theory, a toy model of Lorentz-symmetry breaking proposed by Jacobson and Mattingly[6,7];and TeVeS,a covariant theory of MOND proposed by Bekenstein[8].We apply our techniques to these theories in Sections III,IV,and V,respectively.We will use the sign conventions of[9]throughout. Units will be those in which c=G=1.II.REVIEW OF THE GENERALIZEDV ARIATIONAL PRINCIPLEA.Symplectic dynamicsWefirst introduce some necessary concepts and nota-tion.Consider a covariantfield theory with an action of the formS= L= L[Ψ]ǫ(4)where L[Ψ]is a scalar depending on some setΨof dynam-ical tensorfields including the spacetime metric.(For convenience,we will describe the gravitational degrees of freedom using the inverse metric g ab rather than the met-ric g ab itself.)To obtain the equations of motion for the dynamicalfields,we take the variation of the four-form Lǫwith respect to the dynamicalfieldsΨ:δ(Lǫ)=(EΨδΨ+∇aθa[Ψ,δΨ])ǫ(5) where a sum over allfields comprisingΨis implicit in thefirst term.Requiring thatδS=0under this varia-tion then implies that the quantities EΨvanish for each dynamicalfieldΨ.The second term in(5)defines the vectorfield θa[Ψ,δΨ].The three-formθdual to this vectorfield (i.e.,θbcd=θaǫabcd)is the“symplectic potential cur-rent.”Taking the antisymmetrized second variation of this quantity,we then obtain the symplectic current three-formωfor the theory:ω[Ψ;δ1Ψ,δ2Ψ]=δ1θ[Ψ,δ2Ψ]−δ2θ[Ψ,δ1Ψ].(6) In terms of the vectorfieldωa dual toω,this can also be written asωaǫabcd=δ1(θa2ǫabcd)−δ2(θa1ǫabcd).(7) The symplectic form for the theory is then obtained by integrating the pullback of this three-form over a space-like three-surfaceΣ:Ω[Ψ;δ1Ψ,δ2Ψ]= Σ¯ω[Ψ;δ1Ψ,δ2Ψ].(8)If we define n a as the future-directed timelike normal toΣand e to be the induced volume three-form onΣ(i.e.,e bcd=n aǫabcd),this can be written in terms ofωa instead:Ω=− Σ(ωa n a)e.(9)In performing the calculations which follow,it is this sec-ond expression forΩwhich will be most useful to us.B.Obtaining a variational principleIn[1],we presented a procedure by which a varia-tional principle for spherically symmetric perturbations of static,spherically symmetric spacetimes could gener-ally be obtained.For our purposes,we will outline this method;further details can be found in the original pa-per.The method described in[1]consists of the following steps:1.Vary the action to obtain the equations of motion(E G)ab corresponding to the variation of the met-ric,as well as any other equations of motion E Acorresponding to the variations of any matterfieldspresent.This variation will also yield the dualθaof the symplectic current potential;take the anti-symmetrized variation of this quantity(as in(7))to obtain the symplectic form(9).2.Fix the gauge for the metric,and choose an ap-propriate set of spacetime functions to describethe matterfields.Throughout this paper,we willchoose our coordinates such that the metric takesthe formd s2=−e2Φ(r,t)d t2+e2Λ(r,t)d r2+r2dΩ2(10)for some functionsΦandΛ.1Our spacetimes willbe static at zero-order(i.e.,(∂/∂t)a is a Killing vec-torfield at zero-order),but non-static infirst-orderperturbations.We will therefore haveΦ(r,t)=Φ(r)+φ(r,t),whereφis afirst-order quantity;sim-ilarly,we define thefirst-order quantityλsuch thatΛ(r,t)=Λ(r)+λ(r,t).In other words,δg tt=2e−2Φ(r)φ(r,t)(11) andδg rr=−2e−2Λ(r)λ(r,t).(12) Similarly,all matterfields will be static in the back-ground,but possibly time-dependent atfirst order.33.Write the linearized equations of motion and the symplectic form in terms of these perturbational fields (metric and matter.)4.Solve the linearized constraints.Oneofthe mainresults of [1]was to show that this can be done quite generally for spherically symmetric perturba-tions offof a static,spherically symmetric back-ground.Specifically,suppose the field content Ψof our theory consists of the inverse metric g ab and a single tensor field A a 1...a n b 1...b m .(The general-ization to multiple tensor fields is straightforward.)Let (E A )a 1...a n b 1...b m denote the equation of motion associated with A a 1...a n b 1...b m .We define the con-straint tensor asC cd =2(E G )cd−g ceiA a 1...a n b 1...d...b m (E A )a 1...a n b 1...e...b m +g cei A a 1...e...a n b 1...b m (E A )a 1...d...a n b 1...b m(13)where the summations run over all possible index “slots”,from 1to n and from 1to m for the first and second summation respectively.It can then be shown that if the background equations of motion hold,and the matter equation of motion also holds to first order,the perturbations of the tensor C ab will satisfy∂F∂r=−r 2e Λ−ΦδC tt(15)for some quantity F which is linear in the first-order fields.Moreover,the first-order constraint equations δC tt =δC tr =0will be satisfied if and only if F =0.We can then solve this equation algebraically for one of our perturbational fields,usually the metric perturbation λ.We will refer to the equation F =0as the preconstraint equation .5.Eliminate the metric perturbation φfrom the equa-tions.As φcannot appear without a radial deriva-tive (due to residual gauge freedom)2,we must find an algebraic equation for ∂φ/∂r .The first-order equation δC rr =0,will,in general,serve this pur-pose [1].Use the above relations for λand ∂φ/∂r∂tψβ2−∂ψα23If W αβis negative definite in this sense,we use the negative of this quantity as our inner product,and the construction proceeds identically.4 III.f(R)GRA VITYA.TheoryIn f(R)gravity,the Ricci scalar R in the Einstein-Hilbert action is replaced by an arbitrary function of R,leaving the rest of the action unchanged;in other words,the Lagrangian four-form L is of the formL=12f(R)g ab−∇a∇b f′(R)+g ab f′(R)=8πT ab(21)where T ab,given byδL mat=−116π(f′(α)R+f(α)−αf′(α))ǫ+L mat[A,g ab].(23)Varying the gravitational part of action with respect to g ab andαgives usδL= (E G)abδg ab+Eαδα+E AδA+∇aθa ǫ(24) where E A denotes the matter equations of motion,(E G)ab=12g ab(f(α)−αf′(α))−8πT ab (25) andEα=f′′(α)(R−α),(26)16π (∇b f′(α))δg ab−(∇a f′(α))g bcδg bc .(27) The vectorθa mat above is the symplectic potential current resulting from variation of L mat,andθa Ein is the symplec-tic potential current for pure Einstein gravity,i.e.,θa Ein=15In certain scalar-tensor theories[20],the scalar perturbations decouple from the metric and matter perturbations;it is then legitimate in such theories to“ignore”the constraints if we con-cern ourselves only with the scalarfield.However,this de-coupling does not occur in f(R)gravity theory.To see this,5correct:stars in CDTT f (R )gravity do in fact have an ultra-short timescale instability.To describe the fluid matter,we will use the “La-grangian coordinate”formalism,as in Section V of [1].In this formalism,the fluid is described by considering the manifold M of all fluid worldlines in the spacetime,equipped with a volume three-form N .If we introduce three “fluid coordinates”X A on M ,with A running from one to three,then the motion of the fluid in our spacetime manifold M is completely described by a map χ:M →M associating with every spacetime event x the fluid worldline X A (x )passing through it.The mat-ter Lagrangian is then given byL =−̺(ν)ǫ(30)where ν,the “number density”of the fluid,is given byν2=1∂r 2f ′(α)+∂Λr∂r ∂Λr2(e 2Λ−1) f ′(α)−1we can rewrite the Lagrangian in terms of a rescaled metric ˜g ab =Ω−2(σ)g ab =eσ/2√16π˜R−13π˜g ab ](29)where σis related to the scalar σby f ′(α)=e σ/2√∂r +2∂r f ′(α)+ 2∂r−12e 2Λ(αf ′(α)−f (α))−e 2Λ(̺′ν−̺)=0,(34b)(E G )θθ=r 2e −2Λ∂2∂r −∂Λr∂∂r 2−∂Φ∂r +∂Φr∂Φ∂rf ′(α)−r 2∂r 2+∂Φ∂r−∂Φr∂Λ∂r +1∂r+∂Φ2δ1g bc δ2g ad g bc ∇d (f ′(α))+δ1(f ′(α))∇b δ2g cd −∇b (δ1(f ′(α)))δ2g cd×(g ab g cd −δa c δb d )−[1↔2](35)where ωa Ein is defined,analogously to θaEin ,to be the symplectic current associated with pure Einstein grav-ity.This is equal to [21]ωa Ein =S a bc d ef (δ2g bc ∇d δ1g ef −δ1g bc ∇d δ2g ef ),(36)where S a bc d ef =12g ad g be g cf−12δa e δd f g bc +16The contribution to the symplectic current from the matter terms inthe Lagrangiancoordinateformalismwas calculated in [1];we simply cite the result for the t -component of ωa in a static background here:t a ωa mat =−t a̺′∂t ξ2−∂ξ2∂t−∂b 1∂t+∂b 2∂r +∂Φ∂r −e−2Φ∂2b∂r +2∂r−e −2Λ 2∂ΦrS −6r ∂Φr 21−e−2Λ+α∂r +∂Λr +1∂rξ−λ ,(43)where we have introduced the new quantityS =∂rf ′(α)(44)for notational convenience.Equation (42)then impliesthat our algebraic equation for λisλ=S −1∂b∂rb +8πe 2Λ̺′νξ.(45)We could plug this result into (43)to obtain an alge-braic equation for ∂φ/∂r ;however,the resulting expres-sion is somewhat complicated.In fact,we can derive a simpler expression for ∂φ/∂r .To do so,we note that (E G )ab −12g ab f ′(α)−∇a ∇b f ′(α)+12g ab T c c=0.(46)Using the trace of this equation to eliminate the f ′(α)term,along with the equation R =α,we can show thatf ′(α)R ab +13g ab Tcc=0.(47)To zero-order,the θθ-component of this equation is1∂r−∂Φre 2Λ−1+1r∂3e 2Λ̺=0,(48)and to first order,it is71∂r −∂φre2Λλ + 1∂r−∂Φr2 e2Λ−1 −1f′′(α)+2α b−1∂r+13̺′νe2Λ ∂ξ∂r+2ν∂ν∂r2+∂Φ∂r+∂Λ∂r−2∂Φ∂r+2∂r−∂φr2e2Λλ+e2Λ−2Φ∂2λ2f′′(α) =0(50)and the matter equation of motion,which as in the case of pure Einstein gravity is̺′ −e2Λ−2Φ∂2ξ∂r+ ∂∂r ̺′′ν ∂ξν∂ν∂r+2∂t2=A1∂2ζ∂r+A3ζ+A4∂b∂t2=B1∂2b∂r+B3b+B4∂ζr2f′(α)∂b1∂tζ2 −[1↔2].(55)As noted above(16),this reduced symplecticform de-fines a three-form Wαβ.For a valid variational principleto exist,this Wαβmust be positive definite in the senseof(18).We can see that the Wαβdefined by(55)ispositive definite if and only if f′(α)>0and̺′ν>0in our background solutions.The latter condition willhold for any matter satisfying the null energy condition,since̺′ν=ρ+P;however,the former condition mustbe checked in order to determine whether a valid varia-tional principle exists.6For the particular f(R)chosenby Carroll et al.[5],we have f′(R)=1+µ4/R2>0,andso Wαβis always positive definite in the required sensefor this choice of f(R).All that remains is to actually write down the varia-tional principle for f(R)gravity.As noted above(3),thevariational principle will take the formω20≤(ψ,Tψ)6Should f′(α)fail to be positive in the background solutions,all isnot necessarily lost;we can still attempt to analyze the reducedequations that we have obtained.See Section V for an exampleof such an analysis.8denominator of (56)will be(ψ,ψ)=4πd r r 2eΛ−ΦS −2f ′(α)6∂r2+C 2∂ζ∂r −b∂ζf ′′(α)≈−e Φ+ΛS −2ρ3µ4b 2+6e−2Λ ∂b6d re Λ−ΦS −2b 2(60)(note that f ′(α)=1+µ4/ρ2≈1in the stellar inte-rior.)As a representative mass distribution,we take the Newtonian mass profile of an n =1polytrope:ρ(r )=ρ0R sin rr (61)where R is the radius of the star and ρ0is its central density.We take ρ0to be of a typical stellar density,ρ0≈10−24metres −2.Numerically integrating (59)with a test function of the formb =1−r|f ′′(R )/f ′(R )|,the sametime-scale found in the present work.IV.EINSTEIN-ÆTHER THEORYA.TheoryEinstein-æther theory [6,7]was first formulated as a toy model of a gravitational theory in which Lorentz sym-metry is dynamically broken.This theory contains,along with the metric g ab ,a vector field u a which is constrained (via a Lagrange multiplier Q )to be unit and timelike.The Lagrangian four-form for this theory isL =17Note that our definitions of the coefficients c i differ from those in [6,7,24,25],as do the respective metric signature conventions.The net result is that to translate between our results and those of the above papers,one must flip the signs of all four coefficients.9that in the case c3=−c1>0and c2=c4=0,we have the conventional kinetic term for a Maxwellfield;this special case was examined in[26],prior to Jacobson and Mattingly’s work.)In the present work,we will work in the“vacuum theory”,i.e.,in the absence of matterfields A.Performing the variation of the Lagrangian four-form, wefind thatδL= (E G)abδg ab+(E u)aδu a+E QδQ+∇aθa ǫ(66) where(E G)ab=12g ab J c d∇c u d,(67a)(E u)a=−2∇b J b a−2c4˙u b∇a u b+2Qu a,(67b)(E Q)=u a u a+1,(67c) andθa=θa Ein+2J a bδu b+(J b a u c−J a b u c−J bc u a)δg bc.(68) In the above,we have introduced the notation˙u a= u b∇b u a and J a b=K ac bd∇c u d;θa Ein is defined by(28), as above.Equation(67c)is the constraint that u a is unit and timelike,while(67a)and(67b)are the equations of motion for g ab and u a,respectively.If desired,we can eliminate the Lagrange multiplier Q from these equations by contracting(67b)with u a,resulting in the equationQ=−u a∇b J b a−c4u a˙u b∇a u b.(69) We now take the variation of the symplectic potential current to obtain the symplectic current.The resulting expression can be written in the formωa=ωa Ein+ωa vec(70) whereωa Ein is the usual symplectic current for pure Ein-stein gravity(given by(36)),andωa vec is obtained by tak-ing the antisymmetrized variation of the last two terms in(68).The exact form of this tensor expression is rather complicated,and can be found in Appendix B.B.Obtaining a variational principleAs before,we are primarily concerned with the t-component ofωa,and its form in terms of the pertur-bational variables.We will use the usual spherical gauge (10)for g ab.We will further assume that u a∝t a in the background solution(the so-called“staticæther”as-sumption),and that tofirst order we haveuµ=(e−Φ−φe−Φ,υ,0,0),(71)i.e.,thefirst-order perturbation to u t is−φe−Φwhile thefirst-order perturbation to u r isυ.(Note that un-der these conventions,g ab u a u b=−1tofirst order,as required.)The background equations of motion can then be calculated to be(E G)tt=e2Φ−2Λ 2∂r+1∂r2+1∂r 2−∂Φ∂r−2r∂Φr2 e2Λ−1 −c14∂r2,(73) where we have defined c14=c1+c4and used the equation Q=e−2Λ c3∂Φ∂r−2∂r 2−c3∂2Φ∂tλ2+c123eΦ∂υ1∂r+(c123−c14)∂Φr υ1λ2+c14eΦυ1∂φ2∂t −[1↔2](75) where we have defined c123=c1+c2+c3.In what follows, we will assume that c14=0,and,except where otherwise noted,that c123=0as well.Our next step,as usual,is to solve the constraints. However,in this theory we have the added complication of the presence of a vectorfield.This means,in particu-lar,that the tensor C ab(as defined in(13))is not merely proportional to(E G)ab,as in the previous section,but is insteadC ab=2(E G)ab+u a(E u)b(76)=110withF =2r 2e Φ−Λ2∂r λ+c 14e 2Λ−Φ∂υ∂r .(78)Note that this quantity depends on φ;φand its deriva-tives can appear in the preconstraint equation F =0when our background tensor fields have non-vanishing t -components,asnotedin Footnote2.The remaining non-trivial equations of motion are(δE G )rr =0and (δE u )r =0;in terms of the pertur-bational variables,these are (δE G )rr =−2∂t 2+ 2∂r∂φ∂r∂t +e 2Λ−Φ2c 2∂r+(c 123−c 14)∂Φ∂t(79)and1∂t 2−c 123∂2υ∂r∂t +c 14e−Φ∂2φr+∂Λ∂r ∂υ∂r−(c 1+c 3)2∂t −(c 123−c 14)∂2Φ∂r 2+c 14∂Λ∂r+2∂r+(c 3−c 4)∂Φr 2υ.(80)While we could follow the methods outlined in Section II to reduce these equations to the basic form (1),it is actually simpler to pursue a different path.If we solve (78)for ∂φ/∂r ,rather than λas usual,and plug the resulting expressions into (79)and (80),there result the equations∂ψr 22∂r+c 1+c 3+1r −∂Φc 123υ=0,(82)where we have introduced the new variable ψ,defined asψ=c 123∂λ∂r+c 123∂Λ∂r+(c 2−1)2∂t+r22∂2ψ∂r −∂Φr∂ψ∂r∂Φc 14+c 123−c 2+11∂r+c 123r∂Φ2+c 14∂2ψac 123∂2ψ∂r −∂Φr ∂ψ∂r ∂Φc 14+c 1+c 3+11∂r+c 123r∂Φ2+c 14∞d r r 4e 3Λ−3Φ∂ψ1∂tψ1(88)The form W αβthus defined may be positive or neg-ative definite,depending on the signs of c 123and c 14(recall that we are assuming that c 123and c 14are non-vanishing);however,it is never indefinite.Thus,the sym-plectic form is in fact of the required form (16),and the11 equations of motion can be put in the form(1).We cantherefore write down our variational principle of the form(3);the denominator is(ψ,ψ)=−4πc123c14a ∞0d r r4eΛ−Φ c123 ∂ψ∂r∂Φc14−c2+1 1∂r− c123r∂Λ∂r ∂Φc14−c2+1 1∂r− c123r∂Λc14a∞d r r4 ∂ψ∞0d r r4ψ2.(92)Since both integrands are strictly positive,we conclude thatflat space is stable to spherically symmetric pertur-bations in Einstein-æther theory if and only ifc123(2+c14)Y Y−Y−2+Y+(94)Φ(Y)=−Y+1−Y/Y+ (95)Λ(Y)=18(Y−Y+)(Y−Y−) (96) where the constants Y±are given byY±=−41+c14−Y+(−1−Y+)(1+Y+)/(2+Y+)(98)We can then obtain Z(Y)by writing out(91)in terms of these functions of Y(noting that,for example,∂Λ/∂r= (∂Λ/∂Y)/(∂r/∂Y)),and then plot Z and r parametri-cally.The resulting function is shown in Figure1,for c123=±c14and c14=−0.1.Thefirst thing we see is that in the asymptotic region, the sign of Z(r)is determined by the sign of c123.To quantify this,we can obtain an asymptotic expansion of Z(r)as r→∞,noting that r→∞as Y→0.Doing this,wefind that to leading order in M/r,Z(r)=c123M8Note that if the leading-order coefficient in(99)had been differ-ent from that of the(∂ψ12345678910r r min0.010.0075 0.005 0.00250.00250.0050.00750.01Z r FIG.1:Representative plots of Z (r )versus r for the solu-tions described in [24].The solid and dashed lines corre-spond to c 123=±0.1,respectively;in both cases,c 2=0and c 14=−0.1.Note that the choice of parameters leading to the dashed plot would lead to instability in the asymptotic region.spacetime,of course,there will be some ball of matter in the central region,and the region where the vacuum solution holds will be precisely the region where r ≫M ;thus,we can conclude that for a normal star in Einstein-æther theory,the exterior is stable if and only if (93)holds.(Henceforth,we will assume that the c i coefficients have been chosen with this constraint in mind,unless otherwise specified.)We also note that as r approaches r min ,Z (r )diverges negatively.One might ask whether a test function in this region (in a spherically symmetric spacetime surrounding a compact object,say)could lead to an instability.We investigated this question numerically using our varia-tional principle;however,our results for (ψ,T ψ)were positive for all test functions ψthat we tried.Roughly speaking,the derivative terms in (89)always won out over the effects of the negative Z (r ).9This is,of course,far from a definitive proof of the positivity of (90);and it should be emphasized that the above analysis has not considered the effects of matter on the stability of such solutions.Nevertheless,the above results are at least in-dicative that the spherically symmetric vacuum solutions Einstein-æther theory do not possess any serious stability problems.r 22r (c 2−1)∂υr(1−c 2)∂λ2∂Φrc 2∂Λ∂r+2∂r+12+c 14∂2υZ 0(r )2c 14+1)Z 0(r ).We can plot Z 0(r )parametrically,as was done in the general case;the re-sulting function,shown in Figure 2,falls offrapidly in the asymptotic region.We can also perform an asymp-totic expansion similar to that done in the general case to find the large-r behaviour of Z 0(r );the result is thatZ 0(r )≈(1−c 2)c 14M 213345678910r r min0.10.20.30.40.5Z 0 r FIG.2:Plot of (2∂r=−2r −c 14∂Φrc 2−c 14∂Φr2c 14e 2Λ+ 2∂r2 −1e 2Λ−Φ∂υ∂r≈−c 14+2c 2∂t.(108)Applying this to our solution (103),we find that for suf-ficiently large t∂2φ(c 2−1)3(c 14+2c 2)M ∂rt (109)We can estimate typical scales for ∂φ/∂r by looking at other sources,such as planets.The scale of perturbations due to the Earth’s gravitational field (at its surface)isgiven by ∂φ/∂r ≈M ⊕/r 2⊕,where M ⊕and r ⊕are the mass and radius of the Earth,respectively.Thus,theperturbation to ∂2φ/∂r 2near the Earth’s surface will be of the order∂2φ(c 2−1)3(c 14+2c 2)M ⊙M ⊕(c 2−1)3(c14+2c 2)≪r 2⊕16πR ;(113)L s is the “scalar part”of the action,L s =−14ℓ−2σ4F (kσ2),(114)with k and ℓpositive constants of the theory (with“length dimensions”zero and one,respectively),and F (x )a free function;L v is the vector part of the action,L v =−K14ignore the scalar and matter portions of the Lagrangian,this Lagrangian is the same as that for Einstein-æther theory (64),with c 1=−c 3=−K/16πand c 2=c 4=0.In the present work,we will work exclusively with the “vacuum”(L m =0)theory.Taking the variation of (112)to obtain the equations of motion and the symplectic current,we find thatδL =(E G )ab δg ab +(E u )a δu a +E αδα+E σδσ+E Q δQ +∇a θa )ǫ,(117)where (E G )ab =12σ2∇a α∇b α−Qu a u b+K4g ab F cd F cd+12σ2(∇c α∇c α−˙α2)+18π∇b F ba +2Qu a ,(118b)E α=∇a σ2(∇a α−u a ˙α),(118c)E σ=−σ∇a α∇a α−˙α2+ℓ−2σ2F (kσ2)+18πF b a δu b +F a b u c δg bc.(119)In the above,we have defined ˙α≡u a ∇a α.The remain-ing equation E Q =0is identical to that in Einstein-æthertheory,(67c).For a static solution,in our usual gauge,the background equations of motion become(E G )tt =e 2Φ−2Λ 1r ∂Λr 2(e 2Λ−1)+K∂r 2+∂Φ∂r −22∂Φ4σ2∂α8ℓ2σ4F (kσ2) ,(120a)(E G )rr =1r ∂Φr2(e 2Λ−1)+K ∂r2−1∂r 2+e 2Λ1r 2∂∂r,(120c)andE σ=−e−2Λ∂αkℓ2y (kσ2),(120d)where we have defined y (x )=−xF (x )−116πe −2Λ −∂2Φ∂r∂Λr,(121)to simplify.(As in Einstein-æther theory,the background equation (E u )a =0is satisfied trivially by a static æther.)The symplectic form for the theory can now be calcu-lated from (119).The result will be essentially the same as that in Einstein-æther theory (with the appropriate values of the c i ’s),with added terms stemming from the variations of L s :ωa =ωa Ein +ωavec +ωa s(122)where ωa Ein is given by (36),ωavec is given by (B1)with c 1=−c 3=−K/16πand c 2=c 4=0,andωa s=−σ2(∇aα−u a˙α)2δ1σ2g bc δ1gbc+δ1g ab ∇b α+(g ab −u a u b )∇b δ1α−2u (a δ1u b )∇b αδ2α.(123)B.Applying the variational formalismThe next step is to write out the t -component of ωa in terms of the perturbational variables.We will take our metric perturbations to have the usual form,and our vector perturbation to be of the same form as was used for Einstein-æther theory (71).For the two scalar fields,we define δα≡βand δσ≡τ.Calculating ωts in terms of these perturbational variables,and using the results of (75),we find that ωt=e−2Φ2σ2∂β18π∂Φ∂r−e2Λ−Φ∂υ2∂rβ2−[1↔2].(124)We now turn to the question of the constraints.The tensor C ab is again given by C ab =(E G )ab +u a (E u )b ,and thus to first order we have δC rt =(E G )rt +δu r (E u )t (since u r =0in the background.)Calculating δC rt in terms of our perturbational variables,we find that the15preconstraint equation isF =r 2e Φ−Λσ2∂α8π2∂rλ+K∂t +K∂r=0.(125)In principle,we could now use this equation,togetherwith the equation0=δC rr =e 2Λ σ44πr 2λ+σ1∂r 2τ+e 2Λ−ΦK ∂r∂υ∂r ∂β8π 2∂r∂φ∂r2 υ+e −Φσ2∂α∂t+e−ΦK∂r∂λ∂t 2−∂2φ∂r2 υ=∂4πrλ−2σ2∂α4πrλ−2σ2∂α∂t 2+∂2βσ∂σ∂r −∂Λr∂β∂r∂φ∂r+2∂r ∂τσ2∂α∂rτ−e2Λ−Φ∂α∂t=0.(130)Finally,the field τ,being the perturbation of the auxil-iary field σ,can be solved for algebraically in the equation δE σ=0:ℓ−2σy ′(kσ2)τ+e −2Λ∂α∂r∂β∂t 2=W χ+V 3∂β∂t 2=U 1∂2β∂r+U 3β+V 1∂χH∂χ1∂tβ2−[1↔2],(134)whereH =−2e 2ΛQ +σ2∂α4πr∂Λ∂r−2σ2∂α1646810r z c5 10 64 10 6 3 10 6 2 10 6 1 10 6H rFIG.3:Plot of H (r )versus r for TeVeS.In this plot,k and K are both chosen to be 10−2,and m s =z g =1.terms of a parameter z :10r (z )=z 2−z 2cz +z c−z g /4z c(137)e Φ(z )=z −z c(z 2+z 2c)−18πz clnz −z c4πm s2(141)while αc “sets the value of αat ∞.”The constant m s ,which can be thought of as the “scalar charge”of the star,is defined by anintegral over the central mass dis-tribution [8];for a perfect fluid with ρ+3P ≥0,m s is non-negative.Finally,the constant z g is also defined in terms of an integral over the central matter distribution [8].Plotting H (r )parametrically (Figure 3),we see that this function is strictly negative.In fact,in the z →∞limit,we have r/z =1+O (z c /z ),and so we can take r ≈z to a good approximation.Calculating H (r )in terms of Φ(r )and Λ(r ),we find that as r →∞,H (r )≈−1πm 2s +Kr 4(142)f i (r )∂f iω(r )∂ω∂rei (ω(r )t +κr )f i (r )≈iκe i (ω(r )t +κr )f i (r ).(145)Now let us apply the time-evolution operator T im-plicitly defined by (133)and (132)to our ansatz (143).We see that for sufficiently large κ,the highest-derivative terms will dominate the lower-derivative terms.Thus,to a good approximation we will haveTβχ≈ −κ2U 1(r )iκV 1(r )iκV 3(r )W (r ) f β(r )f χ(r ) e i (ω(r )t +κr ).(146)Then,in the limit of large κ,our ansatz (143)will be an approximate eigenvector of T if there exist a f β(r )and f χ(r )such that−ω2(r )f β(r )f χ(r ) = −κ2U 1(r )iκV 1(r )iκV 3(r )W (r ) f β(r )f χ(r ) .(147)In other words,in the limit of large κ,the problem offinding modes of T is a simple two-dimensional eigenvalue problem where the eigenvalues are functions of r .In this limit,the eigenvalues of this matrix are (to leading order in κ)ω2(r )≈κ2U 1(r ),−W (r )+V 1(r )V 3(r )11Note that this also implies that the “perturbational Hamilto-nian”of TeVeS,as defined in equation (54)of [1],has an indefi-nite kinetic term.。