【英文版】【开普勒三定律的数学证明】Demostration of Kepler's Laws
证明开普勒猜想的原文(英文)并附译后汉语
Thus it can computed, the ratio of the volume of the internal tangent sphere of cube N to the volume of cube K is equal to √2π/6, i.e. π /√18. This ratio is just that ratio which the Kepler’s conjecture mentions.
Abstract
Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.
《开普勒三大定律》课件
开普勒三大定律推动了科学革命的发展，改变了人们对宇宙的认识 和理解。
开普勒三大定律的现代应用
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航天器轨道设计
开普勒三大定律在卫星轨 道设计、行星探测等方面 有重要应用。
导航定位
利用开普勒三大定律，可 以进行精确的导航定位， 如GPS系统。
科学研究
开普勒三大定律在天文学 、物理学、数学等领域的 研究中仍然具有重要意义 。
REPORTຫໍສະໝຸດ THANKS感谢观看CATALOG
DATE
ANALYSIS
SUMMAR Y
是太阳中心。
轨道的长轴在单位时间内扫过 的面积相等，即行星的轨道周
期与长轴的长度成正比。
定律的发现过程
开普勒通过对第谷·布拉赫观测数据的分析，发现行星轨道并非如哥白尼所设想的那 样是正圆，而是椭圆。
开普勒经过多次尝试和修正，最终确定了行星轨道定律的数学表达式，并成功解释 了行星运动规律。
开普勒的发现为后来的天文学和物理学发展奠定了基础，成为经典力学的重要基石 之一。
定律的意义与影响
总结词
开普勒第二定律对天文学和物理学的发展产生了深远 的影响，它不仅解释了行星轨道运动的规律，还为后 来的牛顿万有引力定律提供了启示。
详细描述
开普勒第二定律的提出，改变了人们对行星轨道运动 的认识。这一定律揭示了行星轨道运动中速度与轨道 半径之间的关系，为后来的天文学和物理学发展提供 了重要的启示。牛顿在研究万有引力定律时，也受到 了开普勒第二定律的启发，进一步揭示了天体运动的 规律。同时，开普勒第二定律也为后来的航天工程提 供了理论基础，帮助人们更好地理解和预测行星运动 。
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CATALOG
《开普勒三大定律》课件
证明方法：通过数学推导和计算，得出开普勒第三定律的公 式
添加项标题
证明意义：开普勒第三定律的发现，为牛顿的万有引力定律 提供了重要的依据，推动了天文学的发展
添加项标题
证明过程：通过观察和计算行星的轨道数据，发现行星的轨 道半长轴与周期之间存在一定的关系
添加项标题
证明结果：开普勒第三定律的公式为T^2=a^3，其中T为 行星绕太阳运动的周期，a为行星轨道半长轴
开普勒三大定律在天体物理学中的应用：可以用来计算行星的轨道参数，预测行星的位置和运动状态，以及研究太阳系和其他星系的结构和演化。
开普勒三大定律的拓展：在广义相对论中，开普勒三大定律被推广到任意引力场中，成为描述物体在引力场中运动的基本规律。
天文学：用于研 究行星运动和宇
宙结构
物理学：用于研 究天体物理和宇
开普勒第二定律还可以用来预测行星的位置和速度，这对于航天器的轨道设计和导航具有重 要意义。
开普勒第二定律在天文学研究中也具有重要意义，它可以帮助我们理解太阳系和其他星系的 形成和演化。
PART FIVE
开普勒第三定律是开普勒行星运动定律之一，描述了行星绕太阳运动的周期与它们到太阳的平均距离之间的关系。
宙学
数学：用于研究 几何学和代数
计算机科学：用 于研究人工智能
和机器学习
工程学：用于研 究航天器和卫星
的轨道控制
地理学：用于研 究地球自转和公
转规律
汇报人：PPT
开普勒第三定律描述了天体运动的周期与它们到太阳的距离之间的关系
在天体运动中，开普勒第三定律可以用来计算天体的轨道参数，如轨道半长轴、偏心率 等
开普勒第三定律在天体物理学、天文学等领域有着广泛的应用，如计算行星的轨道参数、 预测行星的位置等
开普勒第三定律的推导过程
开普勒第三定律的推导过程天文单位（英文：Astronomical Unit，简写AU）是一个长度的单位，约等于地球跟太阳的平均距离万有引力定律是用开普勒第三定律导出的，因此不能再用万有引力定律来推导开普勒第三定律，循环论证是不严谨的。
开普勒第三定律是开普勒根据第谷的观测数据来计算出来的，没有见过推导，推导过程只能是与万有引力定律的联系，不能叫推导。
把星球作的运动看成匀速圆周运动。
这时，万有引力提供向心力。
用质量、角速度、轨道半径表示出向心力，这样就可以写出一个方程.再将方程中的角速度用周期、圆周率表示。
再用绕同一中心天体运的星体列一个方程，两式相比就可证明开普勒第三定律：万有引力（1）向心力（2）（1）=（2），求出（3）又（4）将（3）代入（4）即可R为运行轨道半径，T=行星公转周期，常数这种方法只局限于匀速圆周运动的轨道模型，但现实中的星体运动的轨道都为椭圆，于是便有以下推导：利用微元，矢径R在很小的Δt时间内，扫过面积为ΔS，矢径R与椭圆该点的切线方向夹角为α，椭圆的弧长为ΔR。
在Δt→0时，扫过面积可以看作为三角形，R为半长轴面积速度为各行星绕太阳运行周期为T设椭圆半长轴为a、半短轴为b、太阳到椭圆中心的距离为c则行星绕太阳运动的周期选近日点A和远日点B来研究，由ΔS相等可得从近日点运动到远日点的过程中，根据机械能守恒定律得：得：，由几何关系得：，，所以整理得水星0.998 ，金星0.995 ，地球1，火星0.996，木星0.994，土星0.990 ，天王星1.00 ，海王星0.990。
开普勒三大定律的内容及意义
开普勒三大定律的内容及意义开普勒三大定律是什么，有什么重要的意义？想知道的小伙伴看过来，下面由小编为你精心准备了“开普勒三大定律的内容及意义”仅供参考，持续关注本站将可以持续获取更多的内容!开普勒三大定律的内容开普勒在1609年发表了关于行星运动的两条定律，一条是开普勒第一定律，也叫轨道定律，内容是所有的行星绕太阳运动的轨道都是椭圆的，太阳处在椭圆的一个焦点上。
开普勒第二定律，也叫面积定律，对于任何一个行星来说，它与太阳的连线在相等的时间扫过相等的面积。
用公式表示为：SAB=SCD=SEK到了1619年时，开普勒又发现了第三条定律，也就是开普勒第三定律，也称为周期定律，内容为所有的行星的轨道的半长轴的三次方跟公转周期的二次方的比值都相等。
开普勒不仅为哥白尼的日心说找到了数量关系，更找到了物理上的依存关系，使天文学假说更加的符合自然界本身的真实。
行星运动三大定律的发现为经典天文学奠定了基石，并导致数十年后万有引力定律的发现。
开普勒全名约翰尼斯开普勒，出生于1571年，死于1630年，开普勒是德国近代著名的天文学家，数学家，物理学家和哲学家。
开普勒以数学的和谐性探索宇宙，在天文学方面作出了巨大的贡献，开普勒是继哥白尼之后第一个站出来捍卫太阳中心说，并在天文学方面有突破性的成就的人物，被后世的科学家称为天上的立法者。
开普勒是哥白尼日心说的忠实信徒，为此开普勒做了不少天文测量，并在天文学方面作出了许多积极的贡献，1604年他观察到了银河系内的一颗超新星，历史上称它为开普勒新星，1607年，开普勒观测了一颗大慧星，就是后来的哈雷慧星，到了1609年，开普勒发表了多项有关行星运动的理论，当中包括了开普勒第一定律和开普勒第二定律，1618年，开普勒再次发表了有关行星运动的开普勒第三定律的论文。
开普勒三大定律的意义开普勒的三定律是天文学的又一次革命，它彻底摧毁了托勒密繁杂的本轮宇宙体系，完善和简化了哥白尼的日心宇宙体系。
开普勒第三定律的数学证明
开普勒第三定律的证明开普勒第三定律说轨道周期的平方 T 2 和轨道的半长轴的三次方 a 3 之比为常数。
我们从行星轨道所围成的面积来推导这个结论。
这里顺便推导椭圆的面积公式：如图，椭圆关于x 轴，y 轴均对称，故所求面积为第一象限部分面积的4倍，即S =4S 1=4∫ydx a利用椭圆的参数方程{x =a costy =b sint应用定积分的换元法dx =−a sint dt ,当x =0时 t =π2;当x =a 时，t =0 于是S =4∫bsint (−asint )dt =4ab ∫sin 2tdt =4ab ∫1−cos2t dt =π20π200π24ab (t −1sin2t )|π20=π∙a ∙b 几何公式：椭圆面积=π∙a ∙b (a 为半长轴，b 为半短轴) 定积分公式：椭圆面积=∫d =∫1| 0⃗⃗⃗⃗ || 0⃗⃗⃗⃗ |dt =1 | 0⃗⃗⃗⃗ || 0⃗⃗⃗⃗ | 0 0 两个面积相等可得=2πab | 0|| 0⃗⃗⃗⃗ |=2πa 2| 0|| 0⃗⃗⃗⃗ |∙√1−e 2 (b =a √1−e 2) (式1) 为求a ，利用|⃗ |=(1 e)∙| 0⃗⃗⃗⃗ |令 =π ，得椭圆极半径最大值|⃗|=1e|0⃗⃗⃗⃗ |而长轴2a=|0⃗⃗⃗⃗ ||⃗|=|0⃗⃗⃗⃗ |1e|0⃗⃗⃗⃗ |=2|0⃗⃗⃗⃗ |半长轴a=|0⃗⃗⃗⃗ | 1−e由（式1）平方后，可得2=4π2a4|0⃗⃗⃗⃗ |2|0⃗⃗⃗⃗ |2∙(1−e2)即2a3=4π2|0⃗⃗⃗⃗ |2|0⃗⃗⃗⃗ |2∙a∙(1−e2)=4π2|0⃗⃗⃗⃗ ||0⃗⃗⃗⃗ |2(1e)=4π2GM(a=|0⃗⃗⃗⃗ |1−e)对于特定的太阳系，等式a =4πGM右端是常数，这就是开普勒第三定律所要证明的结论。
开普勒三大定律公式及内容
开普勒三大定律公式及内容开普勒三大定律在天文学中可是超级重要的存在呀！这三大定律就像是解开宇宙奥秘的三把神奇钥匙。
咱们先来说说开普勒第一定律，也叫轨道定律。
它说的是所有行星绕太阳运动的轨道都是椭圆，太阳处在椭圆的一个焦点上。
想象一下，行星们就像一群调皮的孩子，绕着太阳这个“大家长”在椭圆轨道上欢快地奔跑。
我记得有一次在学校给学生们讲解这个定律的时候，有个小同学瞪着大眼睛问我：“老师，那为啥行星的轨道不是正圆呢？”我笑着回答他：“这就好像你跑步，不一定每次都沿着一个完美的圆形跑道跑，可能会有点偏差，行星们也是这样啦。
”这个小家伙似懂非懂地点点头，那模样可爱极了。
开普勒第二定律，又叫面积定律。
说的是行星和太阳的连线在相等的时间内扫过相等的面积。
这就好比行星在“赶路”的时候，离太阳近就跑得快，离太阳远就跑得慢，但是它们很努力地保证在相同时间里走过的“路程”是公平的。
说到这儿，我想起曾经在天文馆看到过一个演示模型，那模型清楚地展示了行星如何按照这个定律运动。
当时周围的小朋友们都看得入了神，嘴里还不停地念叨着：“太神奇啦！”最后是开普勒第三定律，也被称为周期定律。
它指出所有行星绕太阳运动的轨道半长轴的立方与公转周期的平方的比值都相等。
这有点复杂是不是？简单来说，就是不同的行星，它们的轨道大小和绕太阳一圈的时间之间有着固定的数学关系。
记得有一次我带着学生们到操场上，让他们模拟行星的运动，通过实际的体验来感受这些定律。
看着他们兴奋又认真的样子，我知道，他们对这些知识的理解更加深刻了。
在我们探索宇宙的过程中，开普勒三大定律为我们指明了方向。
它们让我们能够更好地理解行星的运动规律，预测天体的位置，甚至为我们探索更遥远的星系提供了基础。
所以呀，别小看这三个定律，它们可是天文学中的瑰宝，带领着我们不断去探索宇宙那无尽的奥秘！。
mathematical proof
Mathematical proofIn mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproven proposition that is believed to be true is known as a conjecture.The statement that is proved is often called a theorem. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice,quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.History and etymologyThe word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),[3] Italian "provare" (to try), and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. Thales (624–546 BCE) proved some theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek “axios” meaning“something worthy”), and used these to prove theorem s using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for ”lines.” He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematicaltheories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).Nature and purposeThere are two different conceptions of mathematical proof. The first is an informal proof, a rigorous natural-language expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof typically encountered in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet. This allows the definition of a formal proof to be precisely specified without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic-synthetic distinction, believed mathematical proofs are synthetic.Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.Methods of proofMain article: Direct proofIn direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two even integers is always even:Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a andb. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clearx+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.This proof uses definition of even integers, as well as distribution law.Proof by mathematical inductionMain article: Mathematical inductionIn proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number.A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, ... } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that∙(i)P(1) is true, i.e., P(n) is true for n = 1∙(ii)P(n + 1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n + 1) is true.Then P(n) is true for all natural numbers n.Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction. However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.Proof by transpositionMain article: Transposition (logic)Proof by transposition or proof by contrapositive establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".Example:∙Proposition: If x² is even then x is even.∙Contrapositive proof:If x is odd (not even) then x = 2k + 1 for an integer k. Thus x² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, where (2k² + 2k) is integer. Therefore x² is odd (not ev en).To see the original proposition, suppose x² is even. If x were odd, then we just showed x² would be odd, even though it is supposed to be even; so this case is impossible. The only other possibility is that x is even.Proof by contradictionMain article: Proof by contradictionIn proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction toward the absurd"), it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. This method is perhaps the most prevalent of mathematical proofs. A famousexample of proof by contradiction shows that is an irrational number:Suppose that were a rational number, so by definitionwhere a and b are non-zero integers with no common factor.Thus, . Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a2 is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even.However, if a and b are both even, they share a factor, namely 2.This contradicts our assumption, so we are forced to conclude that is an irrational number.Students can easily fall into erroneous proofs with this method. In searching for a direct proof, a mistake in reasoning will lead to false conclusions, which can often be detected as absurd, alerting the student to his or her error. But in constructing a proof by contradiction, a mistake in reasoning which implies absurd statements tends to be seen as the successful end of the proof.Proof by constructionMain article: Proof by constructionProof by construction, or proof by example, is the construction of a concrete example with a property to show thatsomething having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.Proof by exhaustionMain article: Proof by exhaustionIn proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.Probabilistic proofMain article: Probabilistic methodA probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not to be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.Combinatorial proofMain article: Combinatorial proofA combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.Nonconstructive proofMain article: Nonconstructive proofA nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b is a rational number:Visual proofAlthough not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historicvisual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.∙Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.∙Visual proof for the Pythagorean theorem by rearrangement. Elementary proofMain article: Elementary proofAn elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.Two-column proofA two-column proof published in 1913A particular form of proof using two parallel columns is often used in elementary geometry classes in the United States. The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typicallyheaded "Statements" and the right-hand column is typically headed "Reasons".Statistical proofs in pure mathematicsMain article: Statistical proofThe expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic or analytic number theory. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also "Statistical proof using data" section below.Computer-assisted proofsMain article: Computer-assisted proofUntil the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced byincorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Furthermore, although a computer might make a mistake when checking a proof, errors can never be completely ruled out in case of a human proof verifier as well, especially if the proof contains natural language and requires mathematical insight.Undecidable statementsA statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see list of statements undecidable in ZFC.Gödel's (first) incomplete ness theorem shows that many axiom systems of mathematical interest will have undecidable statements.Heuristic mathematics and experimental mathematicsMain article: Experimental mathematicsWhile early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs werean essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of theproof-theorem framework, in experimental mathematics. Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so embedded.Related conceptsColloquial use of "mathematical proof"The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument are numbers. It is sometime also used to mean a "statistical proof" (below), especially when used to argue from data.Statistical proof using dataMain article: Statistical proof"Statistical proof" from data refers to the application of statistics, data analysis, or Bayesian analysis to infer propositions regarding the probability of data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empiricalevidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in cosmology. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots, when the data or diagram is adequately convincing without further anaylisis.Inductive logic proofs and Bayesian analysisMain articles: Inductive logic and Bayesian analysisProofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than one certainty. Bayesian analysis establishes assertions as to the degree of a person's subjective belief. Inductive logic should not be confused with mathematical induction.Proofs as mental objectsMain articles: Psychologism and Language of thoughtPsychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibniz, Frege, and Carnap, have attempted to develop a semantics for what they considered to be the language of thought, wherebystandards of mathematical proof might be applied to empirical science.Influence of mathematical proof methods outside mathematicsPhilosopher-mathematicians such as Schopenhauer have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descarte’s cogito argument.Ending a proofMain article: Q.E.D.Sometimes, the abbreviation "Q.E.D."is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" in an oral presentation on a board.。
《开普勒三定律》课件
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开普勒三定律在天文学领域的应用，也促进了数学、物理学等
其他相关学科的发展。
对物理学的影响
开普勒三定律为牛顿力学的发展奠定了基础
牛顿在开普勒三定律的基础上，提出了万有引力定律和三大运动定律，建立了完整的经典 力学体系。
开普勒三定律推动了物理学中理论模型的发展
开普勒三定律的发现促使人们更加重视理论模型在科学研究中的作用，推动了物理学中理 论模型的发展。
推导过程
总结词
通过观察和计算行星的轨道数据，开普勒提出了开普勒第二定律的数学表达式，并进行了详细的推导 。
详细描述
开普勒通过对行星轨道数据的观察和计算，提出了开普勒第二定律的数学表达式。他通过几何学和代 数学的方法，对行星轨道半径和时间的关系进行了详细的推导。推导过程中，开普勒考虑了行星在椭 圆轨道上的运动规律，以及时间与行星位置之间的关系。
实例和行和科 学研究等领域有着广泛的应用。
详细描述
通过开普勒第三定律，我们可以计算出行星 的公转周期，进而了解行星的运动规律和轨 道参数。这对于研究行星运动、太阳系演化 以及探测外太空等领域具有重要意义。此外 ，开普勒第三定律也是研究其他天体系统的
基础之一，如恒星、星系等。
开普勒三定律的提出是科学革命的重要里程碑，它标志着人们对宇宙的理解从地心 说转向日心说。
开普勒的生平
约翰尼斯·开普勒出生于德国威 斯巴登，是文艺复兴时期的天文
学家、数学家和哲学家。
开普勒在年轻时曾追随第谷·布 拉赫进行天文观测，并成为其门
徒。
开普勒提出了行星运动的第一和 第二定律，即椭圆轨道定律和面 积定律。他还在数学和光学方面
做出了重要贡献。
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开普勒第一定律：椭圆轨道定 律
开普勒三大定律证明过程
开普勒三大定律证明过程开普勒三大定律是天体运动的基本规律之一，它们提供了描述行星运动的数学模型。
这三个定律分别被命名为第一定律（椭圆轨道定律）、第二定律（面积速度定律）和第三定律（调和定律）。
下面我将一步一步回答关于开普勒三大定律的证明过程。
首先，我们来看第一定律，即椭圆轨道定律。
开普勒认为，所有行星绕太阳运动的轨道都是椭圆形的，而太阳位于椭圆的一个焦点上。
这个定律可以通过下面的步骤来证明。
第一步：假设太阳是位于椭圆轨道的一个焦点上。
我们将焦点记为F，椭圆上某一点记为P，椭圆长轴的一半长度记为a，焦点到椭圆上某一点的距离记为r。
第二步：根据几何性质，我们可以得知定义椭圆的特点是焦点到椭圆上每一点的距离之和等于2a。
即，FP + FP' = 2a，其中P'是椭圆上对称点。
第三步：由于太阳在焦点F上，所以FP即为太阳到行星的距离，记作r。
然后我们可以得到FP' = r - FP。
第四步：将第二步和第三步的结果代入，可以得到2a = r + (r - FP)，整理得到FP = 2a - r。
第五步：由于椭圆的定义，太阳到椭圆上每一点的距离之和等于2a。
即，太阳到行星的距离即为2a。
所以，我们得到FP = 2a - r = 2a - 2a + FP'，即FP = FP'。
第六步：根据几何性质，椭圆的定义中，焦点到椭圆上每一点的距离之和等于2a，并且椭圆上对称的两个焦点到该点的距离相等。
所以，FP = FP' = r。
第七步：通过以上步骤的推导，我们证明了太阳到行星的距离r等于行星到椭圆的另一个焦点的距离FP'，也就是说，所有行星绕太阳运动的轨道都是椭圆形的。
接下来，我们来看第二定律，即面积速度定律。
开普勒认为，行星在椭圆轨道上任意两点之间的扫面面积相等，且行星和太阳的连线在同等时间内扫过的面积相等。
下面是证明过程。
第一步：假设行星距太阳的距离为r，太阳到该行星的连线在时间间隔dt 内扫过的面积为dA，过太阳的线在该时间间隔内扫过的面积为dA'。
证明开普勒第三定律
证明开普勒第三定律开普勒第三定律是描述行星运动规律的重要定律之一，它被广泛应用于天文学和航天技术中。
本文将从历史背景、定律表述、数学证明等方面详细介绍开普勒第三定律。
一、历史背景开普勒第三定律是由德国天文学家约翰内斯·开普勒于17世纪初期发现的。
当时，他通过对地球和其他行星的观测数据进行分析，发现了一些规律性的现象。
其中最重要的就是行星公转周期与其轨道半长轴之间存在着固定的比例关系。
二、定律表述开普勒第三定律可以简单地表述为：行星公转周期的平方与其轨道半长轴的立方成正比。
具体地说，如果用T表示一个行星公转一周所需的时间（也就是周期），用a表示该行星与太阳之间距离的平均值（也就是轨道半长轴），那么这个关系可以写成：T^2 = k a^3其中k是一个常数，它对于每个行星都有不同的值。
这个式子告诉我们，无论任何一个行星，其公转周期的平方与轨道半长轴的立方之比都是相等的。
三、数学证明为了证明开普勒第三定律，我们需要运用牛顿万有引力定律和牛顿第二定律。
这里简要介绍一下证明过程。
首先，根据万有引力定律，我们可以得到：F =G m M / r^2其中F是两个物体之间的引力，G是万有引力常数，m和M分别是两个物体的质量，r是它们之间的距离。
然后，我们可以将上式中的F代入牛顿第二定律中：F = m a其中a是物体所受合外力产生的加速度。
将前面得到的引力公式代入上式中，并整理一下得到：a = G M / r^2接下来，我们考虑一个行星在其椭圆轨道上运动时所受合外力。
此时，行星所受合外力可以分解成一个向心力Fc和一个切向力Ft。
由于行星在椭圆轨道上运动时速度不断变化，因此切向力也在不断变化。
但由于开普勒第二定律（即面积速率相等定律），我们可以知道单位时间内行星扫过的面积是相等的。
因此，我们可以得到：Fc = m v^2 / r其中v是行星在轨道上的速度。
将上式代入前面得到的a的公式中，我们可以得到：a = G M / r^2 = v^2 / r这个式子告诉我们，行星所受向心加速度与它在轨道上的速度平方成正比，与它距离太阳的距离平方成反比。
开普勒第二定律证明开普勒第三定律
开普勒第二定律是描述行星在其椭圆轨道上运动的规律。
它可以用以下方式来表述：在相同时间内，行星与恒星连线所扫过的面积是相等的。
这个定律表明了行星的轨道速度并非始终保持不变，而是根据其离恒星的距离而变化的。
那么，如何证明开普勒第二定律呢？我们需要先从开普勒第三定律出发，深入探讨开普勒运动定律的数学原理。
1. 开普勒第三定律的数学描述开普勒第三定律可以用数学公式来表示：T^2/a^3 = 常数，其中T代表行星绕恒星一周的周期，a代表行星轨道的半长轴。
这个公式告诉我们，不同行星的轨道特征之间存在着某种关联，而这种关联是用一个常数来描述的。
在这里，我们可以假定这个常数为K。
2. 推导出开普勒第二定律根据椭圆的性质，其面积可以用数学公式进行描述。
假设在时间Δt内，行星在其椭圆轨道上移动了Δθ角度，我们可以推导出行星与恒星连线所扫过的面积为：ΔS = (1/2) * a * b * Δθ，其中a和b分别代表椭圆的半长轴和半短轴。
又因为椭圆面积公式为：S = π * a * b，我们可以进一步得到：ΔS/Δt = (1/2) * a * b * (Δθ/Δt) = (1/2) * r^2 *(Δθ/Δt)，这里r代表行星与恒星的距离。
由开普勒第三定律我们知道T^2/a^3 = K，即T^2 = K * a^3。
将这个式子代入ΔS/Δt的公式中，我们可以得到：ΔS/Δt = (1/2) * K * a^3 * (Δθ/Δt)。
3. 结论与个人观点通过以上推导，我们可以看出行星与恒星连线所扫过的面积与时间有关，而且根据开普勒第三定律，这种关联是用一个常数来描述的。
这就证明了开普勒第二定律：在相同时间内，行星与恒星连线所扫过的面积是相等的。
这个定律的发现，使我们对行星运动的规律有了更深入的理解，也为之后牛顿的万有引力定律奠定了基础。
在我的个人观点中，我认为开普勒定律的提出和证明是人类理解宇宙运动规律的重要里程碑。
它不仅推动了天文学的发展，也深刻影响了整个科学领域。
开普勒三大定律的数学证明
开普勒三大定律的数学证明开普勒是17世纪德国的一位天文学家和数学家，他提出了著名的开普勒三大定律，这些定律描述了行星运动的规律。
开普勒的三大定律为：第一定律（行星轨道为椭圆）、第二定律（面积定律）和第三定律（调和定律）。
本文将逐一证明这三个定律。
首先，我们来证明开普勒第一定律，即行星轨道为椭圆。
为了证明这个定律，我们需要引入一些数学工具，其中最重要的是椭圆的定义和焦点定理。
椭圆是一个平面上的几何图形，它由一个定点F（焦点）和一条定长线段的两个端点P和P'（两个焦点的连线）组成。
椭圆的定义是，对于平面上的任意一点P，P到焦点F的距离与P到直线PP'的距离的和是一个常数。
这个常数被称为椭圆的离心率。
根据椭圆的定义，我们可以推导出行星轨道为椭圆的结论。
假设行星的轨道是一个圆，我们可以找到一个点F，使得行星在这个点F附近运动。
此时，行星到点F的距离和行星到圆的圆心的距离之和是一个常数。
然而，根据圆的性质，行星到圆心的距离是一个常数，所以行星到点F的距离也必须是一个常数。
这意味着点F必须是焦点，而行星的轨道是一个椭圆。
接下来，我们证明开普勒第二定律，即面积定律。
这个定律描述了行星在椭圆轨道上的运动速度和位置的关系。
为了证明面积定律，我们需要先介绍一下角动量和守恒定律。
角动量是一个物体绕某一点旋转时的动量，它的大小等于物体的质量乘以它的角速度和到旋转轴的距离的乘积。
守恒定律是指在一个封闭系统中，如果没有外力作用，系统的某个物理量将保持不变。
假设行星在椭圆轨道上的某一时刻距离焦点F的距离为r，行星的速度为v，角动量为L。
根据角动量守恒定律，L的大小是恒定的。
在相同时间间隔内，行星在轨道上扫过的面积是相等的，即行星在相同时间内在椭圆轨道上扫过的面积是相等的。
根据角动量和面积的关系，我们可以得到行星在椭圆轨道上的运动速度和位置的关系。
行星在椭圆轨道上的速度是不断变化的，当行星离焦点F越近时，它的速度越快，当行星离焦点F越远时，它的速度越慢。
开普勒三大定律的数学推导
开普勒三大定律的数学推导开普勒三大定律，哎呀，这可不是什么天文难题。
反而，如果你有点耐心，捋清楚了，你会发现这些定律既简单又有趣，像是解谜一样。
我们今天就来聊聊这些定律的数学推导，搞明白它们是怎么一步步从天文现象中浮现出来的。
你说开普勒能不能算个“天文大厨”？他可是把这些复杂的宇宙规律煮得像小菜一样简单，怎么做呢？我们慢慢看。
开普勒定律的第一条，听起来有点牛逼，“行星绕太阳的轨道是椭圆形的，太阳在椭圆的一个焦点上。
”别急，别急，咱不讨论焦点这个词是不是有点专业了。
你想象一下，一颗行星绕着太阳转，太阳就像是舞台的明星，行星是明星的粉丝，围着它转。
可是，这个轨道不是圆形的哦，而是椭圆形的，这个椭圆的形状，直接决定了行星的运行方式。
要说推导，嗯，咱得先了解一下啥叫椭圆了。
椭圆就是圆形的“远房亲戚”，它的形状比圆稍微拉长了点。
开普勒是怎么得出这条定律的呢？他研究了大量的观测数据，尤其是当时天文学家对火星的观察数据。
通过分析这些数据，他发现火星并不是在完美的圆形轨道上运行，而是走了一条有点拉长的椭圆轨道。
所以，他推测，其他行星也应该差不多。
然后再通过牛顿的万有引力定律，才终于理顺了这个规律——行星绕太阳转，轨道是椭圆，而且太阳的位置就在这个椭圆的一个焦点上。
嗯，明白了吧？开普勒的第二条定律，别小看这个，它可真不简单。
“行星与太阳连线在相等的时间内扫过相等的面积。
”听起来是不是有点晦涩？但其实挺简单的。
你想想，行星绕太阳跑，它不是匀速的吗？当然不是！行星离太阳近的时候，跑得快；离太阳远的时候，慢得像个老爷车。
开普勒用这个定律就是要说明，行星在靠近太阳时跑得快，远离太阳时跑得慢，但无论快慢，行星和太阳之间的“连线”每经过相同的时间，所扫过的面积都是一样的。
你就把它想成，行星好像在绘制一个巨大的扇形图，太阳在扇形的圆心，无论行星跑多快，画的扇形面积总是相同的。
你别看这个看起来挺抽象的，但实际上，开普勒是通过大量的精确测量数据，得出这个规律的。
开普勒三定律和牛顿三定律的关系
开普勒三定律和牛顿三定律的关系开普勒三定律和牛顿三定律，这两者听起来好像是天文物理学的大块头，实际上，它们之间有着紧密的关系，就像是两兄弟，虽然各自有各自的特点，但站在一起的时候，简直能让人眼花缭乱，恍若天书。
首先说到开普勒，他是那个把太阳系的行星运动规律捋得清清楚楚的家伙。
简单来说，他告诉我们，行星围着太阳转，不是那种随便走的直线，而是那种优雅的椭圆形轨道。
你想，行星们就像是跳舞的舞者，在大舞台上旋转跳跃，飞扬的裙摆，舞步之间的优雅，又让人忍不住为之点赞。
不过，开普勒的三定律嘛，其实只是找到了行星们如何运动的规则，但他并没有深入挖掘为什么会这样。
牛顿呢，嗯，他就像是那位老顽童，天生就好奇，想要弄明白行星运动背后的“真相”。
于是他在开普勒的基础上，给我们带来了重磅消息——引力的定律！哇，牛顿简直是那种“嘿，你们以为这些是巧合吗？”的角色。
他说，太阳和行星之间有一股无形的力量把它们紧紧拉在一起，叫做“引力”。
这引力是个神奇的东西，越大质量的物体，越有吸引力。
离得越远，引力就越小。
这就解释了，为什么地球不至于被太阳吸进去，也不会自己乱跑，而是安稳地围绕太阳转悠。
牛顿三定律可是牛得不得了！第一条是惯性定律，意思是物体不受外力影响的话，它就会一直保持原样，运动或静止，自己不动，除非你给它点动力。
就像你坐在车里，车停了你还会感觉自己向前晃动一下。
第二条是加速度定律，说的其实是力和加速度之间的关系。
简而言之，就是你给物体施加的力越大，它加速得就越快。
第三条是作用与反作用，简单来说，就是“打人一巴掌，你也得挨回一巴掌”。
也就是说，你对地球施加一股力，地球也会给你回敬一股力，虽然你感觉不到，地球巨大的体积让你没啥感觉。
你一定会想，开普勒和牛顿的定律有什么联系嘛？牛顿的引力定律就是开普勒定律的背后推力。
开普勒一开始发现，行星的运动有规律，行星在围绕太阳转的时候，轨道和速度都有固定的数学关系。
可是他并没有解释为什么会这样。
开普勒第三定律的证明
T 2 ka 3 。
关键词：开普勒；轨道；周期；角动量 中图分类号：O311.1 文献标识码：B 文章编号：1009-9115（2004）02-0056-02
开普勒第三定律的内容是：行星运动周期 T 的 平方与其椭圆轨道半长轴 a 的立方之比为常量，即 T2 = ka3，其中常量 k 对所有行星都相同。下面证明 该定律。 行星在太阳的作用下沿椭圆轨道运动，且行星 对太阳的角动量保持不变。因此，行星总在一个平 面内运动，它的轨道是一个平面椭圆轨道。设行 星的质量为 m，其轨道角动量的大小为：
2004 年 3 月 Mar. 2004
开普勒第三定律的证明
刘维生
（唐山师范学院 物理系，河北 唐山 063000） 摘 要：行星在太阳的作用下沿椭圆轨道运动，且行星对太阳的角动量保持不变。因此，行星总在一个平面
内运动，它的轨道是一个平面椭圆轨道。行星运动周期 T 的平方与其椭圆轨道半长轴 a 的立方之比为常量，即
图1 行星的椭圆轨道运动
p ；椭圆轨道上行星距日心最远点 1 e p ； 1 e
称为远日点，远日点与日心的距离为： rmax
以 s 表示阴影部分的面积，则 r R r sin 2s 故 ∴
L 2mlim
t 0
s ds 2m t dt
（1）
图2 椭圆长、短轴与、rmax、rmin的关系
4 2 a 3 GM
1 a (rmax rmin ) ， 2
∴
2a rmax rmin
p p 1 e 1 e
将（6）代入（5）得到： T 2 令
k
2 p 2 L2 · 1 e 2 1 e 2 GMm 2
L2 aGMm 2 (1 e 2 )
开普勒三定律的数学证明
开普勒三定律的数学证明摘 要：本文依次对开普勒第二，第三和第一定律进行详细的数学证明，并用物理学中角动量守恒的方法对开普勒第二定律进行证明。
关键字：开普勒定律；角动量守恒Mathematical Proofs of Kepler ’s LawDu Yonghao(Civil Engineering Department of Southeast University, Nanjing 211189, China)Abstract: My paper particularly derives Kepler ’s Second Law, Third Law and First Law in mathematical methods in order. Law of Conservation of Angular Momentum is also applied to derive Kepler ’s Second Law.Key words: Kepler ’s Law; Law of Conservation of Angular Momentum1 前言开普勒第一定律，也称椭圆定律、轨道定律：每一个行星都沿各自的椭圆轨道环绕太阳，而太阳则处在椭圆的一个焦点中。
开普勒第二定律，也称面积定律：在相等的时间内，太阳和运动中的行星的连线（向量半径）所扫过的面积都是相等的。
这一定律实际揭示了行星绕太阳公转的角动量守恒。
开普勒第三定律，也称调和定律、周期定律：各个行星绕太阳的椭圆轨道的半长轴的立方和它们公转周期的平方成正比[1]。
2 开普勒第二定律证明2.1 数学方法令()t r 为行星在t 时刻的位失，令()t t r ∆+为行星在()t t ∆+时刻的位失。
面积A ∆为在t 时刻与()t t ∆+时刻间行星位失扫过的面积，即()t r 与()()t r t t r r -∆+=∆所围成的三角形面积，如图1，得：()r t r A ∆⨯≈∆21所以：()trt r t A ∆∆⨯≈∆∆21 令0→∆t ，得：()()t r t r dt dA '⨯=21()1 图1[2]行星与太阳之间的万有引力是作用在行星上的唯一的力，引力大小为)2t GMm ，其中m 为行星的质量。
关于Kepler第三定律的推导
关于Kepler第三定律的推导
鞠衍清
【期刊名称】《常州工学院学报》
【年(卷),期】2005(018)001
【摘要】Kepler第三定律是研究星体运动的一条重要定律,但在力学教材中,对其证明或者忽视,或者过于繁琐.采用动量矩及机械能守恒定律对其进行推导,数学计算简洁明了,而且物理意义也更为丰富.
【总页数】3页(P7-9)
【作者】鞠衍清
【作者单位】辽东学院科研处,辽宁,丹东,118003
【正文语种】中文
【中图分类】O31
【相关文献】
1.用积分法推导开普勒第三定律的补充 [J], 丁邦建
2.开普勒第三定律的另几种推导 [J], 项红专
3.由玻意耳定律推导盖·吕萨克定律和查理定律 [J], 冯学斌;王述善
4.从Newton定律到Kepler三定律 [J], 保继光
5.开普勒第三定律表达式的一种新的推导方法 [J], 邓麦芹
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perpendicular to it. (Fig.2)
Then we prove Kepler’s second law before the other ones.
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semi-major axis d, period of motion T and distance to focus r, then the value of acceleration is ������ = ������������2, where ������ = 4������������22������3. Through this method Newton proved that the value of a planet can be derived from Kepler’s first law.
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demonstration to explain Newton’s work on demonstrating Kepler’s Laws.
1. Demonstration by Newton: Kepler’s Laws simply described the phenomenon of planets’ motion, which were definitely meaningful in the development of Physics and Astronomy, however lacked with some supports from theorem, which lately became the work of one of the great man in history of humanity, Newton, who mathematically find the essential of Kepler’s Law. In terms of Kepler’s second law, Newton demonstrated that it is equivalent between the second law and central motion, i.e., supposing a particle moving on a plane where the segment between it and a fixed point sweeps a fixed area, then the particle is moving under the central force, and vice versa. From the equivalence of the second law and central force, we know the acceleration vector of a planet always pointing the
Then discuss
Байду номын сангаас
������ = ������ × ������ = ������������̂ × (������̇������̂ + ���������������̇̂���) = ������(������������̇)���̂���
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make ∆������ → 0,
d������ 1 d������ = 2 ������ × ������.
Already knows, ������ × ������ = ������, which is constant. Thus we can conclude the second law, areal
Report
Abstract:
Demonstration of Kepler’s Laws
HAN Fang-Zhou
The mathematical explanation about planet’s motion given by Newton and Kepler are two of
the most important achievement in science history. In this article we have a modern
2. Demonstrating Kepler’s Laws with modern method:
Suppose ������ to be radial vector emitting from sun pointing a planet with mass of ������.
Universal gravitational law told us that the attraction between planet and sun is ������������������
������=0
=
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Fig.5
or
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Substitute the result into the equation above
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