线性Hamilton系数的几个辛格式英文

应 用 数 学 MATHEMATICA APPLICATA 2005 ,18 (1) :46～50
Ξ A Fe w Symplectic Difference Schemes for Linear Hamiltonian Systems
WANG Ren2hong(王仁宏) ,GAO Feng(高峰)
Zn+1 = ehB Zn , n = 0 ,1 ,2 , …
(6)
where h is the time step size. So when constructing difference schemes of (6) ,the problem is to find ap2
proximants to exp ( tB ) which is also symplectic matrix. A matrix S of order 2 n is called to be symplectic
accelerator physics. The evolution of Hamiltonian systems has the important property of being symplectic ,
i . e. the sum of the areas of the canonical variable pairs projected on any two2dimensional surface in a
Let B = J - 1 C , CT = C , and taking A = J , the author in [ 1 ] comes to the conclusion that for suf2
ficiently small | τ| ,ψ(τB ) is symplectic matrix. This is true if we just note Theorem 1 and the fact B TJ
presented. Based on that ,a few symplectic difference schemes for linear Hamiltonian systems are shown.
Key words :Hamilton system ;Symplectic difference scheme ;Quadratic Padéapproximants CLC Number :O175 AMS( 2000) Subject Classification :34C23 Document code :A Article ID :100129847 (2005) 0120046205
Denote
pi
=
zi , qi
=zi+ nFra bibliotek, Z=
( z1
,
…, z2 n)
T
and
5H 5Z
=
(1) becomes
5H 5 Z1
,
…,
5H 5 Z2 n
T
. Then
dZ dt
=
J
-
1
5 5
H Z
,
(2)
where
0 J=
In , J T = - J = J - 1 ,
- In 0
Ξ Received date :November 11 ,2003 Biography :WANG Ren2hong ,male ,Han ,Jiangxi ,professor ,major research is the theory of approximation.
( Institute of Mathematical Sciences , Dalian University of Technology , Dalian 116024 , China)
Abstract :In this paper , some results of quadratic Padéapproximants of the exponential function exp ( z) are
τwith sufficiently small | τ| , then differentiating both sides of the above equation with respect toτ, we
get B T(ψλ(τB ) ) TAψ(τB ) + (ψ(τB ) ) TAψλ(τB ) = 0.
and A be a matrix of order 2 n. Then (ψ(τB ) ) TA (ψ(τB ) ) = A for all τwith sufficiently small | τ| if
48
MATHEMATICA APPLICATA 2005
B TA + AB = 0. Proof The condition (ii) impliesψ2 (0) ≠0. So ψ(0) ≠0. If (ψ(τB ) ) TA (ψ(τB ) ) = A for all
if it satisfies
STJS = J .
It is clear that all symplectic matrices form a group which is denoted by Sp (2 n) . In [ 1 ] and [ 2 ] some types of symplectic difference schemes of (6) are proposed. These schemes are based on the following the2
⁝
^P
=
.
Q
Q^
f2 n ( z)
f is called a symplectic transformation if
5f 5Z
T
J
5f 5Z
≡J ,
i . e. the Jacobian
matrix
5 5
f Z
is
symplectic
everywhere. For
every
solution
Z ( t)
d pi dt
=-
5H
5 qi
,i
= 1 , …, n
(1)
d qi dt
=
5H 5 pi
is called Hamiltonian system with Hamiltonian function H , where H( P , Q) is some real valued smooth
function on R2 n .
scheme can be constructed by replacing the matrix ehB in (6) with φ( hB) .
It is well known that the Hamiltonian system plays a very important role in the mathematical physics.
One needs only to notice a few examples : classical mechanics , quantum mechanics ,plasma physics ,and
Z
H( Z)
=
1 2
ZT CZ , CT
=
C.
Then the Hamiltonian system (1) becomes
dZ dt
=
BZ,B
=
J - 1 C , CT
=
C,
(4)
and its exact solution is
Z ( t) = exp ( tB ) Z (0) ,
(5)
where exp ( tB ) is symplectic independent of t . From (5) ,we get the following difference scheme
+ JB
= 0. The diagonal Padéapproximants φ( x)
=
[
n,
n
]
x
e
to
the
exponential
function
ex
is
such
a
function that satisfies the conditions (i) , (ii) and (iii) in Theorem 1. So a class of symplectic difference
hold for any analytic functionψ. From condition (ii) it follows that ψλ(λ)ψ( - λ) - ψ(λ)ψλ( - λ) =
0 , so
ddτ(ψ(τB ) TAψ(τB ) ) = ddτ(ψ(τB T) Aψ(τB ) ) = B Tψλ(τB T) Aψ(τB ) + ψ(τB T) ABψλ(τB )
symplectic transformations f t
=
f
t H
,
which
is
called
the
phase
flow
of
the
function H , such that the solution Z ( t) of (2) with initial value Z (0) is given by
orem. Theorem 1[1] Let ψ(λ) be a function of complex variableλsatisfying (i) ψ(λ) is analytic with real coefficients in a neighborhood D of λ = 0 ; (ii) ψ(λ)ψ( - λ) = 1 , in D ; (iii) ψ′(0) ≠0 ,ψ(0) = 1.