Symplectic Schemes for Birkhoffian System

19 Pages·0.18 MB·English

by  Hongling Su

Symplectic Schemes for Birkhoffian system ∗
Hongling Su Mengzhao Qin
CAST (World Laboratory),
Institute of Computational Mathematics
and Scientific/Engineering Computing,
Academy of Mathematics and System Sciences,
3
0 Chinese Academy of Sciences, Beijing, 100080, China.
0
2
February 7, 2008

n
a
J

3
Abstract

1
v Auniversalsymplectic structure for a Newtoniansystemincluding
1 nonconservativecasescanbeconstructedintheframeworkofBirkhof-
0
fian generalization of Hamiltonian mechanics. In this paper the sym-
0
plectic geometrystructure ofBirkhoffiansystemis discussed,then the
1
0 symplecticity of Birkhoffian phase flow is presented. Based on these
3 properties we give a way to construct symplectic schemes for Birkhof-
0 fian systems by using the generating function method.
/
h
p
-
h 1 Introduction
t
a
m
Birkhoffian representation is the generalization of Hamiltonian representa-
:
v tion, which can be applied to hadron physics, statistical mechanics, space
i
X
mechanics, engineering, biophysics and so on[1, 2]. All conservative or non-
r
a conservative, self-adjointornon-self-adjoint,unconstrainedornonholonomic
constrainedsystemsalways admitarepresentationofBirkhoff’sequations[2,
3]. In last 20 years, many researchers have studied Birkhoffian mechanics
andobtainedaseries ofresultsinintegraltheory, stability ofmotion, inverse
problem, algebraic and geometric description, and so on.
Birkhoff’sequationsaremorecomplexthanHamilton’sequations,soare
thestudiesoftheircomputational method. Inthepast, therearenotanyre-
sultinthecomputationalmethodsforBirkhoffiansystem. Theknowndiffer-
∗ SupportedbytheSpecialFundsfor Major StateBasic Research Projects ,G1999,
032800
1
ence methodsare notgenerally applicable to Birkhoffian system. As the dif-
ference scheme to solve Hamiltonian system should be Hamiltonian scheme,
the difference scheme to simulate Birkhoffian system should be Birkhoffian
scheme, however, the conventional difference schemes such as Euler center
scheme, leap-frog scheme andso on are notBirkhoffian scheme, soa method
of how to systematically construct a Birkhoffian scheme is necessary. It is
the main context in this paper.
The systems both Birkhoffian and Hamiltonian mentioned in this paper
are usually finite dimensional situation[6], in fact, the definition of infinite
dimensional Birkhoffian system has not proposed before. Ref. [2] described
the Algebraic and Geometric profiles of the finite dimensional Birkhoffian
systems in local coordinates, and general nonautonomous Hamiltonian sys-
tem is considered as autonomous Birkhoffian system. Ref. [4] developed
systematically the symplectic schemes for the standard Hamiltonian system
and general Hamiltonian system on the Poisson manifold which belong to-
gether the autonomous and semi-autonomous Birkhoffian system. Soin this
paper, we just discuss the nonautonomous Birkhoffian system in detail.
Insection2,wesketchoutBirkhoffiansystemviavariationalself-adjoin-
tness, this sketch shows the relationship between Birkhoffian system and
Hamiltonian system more essentially and directly, then we give the basic
geometrical properties of Birkhoffian system.
Section 3extendsthedefinitionsofK(z)-Lagrangian submanifoldtothe
onewithaparametertsuchasK(z,t)-Lagrangian submanifold,thenwedis-
e
cuss the relationship between symplectic mappings and gradient mappings.
e
Section 4 constructs the generating functions for the phase flow of the
Birkhoffian system and gives the method to simulate Birkhoffian systems
by symplectic schemes of any order. The last section shows an illustrating
example. Afirst-order anda second-order schemes for thesystem of a linear
damped oscillator are given.
The Einstein’s summation convention is used in the next sections.
2 Birkhoffian System
The generalization of Hamilton’s equations we shall study is given by
∂F ∂F dz ∂B(z,t) ∂F(z,t)
j i i
− − + = 0, (1)
∂z ∂z dt ∂z ∂t
i j i
(cid:18) (cid:19) (cid:18) (cid:19)
2
following the terminology suggested by Santilli(1978)[2], we called it as
Birkh-
off’s equation or Birkhoffian system. The function B(z,t) is called the
Birkhoffian, because of certain physical difference with Hamiltonian which
is indicated in Ref. [2]. F , i = 1,2,· · ·,2n, are Birkhoffian functions.
i
A representation of Newton’s equations via Birkhoff’s equation is called a
Birkhoffian representation.
Definition 2.1. Birkhoff’s equations (1) are called autonomous when the
F and B functions are free of the time variable, in which case the equations
i
are of the simple form
dz ∂B(z)
j
K (z) − = 0. (2)
ij
dt ∂z
i
They are called semi-autonomous when the F functions do not depend ex-
i
plicitly on time, in which case we have the more general form
dz ∂B(z,t)
j
K (z) − = 0. (3)
ij
dt ∂z
i
Birkhoff’s equations are called nonautonomous when both the F and B func-
i
tions have explicit dependence on time, in which case we rewrite
dz ∂B(z,t) ∂F (z,t)
j i
K (z,t) − − = 0. (4)
ij
dt ∂z ∂t
i
Here they all have
∂F ∂F
j i
K = − . (5)
ij
∂z ∂z
i j
They are called regular when the functional determinant is not null in the
region considered:
det(K )(ℜ)6= 0, (6)
ij
otherwise, degenerate.
e
Given an arbitrary analytic and regular first-order system
dz
i
K (z,t) +D (z,t) = 0, i = 1,2,...2n, (7)
ij i
dt
From the point of the inverse variational problem[5], this system is self-
∗
adjoint iff. it satisfies the following condition in ℜ , i.e.
K +K = 0,
ij ji
e
∂K ∂K ∂K
ij jk ki
+ + = 0,
∂z ∂z ∂z (8)
k i j
∂K ∂D ∂D
ij i j
= − , i,j,k = 1,2,...2n.
∂t ∂z ∂z
j i
3
We now simply introduce the geometric significance of the condition of vari-
ational self-adjointness[8, 9, 10, 11, 12]. Here the region considered is a
star-shaped region ℜ∗ of points of R×T∗M, T∗M the cotangent space of
the M, M a 2n-dimensional manifold.
e
Consider first the case for which K = K (z). Given a symplectic
ij ij
structure written as the 2-form in local coordinates
2n
Ω = K (z,t)dz ∧dz , K = −K , (9)
ij i j ij ji
i,j=1
X
one of the fundamental properties of symplectic form (9) is that dΩ = 0.
The geometric significance of the condition of self-adjointness (8) is then
the integrability conditions for 2-form (9) to be an exact symplectic form
coincident with the first two formulas of condition (8). Because the exact
character of two-form (9) implies following structure
Ω = d(F dz ), (10)
i i
this geometric property is fully characterized by the first two equations of
condition (8), and we can say that the two-form (10) describes the geomet-
rical structure of the autonomous case (2) of the Birkhoff’s equations, even
it sketches out the geometric structure of the semi-autonomous case.
For the case of that K = K (z,t), the full set of condition (8) must
ij ij
be considered and the corresponding geometric structure can be better ex-
pressed by transition of the symplectic geometry on the cotangent bundle
T∗M with local coordinates z to the contact geometry on the manifold
i
R×T∗M with local coordinates z , i= 0,1,2,···,2n, z = t[1]. In this case
i 0
more general formulations of an exact contact 2-form persist, although it is
e e
now referred to as a (2n+1)-dimensional space,
2n
Ω = K dz ∧dz = Ω+2D dz ∧dt, (11)
ij i j i i
i,j=0
X
b b e e
where
0 −DT
K = , (12)
D K
(cid:18) (cid:19)
if the contact form is also ofbthe exact type
−B
Ω= d(F dz ), F = , (13)
i i i F
i
(cid:26)
b e e e
4
the geometric meaning of the condition of the self-adjointness is then the
integrability condition for the exact contact structure (13). Here B can be
calculated from
∂B ∂F
i
− = D + (14)
i
∂z ∂t
i
for
∂ ∂F ∂ ∂F
i j
D + = D + . (15)
i j
∂z ∂t ∂z ∂t
j i
(cid:18) (cid:19) (cid:18) (cid:19)
All the above discussion can be expressed via the following property.
Proposition 2.1. (Self−Adjointness of Birkhoffian System). Necessary a-
nd sufficient condition for a general nonautonomous first-order system given
as above to be self-adjoint in ℜ∗ of points of R×T∗R2n is that it is of the
Birkhoffian type, i.e.,
e
dz ∂F ∂F dz ∂F(z,t)
i j i i
K (z,t) +D (z,t) = ( − ) −(▽B(z,t)+ ). (16)
ij i
dt ∂z ∂z dt ∂t
i j
The functions F and B can be calculated according to the rules[5]
i
1
1
F = z ·K (λz,t)dλ, (17)
i j ji
2
Z
0
1
∂F
i
B = z ·(D + )(λz,t)dλ. (18)
i i
∂t
Z
0
As well as Hamiltonian system, Birkhoffian system provides a symbio-
sis among variational principle, Lie’s algebra and symplectic geometry.[13,
1, 2, 7] Obviously both standard Hamilton’s equations and general Hamil-
ton’s equations on Poisson manifold are recovered from Birkhoff’s equations
as in the particular cases of autonomous and semi-autonomous Birkhoffian
representations.
Due to the self-adjointness of Birkhoff’s equations, the phaseflow of the
system (16) conserves the symplecticity, then we get
d d
Ω = (K dz ∧dz )= 0. (19)
ij i j
dt dt
It means that if we denote the phase flow of the equations (16) with (z,t),
then
b b
K (z,t)dz ∧dz = K (z,t)dz ∧dz , (20)
ij i j ij i j
b b b b 5
or
∂zT ∂z
K(z,t) = K(z,t). (21)
∂z ∂z
Inthenextsectionswewillcbonstructthbealgorithmpreservingthisgeometric
b b
property of the phase flow in discrete space.
3 Generating Functions for K(z,t)-Symplectic Ma
ppings
In this section we consider general k(z,t)-symplectic mappings and their
relationship with the gradient mappings and their generating functions.
Definition 3.1. Denote
0 I 0 I
J = n , J = 2n ,
2n −I 0 4n −I 0
n 2n
(cid:18) (cid:19) (cid:18) (cid:19) (22)
J 0 K(z,t) 0
J = 2n , K(z,z,t,t ) = .
4n 0 −J 0 0 −K(z,t )
2n 0
(cid:18) (cid:19) (cid:18) (cid:19)
b
eA 2n-dimensional submanifeoldbL ⊂ R4n
z
L = ∈ R4n|z = z(x,t ),z = z(x,t),x ∈ U ⊂ R2n,open set
z 0
(cid:26)(cid:18) (cid:19) (cid:27)
b (23)
b b
is a J - or J - or K(z,z,t,t )-Lagrangian submanifold if
4n 4n 0
T
e e b (TxL) J4n(TxL) = 0 (24)
or
T
(T L) J (T L) = 0 (25)
x 4n x
or
e
T
(T L) K(z,z,t,t )(T L)= 0, (26)
x 0 x
where T L is the tangent space to L at x.
x e b
A mapping with parameters t and t is z −→ z = g(z,t,t ) : R2n −→
0 0
R2n which iscalled acanonical map ora gradient map or aK(z,t)-symplectic
b
map if its graph
z
Γ = ∈ R4n|z = g(z,t,t ),z = z ∈ R2n (27)
g z 0
(cid:26)(cid:18) (cid:19) (cid:27)
b
is a J - or J - or K(z,z,t,t )-Labgrangian submanifold.
4n 4n 0
e e b 6
Definition 3.2. A differentiable mapping g :M → M is K(z,t)-symplectic,
if
T
∂g ∂g
K(g(z,t,t ),t) = K(z,t ). (28)
0 0
∂z ∂z
A difference scheme approximating the Birkhoff’s system (16)
zk+1 = gk(zk,t +τ,t ), k > 0, (29)
k k
either explicit scheme or worked out from a inexplicit scheme, is called a
K-symplectic scheme, when gk is K-symplectic for every k > 0, i.e.
T
∂gk ∂gk
K(zk+1,tk+1) = K(zk,tk). (30)
∂zk ∂zk
The graph of the phase flow of the Birkhoffian system (1) is gt(z,t ) =
0
g(z,t,t ) which is a K(z,z,t,t )-Lagrangian submanifold for
0 0
gzt(z,et0b)TK(gt(z,t0),t)gzt(z,t0) = K(z,t0). (31)
Similarly the graph of the phase flow of standard Hamiltonian system is a
J -Lagrangian submanifold.
4n
Define nonlinear transformation with two parameters t and t from R4n
0
e
to itself,
z w α (z,z,t,t )
α(t,t ): −→ = 1 0 ,
0 z w α (z,z,t,t )
2 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
b b b
α−1(t,t ): w −→ z = α1(wb,w,t,t0) . (32)
0 w z α2(w,w,t,t )
0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Denote b b b
A B b
α∗(z,z,t,t0) = Cα Dα ,
α α
(cid:18) (cid:19)
b Aα Bα
α−1(w,w,t,t ) = . (33)
∗ 0 Cα Dα
(cid:18) (cid:19)
α∗ is the Jacobian of α. Lbet α be a diffeomorphism from R4n to itself,
then it follows that α carries every K-Lagrangian submanifold into a J -
4n
Lagrangian submanifold, iff.
e
αT∗J4nα∗ = K, (34)
e
7
i.e.,
T
A B J 0 A B K(z,t) 0
α α 2n α α = . (35)
C D 0 −J C D 0 −K(z,t )
α α 2n α α 0
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
b
Converselyα−1 carrieseveryJ -LagrangiansubmanifoldintoaK-Lagrang-
4n
ian submanifold.
e
Proposition 3.1. Let M ∈ R2n×2n, α be defined as above. Define a frac-
tional transformation
σ :M −→ M
α
(36)
M −→ N = σ (M) = (A M +B )(C M +D )−1
α α α α α
under the transversality condition
|C M +D |=6 0. (37)
α α
Then the following four conditions are equivalent mutually:
|C M +D |=6 0,
α α
|MCα−Aα|=6 0,
(38)
|CαN +Dα| =6 0,
|NC −A | =6 0.
α α
The proof is direct and simple, so it is omitted here.
Theorem 3.1. Let α be defined as above. Let z −→ z = g(z,t,t ) be a
0
K(z,t)-symplectic mapping in some neighborhood R of R2n with Jacobian
b
g (z,t,t )= M(z,t,t ). If M satisfies the transversality condition in R
z 0 0
e
|C (g(z,t,t ),z,t,t )M(z,t,t )+D (g(z,t,t ),z,t,t )|=6 0, e(39)
α 0 0 0 α 0 0
then there exists uniquely in R a gradient mapping w −→ w = f(w,t,t )
0
with Jacobian f (w,t,t ) = N(w,t,t ) and a scalar function-generating
w 0 0
e b
function-φ(w,t,t ) such that
0
f(w,t,t ) = φ (w,t,t ), (40)
0 w 0
α (g(z,t,t ),z,t,t ) = f(α (g(z,t,t ),z,t,t ),t,t )
1 0 0 2 0 0 0
(41)
= φ (α (g(z,t,t ),z,t,t ),t,t ),
w 2 0 0 0
8
identically in z and t,
N = (A M +B )(C M +D )−1, (42)
α α α α
M = (AαN +Bα)(CαN +Dα)−1. (43)
Proof. Under the transformation α, the image of the graph Γ is
g
w
α(Γ ) = ∈ R4n|w = α (g(z,t,t ),z,t,t ),
g w 1 0 0 (44)
(cid:26)(cid:18) (cid:19)
b w = α (g(z,t,t ),z,t,t )}.
b 2 0 0
By the inequality (39),
∂w ∂α ∂z ∂α
2 2
= · + = C M +D 6= 0, (45)
α α
∂z ∂z ∂z ∂z
b
so w = α (g(z,t,t ),z,t,t ) is invertible, the inverse function is denoted by
2 0 0
b
z = z(w,t,t ). Set
0
w = f(w,t,t ) = α (g(z,t,t ),z,t,t )| , (46)
0 1 0 0 z=z(w,t,t0)
then b
∂f ∂α ∂g ∂α ∂z
N = = ( 1 + 1)( )= (A M +B )(C M +D )−1. (47)
α α α α
∂w ∂z ∂z ∂z ∂w
Notice that the tangent space to α(Γ ) at z is
g
b
∂w A M +B
T (α(Γ )) = ∂z = α α . (48)
z g ∂w C M +D
(cid:18) ∂zb (cid:19) (cid:18) α α (cid:19)
we have a conclusion that α(Γ ) is a J -Lagrangian submanifold for
g 4n
T (α(Γ ))TJ T (α(Γ ))
z g 4n z g
A M +B
= (A M +B )T,(C M +D )T J α α
α α α α 4n C M +D
α α
(cid:18) (cid:19)
(cid:0) M (cid:1) (49)
= MT,I αT∗J4nα∗ I
(cid:18) (cid:19)
(cid:0) (cid:1)
M
= MT,I K = 0.
I
(cid:18) (cid:19)
(cid:0) (cid:1)
So e
(A M +B )T(C M +D )−(C M +D )T(A M +B ) = 0, (50)
α α α α α α α α
9
i.e., N = (A M + B )(C M + D )−1 is symmetric. It implies that w =
α α α α
f(w,t,t ) is a gradient mapping. By the Poincare´ lemma, there is a scalar
0
b
function φ(w,t,t ) such that
0
f(w,t,t ) = φ (w,t,t ). (51)
0 w 0
The equation (41) follows from the construction of f(w,t,t ) and z(w,t,t ).
0 0
Sincez(w,t,t )◦α (g(z,t,t ),z,t,t ) ≡ z, sosubstitutingw = α (g(z,t,t ),
0 2 0 0 2 0
z,t,t ) in the equations (46) and (51), we get the equation (41).
0
Proposition 3.2. f(w,t,t ) obtained in Theorem 3.1 is also the solution
0
of the following implicit equation
α1(f(w,t,t ),w,t,t ) = g(α2(f(w,t,t ),w,t,t ),t,t ). (52)
0 0 0 0 0
The proof is similar to the above proof.
Theorem 3.2. Let α be defined as in Theorem 3.1. Let w −→ w =
f(w,t,t ) be a gradient mapping in some neighborhood R of R2n with Jaco-
0
b
bian f (w,t,t )= N(w,t,t ). If in R, N satisfies the condition
w 0 0
e
|Cα(f(w,t,t0),w,t,t0)N(w,t,t0)e+Dα(f(w,t,t0),w,t,t0)| =6 0, (53)
then there exits uniquely in R, a K(z,t)-symplectic mapping z −→ z =
g(z,t,t ) with Jacobian g(z,t,t ) = M(z,t,t ) such that
0 0 0
e b
α1(f(w,t,t ),w,t,t ) = g(α2(f(w,t,t ),w,t,t ),t,t ),
0 0 0 0 0
M = (AαN +Bα)(CαN +Dα)−1, (54)
N = (A M +B )(C M +D )−1.
α α α α
Similarly to Proposition 3.2, g(z,t,t ) is the solution of the implicit equation
0
α (g(z,t,t ),z,t,t ) =f(α (g(z,t,t ),z,t,t ),z,t,t ). (55)
1 0 0 2 0 0 0
The proof is similar to that of Theorem 3.1 and is omitted here.
4 Symplectic Difference Schemes for Birkhoff’s Eq
uations
In Section 2 it is indicated that for a general Birkhoff’s system, there is
a common property that its phase flow is symplectic. Through the result
10