ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 12, с. 1808-1812
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
THE AVERAGING IN HAMILTONIAN SYSTEMS ON SLOW-FAST PHASE
SPACES WITH S1 SYMMETRY
©2011 Yu. M. Vorobiev*
Department of Mathematics, University of Sonora, Hermosillo, Mexico
Received July 19,2010
Using the averaging procedure for Poisson brackets, we study the nonadiabatic dynamics of perturbed Hamiltonian systems on slow—fast S1-spaces.
1. INTRODUCTION
In this paper, we study a perturbed Hamiltonian dynamics on a phase space M which separates into "slow" and "fast" Poisson manifolds S and P, respectively. This splitting corresponds to the including of the perturbation parameter e into the product Poisson structure on M = S x P by rescaling the Poisson bracket on P by the factor 1/e. Phase spaces of such a kind appear in the theory of adiabatic approximation as well as in various physical applications (see, for example, [1—3]). Here, we are interested in a nonadiabatic perturbed Hamiltonian dynamics on M, considering the case when the leading term H0 in the perturbed Hamiltonian H = H0 + eH1 is independent of the fast variables. This hypothesis, together with the singular dependence of the Poisson bracket on e, leads to the following effect [4, 5]: the limiting (unperturbed) dynamics at e = 0 is no longer Hamiltonian. Viewing M as the total space of a fiber bundle over S with fiber P, we assume that the unperturbed dynamical system admits an integral of motion which represents a parametrized momentum map of a family of Hamiltonian S1 actions on M in the sense of [6, 7]. This circle action only affects the fast variables and does not respect the Poisson bracket on M. We admit that the S1 action is not necessarily free. By using the action—angle variables, the nonadiabatic dynamics of Hamiltonian systems with two degrees of freedom on slow—fast phase spaces was studied in [8].
We show that for small enough e, there exists a near identity mapping T£ on M such that the original Poisson bracket on M is transformed by T£ to a S1-invariant Poisson bracket and the deformed Hamiltonian H o T£ is e2-close to the S1 average of H. Moreover, under an appropriate condition, called the adiabatic hypothesis [6, 7], the S1 action becomes Hamiltonian relative to the new Poisson bracket. Therefore, we obtain that the original
perturbed Hamiltonian system is approximated by the S1-symmetric Hamiltonian model which involves the averaged Poisson bracket [9]. The normalization transformation T£ is constructed by means of the homotopy method for weak coupling Poisson structures [4, 5] associated with Hannay—Berry connections [6, 7, 10]. As an example, we consider the Hamiltonian system describing the motion of a nonrelativistic particle with spin in a slow varying magnetic field.
2. MAIN RESULTS
Let S and P be symplectic and Poisson manifolds, respectively. Consider the product manifold M = S x x P and denote by {, }s and {, }p the canonical lifts of the Poisson brackets on S and P to M. On the phase space M = S x P, we introduce the "slow—fast" product Poisson bracket {, }m given by
{/l,/2}M = {/l,/2}s + ^{/l,/2}p, (1)
where e > 0 is a small parameter. Consider the Hamiltonian system on M associated with Hamilto-nian
H (£,x,e)= Hq(£ )+eHi(£,x) (2)
for some Hq e C™(S) and Hi e C™(M). Here, £ = = (£i) e S and x = (xa) e P. Denote by Xh the Hamiltonian vector field of H relative to the bracket {, }m. Then dynamical system corresponding to XH is written as
de dt
= {Ho,è}s + e{H1,C}s,
dxa
Ht
= {H1,xa}p.
(3)
(4)
E-mail: yurimv@guaymas.uson.mx
Because of singularity of Poisson bracket (1) at e = 0, the limiting vector field V = lime^Q XH is no longer
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THE AVERAGING IN HAMILTONIAN SYSTEMS
1809
Hamiltonian relative to {, }m, in general. Therefore for small e, we have XHs = V + O(e) and hence, the original Hamiltonian system is a perturbation of a non-Hamiltonian one. We assume that the unperturbed system V admits an integral of motion J : M — R,
{Ho, J}s + {Hi, j}p = 0. (5)
Denote by Vj the Hamiltonian vector field of J relative to {, }p, Lvj f = {J, f}p for f e C™(M). Let $T be the flow of Vj,
d_
dr
iL=o = J
We also assume that $T is complete and 2n-periodic. Therefore, the phase portrait of VJ consists of 2n-trajectories and equilibria. Consider M = S x x P as the total space of the trivial Poisson bundle n : M — S (the canonical projection on the first factor) with typical fiber P. Then, according to the terminology introduced in [6, 7], the fiber bundle n is endowed with a family of Hamiltonian actions $T : M — M of the circle S1 = R/2nZ with fiberwise momentum map J. This means that the S1 action leaves each fiber P^ = {£}x P (£ e S) invariant, n o $T = n, and the restriction $T |p is a canonical
S1 action with momentum map J|p.
Note that, in general, the Poisson structure {, }M on the total space is not S1 invariant. It is easy to see that the circle action is canonical on M only in the case when the variations of the infinitesimal generator along S are zero, Vqj/q^i = 0.
For any tensor field F on M its S1 average {F) is a tensor field of the same type on M defined by the usual formula
2n
F ) : =
1
2tt
J ($r )* Fdr.
It is clear that {F) is S1 invariant.
Theorem 1. Let N be a S1-invariant open domain in M with compact closure. Then, for small enough e, there exists a near identity dif-feomorphism T£ : N — M, T0 = id onto its image such that
(a) the original Poisson bracket {, }M is pulled back by T£ to a S1 -invariant Poisson bracket {, }™v on N,
{f1 o $T, f2 o $T}%v = {f1, f2}jnv o $T, (6)
which is a deformation of {, }M as e — 0, {/i, /2}%v = {/i, f2}s + ~£{fi, /2}p + 0(e)-, (7)
(b) the transformed Hamiltonian H o T£ is e2-close to the S1-average H := (H) = H0 + + s (Hi) of H,
H o T£ = H + O(e2). (8)
Moreover, if there exists a smooth function C : n(N) ^ R such that JC := J + C o n satisfies the condition
(ds JC ) = 0 on N (9)
(where dS denotes the exterior derivative on M along S), then the S1 action is Hamiltonian relative to {, }™v with momentum map eJC, d
inv
N ,
V f e C™. (10)
Remark 2. Hypothesis (9), called the adiabatic condition, was introduced in [6, 7] in the context of the theory of the Hannay—Berry phase for classical mechanical systems with symmetry. If the domain n(N) c S is simply connected, then (9) holds for a certain C. In the case when M is compact and S is simply connected, one can put N = M.
Therefore, Theorem 1 says that under hypothesis (9) the averaged Hamiltonian system (N, {, }Nv,H) is G symmetric,
{H, jc }Nv = 0, (ii)
and gives a first approximation to the original one. If we denote by X™ the Hamiltonian vector field of H of relative to {, }Nv, then
(T£y Xh£ = X™ + O(e)
(12)
as e — 0.
Let us consider the adiabatic case when H0 = 0. The original dynamical system takes the form
df dt
= e{Hi ,C]s ,
(13)
dxa ~dt
= {Hi,xa}p,
(14)
and can be viewed as a Hamiltonian system relative to the adiabatic type Poisson bracket e{, }M = = e{, }S + {, }P with Hamiltonian H1. The dynamics according to the corresponding limiting system of slow variables £ e S is frozen. Hypothesis (5) reads H1 is S1 invariant. By Theorem 1, after the transformation T£, we get an approximate Hamiltonian S1-symmetric system (N,e{, }Nv,H1). One can think of this approximation as a first step in the classical averaging method [1,2]. On the second step, to separate the coordinates in slow and fast
o
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VOROBIEV
varying variables, in a domain where the S1 action is free, we have to introduce (generalized) action—angle variables associated with the foliation by S1 orbits.
3. SMNVARIANT POISSON brackets
Starting with a phase space equipped with a S1 action and a noninvariant Poisson bracket, the point is to construct a S1-invariant Poisson bracket. If the original bracket is degenerate, then our goal cannot be simply reached by applying the S1 averaging to the Poisson structure (as a 2-tensor field) because of the nonlinearity of the Jacobi identity. Here, to construct the invariant Poisson bracket {, }Nv in Theorem 1, we follow an approach suggested in [9].
In coordinates £ = (£i) e S and x = (xa) e P, the original Poisson bracket (1) is written as follows
{?,? }m = -uij (£ ),
ie,xs }m = o,
{xa,xl3}M = -я>а13(х),
Q0 :=
11 y ко \0
Define also the following e-dependent 2-form F = \Fis(£, x)dé Л with coefficients
— e
Tij (j,x) := Uij (£) -dQj dQi
de dj
+ {Qi,Qj }p
{e,x° }r = rs(exwv (x)
dQs&x)
dxv '
= -4>al3(x) +
(23)
dQi{e x) ^ dQj{e x)
dxv
dxv
(15)
(16)
(17)
with dei(ulj) = 0. Using the integral of motion J, let us introduce the 1 -form Q = Qi(£, x)d£i on M whose components are given by
Qi = QQ -(QQ), (18)
with
(19)
(20)
Here, uis(£)usj = 5j. On can check that the forms Q and F are well defined on the whole M .At e = 0, the form F coincides with the pull-back by n of the symplectic form to = \iOij(£)de A d£j on S.
Pick an open S1-invariant domain N Q M with compact closure. Then, for small enough e, we have
det(Fij) = 0 on N. We introduce the following bracket relations on N:
}Nv = -Fij (£,x), (21)
(22)
where FiS FSj = 5j.
Proposition 3 [9]. For small enough e, formulas (21)—(23) define the Poisson bracket {, }™v on N which is invariant with respect to the S1 action.
Comparing (15)—(17) with (21)—(23), it is easy to see that {, }™v gives a deformation of the original Poisson bracket {, }M in the sense of (7).
Remark 4. The proof of Proposition 3 is based on the geometric method of constructing Poisson brackets via (nonlinear) connections [4, 9] combined with the theory of Hannay—Berry connections [6, 7]. The forms Q and F, involved in the definition of {, }™v, represent the Hamiltonian 1-form of the Hannay—Berry connection on the bundle n : M — S and the Hamiltonian 2-form of the curvature, respectively.
It follows from Proposition 3 that the S1 action is canonical with respect to {, }™v. The next natural question is to find a condition under which this
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