NEKHOROSHEV ESTIMATES IN THE NON-COMMUTATIVE CASE
Notice that the rotational symmetry of the unperturbed system is completely
transparent in the coordinates (g,p) as the unperturbed Hamiltonian Ho(g,p) (i.e.,
the kinetic energy determined by the standard metric on the sphere5) is independent
of the 'symmetry' coordinate g; it depends only on the 'action' coordinate p, in the
spirit of constructions of conventional action-angle coordinates:
Ho(g,p) = h{p) =
-p2
.
The equations of motion take the form
(*)
where
and
g = g
1 dH
U91
dH
dgs,
0 - 6 £2 '
6 o - 6
- 6 6 o
(g,p) = -H(g,p + t)\t=0
(1 j ^ 3)
dp w '- r / dt'
Roughly speaking, the non-canonical contributions to the equations are the terms
I OH IdH
pdg2 ' pdgi
In the unperturbed case, the equations reduce to
g = g(pe3r i P = °
which admit the general solution
P(t)=p(0) , ^(t)=^(0)exp(^e
3
) ,
where ft = p(0).
Nekhoroshev estimates in the non-commutative case
The angular momentum of m in Example B is given by
Jang = qxv= pg(q0 x v0) = pge3 .
If one keeps p 0 constant and runs through all possible values of the symmetry
variable g, then the corresponding locus of J
a n g
hi three-space is a sphere of radius
p. In fact these spheres are intrinsic geometric objects associated with the rota-
tional symmetry of the problem known as co-adjoint orbits. These orbits (which
are zero dimensional in the Abelian case) influence the perturbed motions, and in
particular the fate of the conservation law Jang = 0, in a fundamental way. As we
show, Nekhoroshev type estimates for a generic perturbation apply to 'drifts' in the
momentum J
a n g
in those directions transverse to the co-adjoint orbits (spheres).
That is, Nekhoroshev estimates, under appropriate hypotheses, apply to the action
5
We realize perturbations in the form of aspherical distortions by perturbing this metric, not
by perturbing the sphere with which configurations q are identified.
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