GENERALIZED NEKHOROSHEV ESTIMATES
^
J(*t)
IJ(^)-OI r
J(x-o)
F I G U R E 1. Schematic representation of a solution curve t »- xt of
the perturbed Hamiltonian H1 projected onto 'momentum space'
0* using the momentum map J : P —* g*. The co-adjoint orbit
through the initial point J(xo) (denoted 0 ) is depicted as a sphere.
we deduce a Nekhoroshev-type estimate on the evolution of momentum maps, in
a class of integrable Hamiltonian systems with nearly G-invariant Hamiltonians
(Corollary 7.1, p. 43). This result may be informally described as follows.
Suppose that a system with Hamiltonian Ho possesses a symmetry group G.
Then, under appropriate hypotheses, the system will possess a corresponding con-
servation law. This law is embodied in the existence of a vector-valued function
J : P g* (g denoting the Lie algebra of G), known as a momentum map, that is
constant on solution curves t 1— xt of XH0 '
| j ( x
t
) = 0 .
Next, consider a perturbation to HQ of the form
H = H0 + eF ,
where F is arbitrary. Furthermore, assume that the symmetry is sufficiently large
so as to enforce integrability in the unperturbed system 9 . Let t » xt be a solution
of the perturbed Hamiltonian H, and let O C 0* denote the co-adjoint orbit through
the initial point J(#o) £ 0*- Then we show that there exist positive constants a, 6,
c, to and 7*0, such that for all sufficiently small e ^ 0, one has (see Fig. 1)
\t\ t0 exp(ce- a ) = I J ( z
t
) - 0 | r
0
e 6 .
Here |J(x
t
) G\ denotes the distance of J(xt) from the orbit (9, measured using
some Ad*-invariant inner product on 9*. The estimate holds provided the Hamil-
tonian is real-analytic and satisfies an appropriate 'convexity' condition. One must
also assume that action-group coordinates can be constructed in an appropriate
neighborhood of xo (see below). The constants appearing in the estimate depend
on the unperturbed Hamiltonian # 0 , on the magnitude and analyticity proper-
ties of F , and on characteristics of G. In directions tangential to the orbits, 'fast'
motions, corresponding to those observed by Fasso, are possible.
9
To be precise, we require that the Marsden-Weinstein reduced spaces be zero-dimensional;
see Chap. 3. In some cases the two-dimensional case can be treated also; see Remark 3.9.
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