XIV
OVERTURE
where flj = dh/dpj(p(0)). In our example, we have ttj = Pj(0), and the constancy
of the action variables corresponds to the conservation of momentum Jnn.
We are now ready to apply Nekhoroshev's result:
Nekhoroshev's Theorem. (Nekhoroshev
(1977)3)
Consider a Hamiltonian
system in action-angle coordinates #1,... ,qn, Pi,--- iPn o,nd consider perturbed
Hamiltonians of the form
H(q,p) = h(p)+eF(q,p) .
Restrict attention to values of the action vector p = (pi,... ,pn) lying in some finite
closed ball B, and initial conditions p(0) lying in the interior of B. Assume h and
F are real-analytic, and that the level sets in B of the unperturbed Hamiltonian
h are strictly convex. Then there exist positive constants a, b, c, to and ro such
that for all sufficiently small e ^ 0 all solutions of the perturbed system satisfy the
exponential estimate
\t\ ^ t0 exp(ce-°) = \p(t) - p(0)\
r0eb
.
In Example A we have p = Jnn, so that order e perturbations lead to order
eb
'drifts' in the momentum Jnn of the mass m after times on the order of
exp(ce~a),
at most. If e is small, such times can be very large indeed. For example, in an
application by Giorgilli and Skokos (1997) to Trojan asteroids, the Nekhoroshev-
type stability times derived are on the order of the current estimated age of the
universe!
Non-commutative integrability
Before turning to Example B, let us summarize some key observations about
Example A, which are typical of systems admitting action-angle coordinates:
Al. Unperturbed motions in Example A (which has three degrees of freedom)
are quasiperiodic with three independently
controllable4
frequencies.
A2. The conserved quantity Jnn is a vector with three components.
A3. The underlying translational symmetry is Abelian, meaning that the net
effect of two successive translations is independent of the order in which
they are applied.
We add one final observation which is less obvious but nevertheless important:
A4. The Poisson bracket (see below) {J^, Jj} of any pair of components Ji, J2, J3
of Jlin vanishes.
By definition, the Poisson bracket {/, h] of two functions / and h is computed by
differentiating / along solution curves of the system obtained by taking as Hamil-
tonian the function h.
If we are to apply Nekhoroshev's theorem as above to Example B, then we shall
first need to construct action-angle coordinates. But at odds with this requirement
is the disturbing fact that analogues of the observations A1-A4 follow an altogether
different pattern in Example B:
3For
this formulation see, e.g., Lochak (1992). In. Nekhoroshev's original statement, the
convexity condition is replaced by a 'steepness' condition. This is a weaker (indeed C°°-generic)
condition we will not attempt to consider here or elsewhere.
4
Controllable by varying the initial conditions Pj(0).
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