Th e complexit y of perturbe d motion s
In the absence of perturbations, Examples A and B are both integrable sys-
tems, i.e., systems whose solutions can be written down in an explicit or 'closed'
form. However, integrability is the exception rather than the rule, and generic
perturbations to integrable systems create extraordinarily complicated behavior.
See, e.g., Lichtenberg and Lieberman (1992). In particular, Poincare's nonex-
istence theorem (see, e.g., Benettin, Ferrari, Galgani and Giorgilli (1982)) tells
us that conserved quantities in an integrable system are irrevocably destroyed
by generic perturbations. Despite this, however, there are two important re-
sults establishing, under appropriate hypotheses, some kind of 'stability' for the
conserved quantities. These are the Kolmogorov-Arnold-Moser (or KAM) theo-
rem (Kolmogorov, 1954; Arnold, 1963; Moser, 1962) and Nekhoroshev's theorem
(Nekhoroshev, 1977). For generic perturbations, the KAM theorem does not di-
rectly apply to Example B (or to other examples like it), and we discuss it no
further1. Nekhoroshev's theorem can be applied to both Examples A and B, but
not without significant differences that we now elucidate.
Nekhoroshev' s theore m
Nekhoroshev's theorem applies to systems admitting action-angle coordinates.
Nekhoroshev also stated his result in terms of a generalization called partial action-
angle coordinates, but his argument in the more general setting was incomplete (see
below). These coordinates are relevant in Example B, but not in Example A, which
we describe first.
Action-angle coordinates in Example A are easily constructed. Suppose as be-
fore, that attention is restricted to spatially periodic perturbations, and let L be
a 'lattice of periodicity.' For simplicity, assume that L has an orthogonal basis,
and that the corresponding periods of L are the same in each direction. Non-
dimensionalize all lengths by this period, and all masses by the mass of m. Modulo
the lattice L, the position of m is determined by three numbers qi,q2,Qs with
0 ^ Qj 1; these numbers constitute the 'angles.'2 The associated 'actions' are
the components pi,P2,P3 of the linear momentum Jn
. What makes these coor-
dinates action-angle coordinates is that: (i) they put the equations of motion into
Hamiltonian form,
_0H _ dH
where the appropriate Hamiltonian H is in this case the total energy of the system,
and (ii) the unperturbed value Ho of the Hamiltonian (i.e., the kinetic energy)
depends only on the action variables:
H0(q,p) = h(p) = -p\ + -p\ + -p\ .
In particular, the unperturbed motions of a system in action-angle coordinates are
Pj(t)=Pj(0) , qj(t)=qJ(0) + mj ,
1 The interested reader is referred to, e.g., Arnold, Kozlov and Neishtadt (1988).
The familiar scaling 0 ^ q3 2n produces distracting factors of 2TT later on.
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