NON-COMMUTATIVE INTEGRABILITY xv
Bl. Unperturbed motions in Example B (which has two degrees of freedom) are
periodic, i.e., have only one associated frequency.
B2. The conserved quantity J
a n g
has three components, i.e., one more component
than the system has degrees of freedom.
B3. The underlying rotational symmetry is non-Abelian, since the net effect of
applying successive rotations in three-space depends in general on the order
in which these rotations are applied.
B4. The Poisson brackets of components Ji , J2, J2 of Jang satisfy the cyclic con-
ditions
{Ji,J*} = Js , {J2, J3} = Ji , {J3, J i } = J2
What is remarkable is that (local) action-angle coordinates can be constructed
in Example B although, as one might imagine in view of the incongruence of the
properties listed above, the construction is substantially more complicated than was
the case in Example A. We do not attempt to describe this construction here. At
any rate, the unperturbed Hamiltonian H0(q,p) h(p) in action-angle coordinates
fails to be convex, so that Nekhoroshev's theorem above fails to apply.
Integrable systems whose conserved quantities satisfy nontrivial Poisson bracket
relations, as in B4, are known as non-commutatively integrable systems. In work
preceeding the research on exponential estimates, Nekhoroshev (1972) determined
that a large class of non-commutatively integrable systems admit a generalization
of action-angle coordinates known as partial action-angle coordinates, which are
in some sense more natural. Partial action-angle coordinates consist of two sets
of variables: The first set consists of k angles # 1 , . . . ,qk and k conjugate actions
P i , . . . ,pfc. The second set consists of (n k) variables x i , . . . , xn-k and (n k)
conjugate variables 2/1,... , 2/n-fc5 which are all ordinary (i.e., non-angular) coor-
dinate functions. Here n denotes the total number of degrees of freedom. These
coordinates, like conventional action-angle coordinates, are canonical in the sense
that they put the equations of motion into Hamiltonian form:
. _dH . _ dH . _ dE_ . _ dH
q3~dij' P3~~dg~; Xj~dy~j' Vj~~~dx~'
In partial action-angle coordinates, a non-commutatively integrable Hamiltonian
depends only on the pj variables.
In Nekhoroshev's work on exponential estimates, he realized that most of his
arguments carry over to the non-commutatively integrable case if one uses partial
action-angle coordinates, and assumes convexity of the unperturbed Hamiltonian
with respect to the pj variables only. Even in Example B, however (the simplest ex-
ample of a non-commutatively integrable system imaginable!), constructing partial
action-angle coordinates is not trivial. Furthermore, one cannot construct partial
action-angle coordinates globally in Example B without coordinate singularities.
This is true even if we remove points in phase space corresponding to trivial mo-
tions (the mass m at rest), which constitute a 'natural' singularity of the problem.
Another problem with Nekhoroshev's generalized setting, as Fasso (1995) has
pointed out, is that 'fast' (i.e., order e) motions in the Xj, yj variables take solutions
to the perturbed problem in a non-commutatively integrable system out of locally
defined partial action-angle coordinate charts, before the exponential estimates on
the pj variables can be rigorously established. Fasso overcomes this problem by
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