XVI
OVERTURE
showing how to make intrinsic sense of 'normal forms' for the perturbed Hamil-
tonian, although to express and compute Nekhoroshev type estimates still requires
one to fix an atlas of partial action-angle coordinate charts. The extent to which
these estimates depend on the choice of atlas is an open problem (Fassb, 1995,
Appendix C).
T h e s y m m e t r y point of vie w
The reason for the difficulties presented by action-angle or partial action-angle
coordinates in non-commutatively integrable systems is that the canonical nature
of such coordinates is at odds with the intrinsic non-Abelian symmetry underlying
many such systems. We shall take the point of view that the geometry of an un-
derlying symmetry, as well as its corresponding conservation law, should be built
into whatever geometric framework is employed to carry out an analysis of pertur-
bations. This viewpoint is to take precedence over previous requirements that one
work exclusively with canonical coordinate systems. Rather, one should endeavor
to understand how non-canonical contributions enter the equations of motion, when
these are viewed in a coordinate system intrinsic to the non-Abelian symmetry.
With this kind of understanding in hand, we seek to geometrize and generalize
the commutative version of Nekhoroshev's theorem given above, in such a way
that its application to systems with a perturbed non-Abelian symmetry becomes
transparent. In particular, this generalization should yield estimates that apply
immediately to the conservation law intrinsically associated with such a symmetry.
Action-grou p coordinates
The simplest coordinate system of the kind we are advocating is one that
we shall refer to as action-group coordinates. In Example A, and other systems
whose integrability arises from the existence of an appropriate Abelian symmetry,
action-group coordinates correspond to conventional action-angle coordinates. A
discussion of action-group coordinates in general is postponed to Chap. 2. We pre-
view these coordinates here in the special case of Example B, which we shall revisit
throughout our exposition of the general theory. A more sophisticated demonstra-
tion of this theory (Chap. 11) will be an application to the Euler-Poinsot rigid
body.
Nondimensionalize all lengths in Example B by the radius of the sphere, and
all masses by that of m. The state of the system is determined by the position q
and instantaneous linear momentum vector v of m. We view q and v and vectors
in three-space, subject to the conditions \\q\\ = 1 and v q = 0 consistent with the
physical constraints.
Assume the unit sphere is centered on the origin of an inertial orthonormal
frame with basis ei, e^, e%. It is convenient to express states (q, v) of m with respect
to a reference state (qo,vo), which we choose arbitrarily to be (ei,e2). For an
arbitrary state (q,v) there exists a 3 x 3 orientation preserving rotation matrix g
and a number p ^ 0 such that
q = gqo , v = pgv0 .
If we ignore trivial states (v = 0), then g and p are determined uniquely, and p is
strictly positive. Whence g,p constitute 'coordinates,' in the sense that there is a
one-to-one correspondence between states (q,v) and values for the pair (g,p).
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