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).