12 2. ACTION-GROUP COORDINATES

where

2.8 6f(s,p) = -^(g,p) + \P(T± — (g,p)

and

2.9 rH(g,p) = -a — (g,p) .

Here cr : g — • t denotes the projection along

t"1,

and cr^ : g — •

t-1

that along t.

These formulas are the complexified version of formulas on G x to = G x tg we will

derive as Proposition 9.13, Part 2. The vector valued functions ^p : G

c

x IQ — » g

c

and ^ : G c x t^ — t c appearing in 2.8 and 2.9 are defined implicitly by

~^(9,P) ' £o - (dH, (&,0)pp - ^fr(^exp(^

0

),p)|

t = 0

(Co G 0C)

and

-g^{g,p)-To = (dH,(0,TQ)g,p) = —H(g,p + tT0)\t=0 (r0 G

tc)

.

Here a dot denotes the non-degenerate C-bilinear form on gc.

Equations of motion in action-group coordinates

Recalling that we are identifying G with a linear group, we may think of ele-

ments of G as matrices and use formulas 2.7-2.9 to write the 'equations of motion'

corresponding to a Hamiltonian H as

dH

x

±dH

2-10 % " ( M . ^ x t S ) .

Although one can (abstractly) make sense of these equations without the realization

of G as a linear group, our estimates later on will depend on the vector space

structure of C

n c X n G

in which we have assumed G

c

to be embedded.

2.11 Remark. It is worth comparing the above equations to those corresponding

to Hamiltonian vector fields on T*G = = G x g* (left trivialization, say). These may

be derived from the formula for the symplectic structure in Abraham and Marsden

(1978, Proposition 4.4.1). In the present notation, and identifying g* with g as

above, these equations take the form

dH

OH

» = —=- +

dg

OH

( M e G x j ) .

A Hamiltonian invariant with respect to the cotangent lifted left action of G (see

p. 53) is represented by a function Ho(g, ji) = /z(/i). However, as we will explain in

Remark 4.6, the 'coordinates' Gxg are poorly suited for an analysis of perturbations

to general Hamiltonians of this form (unless G is Abelian).