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).
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