TH E POISSON BRACKE T IN ACTION-GROUP COORDINATES 13

2.12 Example.

Take G = S1. Then G (resp. Gc) is realized as SO(2) (resp. SO(2, C)) and we have

*o

=

S

C

'

t_L =

0- Coordinates (g,p) £

Gc

x t£ are of the form

cos#

sin#

— suit

cos#

6eC

P

0 - J

J 0

leC

For an inner product on g, one chooses pi • P2 = hh- If one writes H'(6,I) =

H(g,p), then 2.10 takes the form

• dH' • dH'

which are the familiar form of Hamilton's equations.

One generalizes the above argument to G =

Tn

= SO(2) x • • • x SO(2) by

realizing G as the set of block diagonal 2n x 2n matrices whose 2 x 2 blocks are of

the form just described for n — 1.

The next example is relevant to the problem of geodesic motions on

S2

(see

3.4).

2.13 Example. Take G = SO(3). Then Q = so(3) can be identified with R3 via the

isomorphism £ ^^ £ : R3 — * so(3) defined by ^u = £ x u (u £ M3). This isomorphism

extends uniquely to a C-linear isomorphism C3 — so(3,C). Let {61,62,63} denote

the standard basis of R3. Choose T C SO (3) to be the rotations about the e%

axis, so that t = span{e3} = R. Then i1- = span{ei,e2} = M2. The adjoint

action is given simply by g • £ = g£ (g G SO(3),£ G R3), so that the regular

adjoint orbits are the spheres centered at the origin of positive radius. Whence

0reg = R3\{0}. The standard inner product a-b = aibi+a2b2 + asbs is Ad-invariant.

A connected component to of t H greg is given by to = {tes \ t 0} = (0, 00). So

Gx t0 = SO(3) x (0,oo).

£xr;. Fixing pe t

c

^ C,

(-pf2,Pfi).

(&/p,-£i/p)

One has g

c

=

C3

with ad^ : g

c

— • g

c

given by ad^ 7 7

one finds that adp : i±c - t ± c (i±cC ^ C2) is given by adp(fi,f2

^ 2

_L C c^JL

Therefore, t£ ^ C\{0} and Ap : t±c -

(p G C\{0}). The projections a : /

^1,6,6) = 6, ^(6,6,6) = (6,6).

As the reader will readily verify, the equations of motion 2.10 take the form of

( * ) on p. xvii.

t±L

is given by Ap(fi,f2)

— » tc and a1- : g c — • t ± c are given by

The Poisson bracket in action-group coordinates

Our convention for defining Poisson brackets is {u, v} =

XVJXUJUJ.

Ift\-+xt is

an integral curve of a Hamiltonian vector field XJJ then, according to this definition,

d

dt

u(xt) = {u,H}(xt)

for any function u.

2.14 Lemma. The Poisson bracket on

Gc

x tg is given by

{u,v}(g,p)

du dv dv du

-9j(9,P)-^(9,P)-ojb,P)--Qj(9,p)

du dv,

P'lXPa

-£-{9,P),\° ^~(9,P)]

dg

dg