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 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 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, ^ 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
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