Symmetry, Reduction, and Phases in Mechanics
11
To illustrate the ideas, we look at the special case m{ = r{ =1 so that
L = |(92 + 92)-V(9
1
-9
2
)
and
ds2
=
d92
+ d9| = \ d(9x -
92)2
+ \ d(9t +
92)2
.
The group S1 acts on configuration space T2 = {(9p 92)} by 9 (91? 92) = (6•+ 9
p
9 + 92)
so L is invariant under this action. Letting cp = (9X + 92)/V2 and \\f = (9j - 92)/V2 , we see that
9 (p, \}/) = (9 + 9, \|/) and hence that the induced momentum map for the lifted action J : T*T2
-»R is given by J(cp, \j/, p , p ) = p . Therefore the reduced space
J'^M-VS1
is diffeomorphic
to T S1 = {(y, p )} with the canonical symplectic structure. The Hamiltonian on T T2 is
H(q.V.PrPy) = \3l + Pp + V(V2V)
and the reduced Hamiltonian is
H^(¥.PV) = | p
v
2 + V ( V 2
V
)
The equations of motion for H are
9 = P9 P
¥
= ° (!)
V = PV = -V2V'(V2
¥
). (2)
Equations (2) are Hamilton's equations for H on the reduced space.
Assume that we have solved (2) with initial conditions (\|/0, p
v
) and are given the initial
conditions (cp0, \j/0, p
9
= |i, p
y
) of a solution for the system (1), (2). To find the solution for
(q(t), \|/(t), p9(t), py(t» of (1), (2) we proceed in two steps:
Step 1 Consider the curve d(t) = ((p0, \|/(t), p., Pv(t)).
Step 2 Solve the equation 9'(t) = \L, 9(0) = 0 yielding 9(t) = fit.
77ien the solution to (1), (2) is given by c(t) = 6-d(t) = (cp0 + |it, \|/(t), |i, pv(t)).
This method is quite general and applies to all Hamiltonian systems. We will discuss it in
§2 and §3. To get a feeling of what is happening, we make some remarks. The principal
S^bundle
J-1(M)
= {(9, V, M- , Py)}
T*S!
, (9, y, |i, p ) !- (\|/, p ) has a connection whose
horizontal space at any point is generated by the vector fields J— , I . Then the curve d(t) in
Step 1 is simply the horizontal lift of the integral curve of the reduced system (\|/(t), p^(t))
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