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 P¥ = -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))