6

Marsden, Montgomery, and Ratiu

Put on Q the (possibly time dependent) metric induced by the mapping mr In other

words, we choose the metric on Q that makes rr^ into an isometry for each t. In many examples

of mechanical systems, such as the ball in the hoop given below, mt is already a restriction of an

isometry to a submanifold of 5, so the metric on Q in this case is in fact time independent. Now

we take the Legendre transform of (2), to get a Hamiltonian system on T*Q. Recall (see, for

example, Abraham and Marsden [1978] or Arnold [1978]), that the Legendre transformation is

3L

given by p = — . Taking the derivative of (2) with respect to v in the direction of w gives:

3v

p-w = (m{y + Zt(q(t))9mt'Vf)^t) = (mt-v +

Zt(#(t))T,

mt-w^t) (3a)

where p-w means the natural pairing between the covector p e TqTt\Q and the vector w e

Tq/txQ, (, )-/tx denotes the metric inner product on the space S at the point q(t) and

T

denotes

the tangential projection to the space mt(Q) at the point q(i). Recalling that the metric on Q,

denoted ( , ) ,t) is obtained by declaring mt to be an isometry, (3a) gives

p.

w

= (v+m-lZt(q(t))T,w)q(t) i.e., p = (v + m-1 Zt(q(t))T)b (3b)

where

b

denotes the index lowering operation at q(t) using the metric on Q. The (in general time

dependent) Hamiltonian is given by the prescription H = pv - L, which in this case becomes

Hmt(q, p) = \ ||p

II2

- ^Z

t

) - \

IIZJJ2

+ V(q) + U(tf(t))

= H

0

(q,p)- %Zt) - \ \\Z\f + U(#t)), (4)

where H0(q, p)= r||p||

2

+ V(q), the time dependent vector field Z

t

e X(Q) is defined by Zt(q)

= 7nt"1[2\(mt(q))]T, the momentum function 2\Y) is defined by (P(Y)(qy p) = p-Y(q) for Y e

X(Q), and where Z\ denotes the orthogonal projection of Zt to mt(Q). Even though the

Lagrangian and Hamiltonian are time dependent, we recall that the Euler-Lagrange equations for

Lm are equivalent to Hamilton's equations for Hm . These give the correct equations of motion

for this moving system. (An interesting example of this is fluid flow on the rotating earth, where it

is important to consider the fluid with the motion of the earth superposed, rather than the motion

relative to an observer.)

Let G be a Lie group that acts on Q. (For the ball in the hoop, this will be the dynamics

of H0 itself). We assume for the general theory that H0 is G-invariant. Assuming the

"averaging principle" (cf. Arnold [1978], for example) we replace Hm by its G-average,

Hmt(q,p) = \ ||p

||2

- 2XZt) _ 1 |j^J-|2 + V(q) + U(^(t)). (5)