Marsden, Montgomery, and Ratiu

Rflq'(s)

Figure 1B-1

The configuration space is diffeomorphic to the circle Q =

S1

with length L the length of the

hoop. The Lagrangian L(s, s, t) is simply the kinetic energy of the particle; i.e., since

j

t

Re(t) q(s(t)) = RG(t)q'(s(t)) s(t) + R9(t)[co(t) x q(s(t))],

we set

L(s, s, t) = 2 m I I q'(s) s+ co x q(s)

||2.

(1)

The Euler-Lagrange equations

become

d_3L _ dL

d t

3s ~ 3s

^ m [ s + q ' ( c o x q)] = m[sq"- (co x q) + sq'(co x q') + (co x q) • (co x q')]

since ||q'||2= 1 . Therefore

s + q"(co x q)s+ q'- (ooxq) = s q " ( c o x q ) + (co x q) • (co x q')

s - (co x q) • (co x q') + q' • (6) x q) = 0.

i.e.,

(2)

The second and third terms in (2) are the centrifugal and Euler forces respectively. We

rewrite (2) as

s = co2 q • q' - co q sin a (3)

where a is as in Figure 1-1 and q = || q ||. From (3), Taylor's formula with remainder gives

s(t) = s0+s

JQ I

co(t')2

q • q'(s(0) - cXOq(s(0) sin a(s(t')) dt'

(4)