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)
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