Symmetry, Reduction, and Phases in Mechanics 9
Now co and cb are assumed small with respect to the particle's velocity, so by the averaging
theorem (see, e.g. Hale [1969]), the s-dependent quantities in (4) can be replaced by their
averages around the hoop:
T L L
s(T) - s0 + s0T + f (T -1')
jcD(t')2
£ J q q' ds - (b(t') ^ J q(s) sin a dsj dt'. (5)
Aside The essence of the averaging can be seen as follows. Suppose g(t) is a rapidly varying
function and f(t) is slowly varying on an interval [a,b]. Over one period of g , say [a, p] , we
have
fp fP
f(t)g(t)dt = f(t)gdt (6)
i
fp
where g = ^~— I g(t)dt is the average of g . The error in (6) is
P
a
J
a
fP
f(t)(g(t)-g)dt
•'a
which is less than (p - a) x (variation of f) x constant constant |f |(p - a) 2 . If this is added
up over [a, b] one still gets something small as the period of g 0 .
The first integral in (5) over s vanishes and the second is 2A where A is the area
enclosed by the hoop. Now integrate by parts:
T T
f (T-t')G(t')dt' = -Tco(0) + f co(t')dt' = -TCO(0) + 2TC,
JO JO
assuming the hoop makes one complete revolution in time T. Substituting (7) in (5) gives
(7)
s(T) = s0+%T + ^ c o
0
T - ^ . (8)
The initial velocity of the ball relative to the hoop is SQ , while that relative to the inertial frame is
(see(l)),
v0 = q'(0) [q'(0) So + co0 x q(0)] = ^ + co0 q(s0) sin a(s0). (9)
Now average (8) and (9) over the initial conditions to get
s(T)-s0-v0T = - ^ (10)
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