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)