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THEORETICAL PH MECHANICS Problem Assignment # 1
Winter 2009
01/07/09 due 01/14/09
1. Pendulum (Part 1) note: This problem will be continued in later assignments. Consider a mathematical pendulum (mass m, length ) with Lagrangian L= m 2
2
2 + mg cos
a) Show that the system can display bounded or unbounded motion, depending on the value of the energy. Show that the motion is always bounded if E < U0 . Determine U0 and the turning points 0 of the oscillation. b) For the case of bounded motion, nd the oscillation frequency as a function of E. Discuss your result as a function of = E/2U0 + 1/2, especially the limiting cases 0 and 1. hint: The result can be expressed in terms of the complete elliptic integral of the rst kind, K(k), dened as /2 1 K(k) = dx 0 1 k 2 sin2 x c) Show that (t) can be expressed in closed form in terms of the Jacobi elliptic function cn: (t) = 20 sin(0 /2) cn(0 t, k) , where 0 = U0 /m 2 .
x
hint: The elliptic integral of the rst kind, F (x, k), is dened as F (x, k) =
0
dz
1 1 k 2 sin2 z
(k 1) .
Its inverse is known as the amplitude, x F = (y, k) and cn is dened as cn(x, k) = cos am(x, k) . Also useful are sn(x, k) = sin am(x, k) , and dn(x, k) = d am(x, k) = dx 1 k 2 sin2 y . (15 points) .../over y = am(x, k) ,
2. Geodesics on Manifolds, Rotational Invariance, and Angular Momentum Conservation Let x( ) = (x1 ( ), x2 ( ), x3 ( )) be a path in I 3 parameterized by some parameter . Then the length R of the path between the points x = x( ) and x+ = x(+ ) is given by the functional
+
S=
d L(x, x, )
with L(x, x, ) = L(x) = x2 + x2 + x2 1 2 3
a) Show that the geodesics in I 3 , i.e. the paths whose lenghts are stationary, are given by straight R lines. Now consider the geod...