Phys601/F08/Problem Set 2 Due 09/19/06
2.1H
A particle of mass m moves in the gravitational field of a fixed massive point object of
mass M. Let k=GMm and p = mv. Let L = r x p be the angular momentum with r(t)
being the position vector from the center of the mass M.
(a) Show by direct differentiation that the vector A = p x L – kmr/r is a constant of
the motion. The relation v = (dr/dt)r^ + r(dφ/dt)φ^, where r and φ are polar
coordinates in the plane of the motion and r^ and φ^ are unit vectors, may be
useful.
(b) Now derive the constancy of A as follows. Starting from the equation of motion,
cross both sides by L and then manipulate the resulting equation, using the
equations of motion, to derive the equation dA/dt = 0.
2.2H
Starting from mdv/dt = q(E + v x B), prove that U = (½)mv2 + qϕ = constant under
certain conditions on and ϕ and A. State precisely under what conditions on the
spacetime behavior of ϕ (r, t) and A(r, t) is U a constant. Here ϕ and A are the
electromagnetic potentials. (The proof is short.)
