Physics 31600 R. Wald Classical Mechanics Autumn, 2002

Problem Set VI

Due in class on Tuesday, November 26 (Note that this problem set is due one week after the midterm.)

1) Consider a space M with coordinates y^{μ} and define the Poisson-bracket-like map on functions on M by

W (f, g) = ∑_{μ,ν} W^{μν} \frac{∂f}{∂y^{μ}} \frac{∂g}{∂y^{ν}}

where W^{μν} is any antisymmetric matrix (i.e., W^{νμ} = −W^{μν}) whose components are constant (i.e., independent of y^{μ}). Show that W satisfies the Jacobi identity, i.e.,

W (W (f, g), h) + W (W (h, f), g) + W (W (g, h), f) = 0

2) Let f(q, p; t) and g(q, p; t) be (possibly time-dependent) observables on a 2ndimensional phase space with Hamiltonian H(q, p; t).

(a) Suppose that both f and g are constants of motion, i.e., df/dt = dg/dt = 0. Show that Ω(f, g) also is a constant of motion.

(b) Suppose that H itself is a constant of motion and f(q, p; t) is a constant of motion. Show that \frac{∂^{n}f}{∂t^{n}} also is a constant of motion.

3) Consider a particle of mass m in ordinary 3-dimensional space. Let L_{x}, L_{y}, L_{z} denote the usual Cartesian components of angular momentum of the particle, viewed as functions on its 6-dimensional phase space.

(a) Show that Ω(L_{x}, L_{y}) = L_{z}. (Thus, according to problem 2, if L_{x} and L_{y} are constants of motion, then so is L_{z}.)

(b) Suppose that an observable f(x⃗, p⃗) depends on x⃗ and p⃗ only in the scalar combinations x⃗ · x⃗, x⃗ · p⃗, and p⃗ · p⃗, i.e., suppose that f can be written in the form

f(x⃗, p⃗) = h(x⃗ · x⃗, x⃗ · p⃗, p⃗ · p⃗)

for some function h. Show that Ω(f, L_{i}) = 0.