Classical Mechanics (Goldstein) · Problem 37

Canonical Transformations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · poisson-brackets

Problem

Set up the problem of the spherical pendulum in the Hamiltonian formulation, using spherical polar coordinates for the

q_i

. Evaluate directly in terms of these canonical variables the following Poisson brackets:

[L_x, L_y], \qquad [L_y, L_z], \qquad [L_z, L_x],

showing that they have the values predicted by Eq. (9.128). Why is it that

p_\theta

and

p_\psi

can be used as canonical momenta, although they are perpendicular components of the angular momentum?