Classical Mechanics (Goldstein) · Problem 36

Canonical Transformations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · poisson-brackets

Problem

(a) Using the theorem concerning Poisson brackets of vector functions and components of the angular momentum, show that if

\mathbf{F}

and

\mathbf{G}

are two vector functions of the coordinates and momenta only, then

[\mathbf{F}\cdot\mathbf{L},\, \mathbf{G}\cdot\mathbf{L}] = \mathbf{L}\cdot(\mathbf{G}\times\mathbf{F}) + L_i L_j [F_i, G_j].

(b) Let

\mathbf{L}

be the total angular momentum of a rigid body with one point fixed and let

L_\mu

be its component along a set of Cartesian axes fixed in the rigid body. By means of part (a) find a general expression for

[L_\mu, L_\nu], \qquad \mu, \nu = 1, 2, 3.

(*Hint:* Choose for

\mathbf{F}

and

\mathbf{G}

unit vectors along the

\mu

and

\nu

axes.) (c) From the Poisson bracket equations of motion for

L_\mu

derive Euler's equations of motion for a rigid body.