Classical Mechanics (Goldstein) · Problem 2

Variational Principles and Lagrange's Equations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · canonical-momentum-rotation-velocity-dependent-potential

Problem

Show that if the potential in the Lagrangian contains velocity-dependent terms, the canonical momentum corresponding to a coordinate of rotation

\theta

of the entire system is no longer the mechanical angular momentum

L_\theta

but is given by

p_\theta = L_\theta - \sum_i \mathbf{n} \cdot \mathbf{r}_i \times \nabla_{\mathbf{v}_i} U,

where

\nabla_{\mathbf{v}}

is the gradient operator in which the derivatives are with respect to the velocity components and

\mathbf{n}

is a unit vector in the direction of rotation. If the forces are electromagnetic in character, the canonical momentum is therefore

p_\theta = L_\theta + \sum_i \mathbf{n} \cdot \mathbf{r}_i \times \frac{q_i}{c} \mathbf{A}_i.