Introduction to Classical Mechanics · Problem 6.5

The Lagrangian method — Morin

Physics — Mechanics · lagrangian-spherical-coordinates-conservation

Problem

In spherical coordinates, consider a potential that depends only on

r

and

\theta

. The convention for spherical coordinates is that

\theta

is the angle down from the north pole, and

\phi

is the angle around the equator. The Lagrangian is

L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2\sin^2\theta\,\dot{\phi}^2) - V(r,\theta).

Identify the cyclic coordinate and the corresponding conserved quantity.