Angular momentum as a conserved charge.
Consider a Lagrangian L that depends only on the magnitude of the velocity q(t ) of a particle which moves in ordinary three-di mensional space.
(a) Write an infinitesimal variation 8q(t) that represents a small rotation of the vector q(t ).
Explain why it leaves the Lagrangian invariant.
(b) Construct the conserved charge associated with this symmetry transformation. Verify that this conserved quantity is the (vector) angular momentum.