Welcome to MHF, supaman5!

There are a couple of typos in your lagrangian, but you did get the proper generalized momentums.

The generalized momentum for a generalized coordinate q is defined as .

If is zero, the langrangian equation tells us that the generalized momentum is conserved (its time derivative is zero).

This is often the case for angular coordinates, meaning both the kinetic energy and the potential energy often do not depend on the angle.

For a coordinate that is angular, the generalized momentum corresponds exactly to the angular momentum.