generalized momentum

Momentum, generalized. For a free particle it is the familiar m v, but when it is paired to an angular coordinate it becomes angular momentum, and for a charged particle in a magnetic field it picks up an extra field term. The generalized momentum is whatever quantity plays momentum's role for a given coordinate.

It is defined as p_i = partial L / partial q_dot_i, the momentum conjugate to the coordinate q_i. For L = (1/2) m x_dot^2 - V it is p = m x_dot as expected; for a rotational coordinate theta it is p_theta = partial L / partial theta_dot, which is the angular momentum. It is exactly the quantity whose time derivative the Euler-Lagrange equation sets equal to the generalized force.

It is the bridge to Hamiltonian mechanics: the Hamiltonian is built by a Legendre transform that trades each velocity q_dot for its conjugate momentum p, and it is what is conserved when a coordinate is cyclic. A key honesty: the canonical momentum is not the mechanical momentum m v in general. For a charge in a magnetic field p = m v + q A, and it is this canonical p, not m v, that gets promoted to the operator -i hbar d/dq in quantum mechanics.