Defining Momentum
We usually think of momentum as \(mv\). But the Generalized Momentum \(p_i\) associated with a coordinate \(q_i\) is defined as:
\[p_i = \frac{\partial L}{\partial \dot{q}_i}\]
We call \(q_i\) and \(p_i\) Conjugate Variables. They are two halves of the information needed to describe a particle's state.
Worked Examples
Example 1: Magnetic Momentum
For a particle in a magnetic field, the generalized momentum is \(p = mv + qA\), where \(A\) is the vector potential. Notice that this is not the same as the velocity. The "Canonical Momentum" includes the effect of the field.
The Bridge to Quantum Mechanics
Quantum Mechanics is built on conjugate variables. The Commutation Relation \([\hat{x}, \hat{p}] = i\hbar\) only applies to variables that are conjugate to each other. If you want to know which variables follow the Uncertainty Principle, you just look at the Lagrangian. If \(p\) is the conjugate momentum to \(q\), then you can never know both perfectly at the same time. The math of classical analytical mechanics defines the "limits" of quantum knowledge.