Mechanics · undergrad

Canonical Momentum

Momentum defined by the Lagrangian, not always equal to mechanical momentum.

Published Apr 26, 2026 · 1 min

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Definition

For generalized coordinate $q_i$ , canonical momentum is

p_i = \frac{\partial L}{\partial \dot q_i}.

Canonical momentum is the variable paired with $q_i$ in phase space. In the presence of a vector potential, it can differ from mechanical momentum.

Dynamics on phase space using coordinates and conjugate momenta.

Equations of motion from the principle of least action — the gateway to analytical mechanics.

Continuous symmetries of the action generate conserved quantities.