Principle of least action

A physical principle stating that the actual path of a system makes its action stationary under small variations of that path, given suitable boundary conditions.

Action (physics): A quantity with units of energy multiplied by time, defined for a system’s history or path. The principle selects paths by requiring this functional to be stationary.
Lagrangian: A function, usually kinetic energy minus potential energy, that encodes a system’s dynamics. Its time integral defines the action in ordinary classical mechanics.
Pierre Louis Maupertuis: An eighteenth-century French mathematician and philosopher who promoted a principle of least action. He helped popularize action-based explanations of motion in the 1740s.
Hamiltonian mechanics: A formulation of mechanics based on a system’s Hamiltonian and phase-space evolution. It is related to the action formulation and supports powerful transformations of dynamics.
Fermat's principle: The principle that light follows paths making optical travel time stationary. It is an optical stationary-time principle analogous to stationary action.
Calculus of variations: A branch of mathematics that finds extrema or stationary values of functionals. It supplies the method for varying paths and deriving motion equations.
Generalized coordinates: Coordinates chosen to describe a system’s independent degrees of freedom. The action can be varied in coordinates adapted to constraints and symmetries.
Leonhard Euler: An eighteenth-century mathematician whose work established foundational methods in calculus and mechanics. His variational work gave mathematical form to action principles.
Noether's theorem: A theorem connecting continuous symmetries of an action to conserved quantities. Symmetries of the action explain conservation of energy, momentum, and angular momentum.
Maupertuis' principle: A variational principle for mechanical trajectories at fixed energy, involving abbreviated action. It is a distinct action principle, not simply Hamilton’s fixed-time formulation.

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