Action (physics)

Action is a quantity with units of energy multiplied by time, usually defined as the time integral of a system’s Lagrangian. In classical mechanics, its stationary value characterizes the physical path between fixed endpoints.

Lagrangian mechanics: A formulation of classical mechanics in which equations of motion follow from a system’s Lagrangian. Its Lagrangian is integrated over time to define classical action.
Lagrangian: A function of a system’s generalized coordinates, velocities, and possibly time, commonly kinetic energy minus potential energy. It is the integrand in the standard definition of mechanical action.
Noether's theorem: A theorem connecting continuous symmetries of an action to conserved quantities. Symmetries of the action reveal conservation laws in physical systems.
Hamiltonian (classical mechanics): A function of generalized coordinates and momenta that often represents total energy in classical mechanics. Unlike action, it generates time evolution in phase space and is not a time-integrated Lagrangian.
Principle of stationary action: The principle that physical trajectories make the action stationary under small variations with fixed endpoints. It specifies how classical motion is selected from possible paths.
Generalized coordinates: Coordinates chosen to describe a system’s configuration using its independent degrees of freedom. The action is a functional of the system’s coordinate history.
Classical field theory: The description of classical physical fields through equations governing their values across spacetime. Field theories use an action integrated over spacetime rather than time alone.
Action–angle variables: Canonical coordinates for integrable systems, including an action variable defined by a phase-space loop integral. Their action variable is related to, but distinct from, the trajectory’s time-integrated Lagrangian.
Euler–Lagrange equation: A differential equation whose solutions make a functional stationary under variations. Applied to the action, it yields the equations of motion.
Functional: A mapping that assigns a scalar or other value to a function or an entire function history. Action assigns one number to each candidate trajectory.

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