Hamilton's principle

Hamilton's principle states that the action integral is stationary for the actual path of a mechanical system under specified endpoint conditions. It yields the Euler–Lagrange equations of motion.

Action (physics): A quantity defined by integrating a system's Lagrangian over time along a path. Hamilton's principle selects paths by requiring this integral to be stationary.
Lagrangian: A function whose kinetic-minus-potential form commonly determines a mechanical system's dynamics. Its time integral defines the action varied in the principle.
Pierre Louis Maupertuis: An eighteenth-century French mathematician and philosopher associated with an early principle of least action. His action-based proposal helped establish the historical setting for variational mechanics.
Geodesic: A locally length-extremizing curve on a curved space or spacetime. Free-particle paths in suitable settings can be derived by extremizing an action.
Hamiltonian mechanics: A formulation of mechanics using coordinates, conjugate momenta, and a Hamiltonian function. It expresses evolution in phase space rather than deriving it directly from stationary paths.
Calculus of variations: The mathematical study of extrema or stationary values of functionals. The principle applies variational calculus to the action functional.
Functional: A mapping that assigns a number to a function or other field. Action is a functional: changing a path changes the value being tested.
Leonhard Euler: An eighteenth-century Swiss mathematician who developed foundational methods in calculus and mechanics. Euler helped formulate and apply variational principles to mechanical motion.
Noether's theorem: A theorem connecting continuous symmetries of an action to conserved quantities. Symmetries of the action yield conservation laws for systems governed by the principle.
Hamilton's equations: First-order differential equations governing canonical coordinates and momenta in Hamiltonian mechanics. They describe motion in phase space, while Hamilton's principle is a variational statement.

Linked from 10 pages