Stationary-action principle

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It has been suggested that this article be merged into Action principles. (Discuss) Proposed since May 2024.

The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are stationary points of the system's action functional.^[1]

The term "least action" is often used^[1] by physicists even though the principle has no general minimality requirement.^[2] Historically the principle was known as "least action" and Feynman adopted this name over "Hamilton's principle" when he adapted it for quantum mechanics.^[3]

The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity, as well as classical electrodynamics and quantum field theory. In these cases, a different action must be minimized or maximized. For relativity, it is the Einstein–Hilbert action. For quantum field theory, it involves the path integral formulation.

The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics.^[4]

The principle remains central in modern physics and mathematics, being applied in thermodynamics,^[5][6][7] fluid mechanics,^[8] the theory of relativity, quantum mechanics,^[9] particle physics, and string theory^[10] and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744^[11] and 1746.^[12]