# Stationary-action principle

## Variational principle in physics / From Wikipedia, the free encyclopedia

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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*. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories.[1]

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${\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})$ |

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The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). In relativity, a different action must be minimized or maximized.

The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics.[2] In 1933, the physicist Paul Dirac demonstrated how this principle can be used in quantum calculations by discerning the quantum mechanical underpinning of the principle in the quantum interference of amplitudes.[3] Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics.[4][5]

The principle remains central in modern physics and mathematics, being applied in thermodynamics,[6][7][8] fluid mechanics,[9] the theory of relativity, quantum mechanics,[10] particle physics, and string theory[11] and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his *Catoptrica* that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection.[12] Hero of Alexandria later showed that this path was the shortest length and least time.[13]

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744[14] and 1746.[15] However, Leonhard Euler also discussed the principle in 1744,[16] and evidence shows that Gottfried Leibniz preceded both by 39 years.[17][18][19][20]