Stationary-action principle

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 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. 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. Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics. The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory 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. Hero of Alexandria later showed that this path was the shortest length and least time. Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746.