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 because the principle has no general minimality requirement.
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. However, Leonhard Euler also discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.
General statement
The action, denoted , of a physical system is defined as the integral of the Lagrangian L between two instants of time and – technically a functional of the generalized coordinates which are functions of time and define the configuration of the system:
where the dot denotes the time derivative, and is time.
Mathematically the principle is
where δ (lowercase Greek delta) means a small change. In words this reads:
Stationary action is not always a minimum, despite the historical name of least action. It is a minimum principle for sufficiently short, finite segments in the path of a finite-dimensional system.
In applications the statement and definition of action are taken together:
The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve , parameterized by time (see also parametric equation for this concept).
Origins, statements, and controversy
Fermat
In the 1600s, Pierre de Fermat postulated that "light travels between two given points along the path of shortest time," which is known as the principle of least time or Fermat's principle.
Maupertuis
Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who felt that "Nature is thrifty in all its actions", and applied the principle broadly:
This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of mechanics.
In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the "vis viva",
which is the integral of twice what we now call the kinetic energy T of the system.
Euler
Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Beginning with the second paragraph:
As Euler states, is the integral of the momentum over distance travelled, which, in modern notation, equals the abbreviated or reduced action