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Thus, instead of thinking about an object accelerating in response to applied forces, one might think of them picking out the path with a stationary action.
The principle is also called the principle of stationary action and also Hamilton's principle or (less general and in fact incorrect) the principle of least action and the principle of minimal action. The action is a scalar (a number) with the dimension of energy time. The principle is a simple, general, and powerful theory for predicting motion in classical mechanics. Extensions of the action principle describe relativistic mechanics, quantum mechanics, electricity and magnetism.
Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations can be derived as conditions of stationary action.
Many problems in physics can be represented and solved in the form of an action principle, such as finding the quickest way to run down the beach for reaching a drowning person. Water running downhill seeks the steepest descent, the quickest way down, and water running into a basin distributes itself so that its surface is as low as possible. Light finds the quickest trajectory through an optical system ( Fermat's principle of least time). The path of a body in a gravitational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
Symmetries in a physical situation can better be treated with the action principle, together with the Euler-Lagrange equations which are derived from the action principle. For example, Noether's theorem which states that with every continuous symmetry in a physical situation there corresponds a conservation law. This deep connection, however, requires that the action principle is assumed.
In classical mechanics (non-relativistic, non-quantum mechanics), the correct choice of the action can be proven from Newton's laws of motion. Conversely, the action principle proves Newton's equation of motion given the correct choice of action. So in classical mechanics the action principle is equivalent to Newton's equation of motion. The use of the action principle often is simpler than the direct application of Newton's equation of motion. The action principle is a scalar theory, with derivations and applications that employ elementary calculus.
The principle of least action was first formulated by MaupertuisPierre Louis Moreau de Maupertuis ( July 17, 1698 July 27, 1759) was a French mathematician and astronomer. He is often credited with having invented the principle of least action. He was born at Saint-Malo, France. At the age of twenty he entered the arm [1] in 1746 and further developed (from 1748 onwards) by the mathematicians EulerLeonhard Euler ( April 15, 1707 September 18, 1783) (pronounced "oiler") was a Swiss mathematician and physicist. He is considered (together with Gauss) to be one of the two greatest mathematicians. Leonhard Euler was the first to use the term " function", LagrangeJoseph Louis Lagrange ( January 25, 1736 April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. Lagrange worked for Frederick II, in Berlin, for twenty years. It was Lagrange who developed the mean value theorem, and HamiltonSir William Rowan Hamilton ( August 4, 1805 September 2, 1865) was an Irish mathematician, physicist, and astronomer. Hamilton's discovery of quaternions is his best known investigation. Hamilton also contributed to the development of optics, dynamics, an. Maupertuis arrived at this principle from a feeling that the very perfection of the universe demands a certain economy in nature and is opposed to any needless expenditure of energy. Natural motions must be such as to make some quantity a minimum. It was only necessary to find that quantity, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" or twice what we now call the kinetic energy of the system.
Euler (in "Reflexions sur quelques loix generales de la nature", 1748) adopts the least-action principle, calling the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.