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In classical mechanics, the Least Action Principle states that the motion of a particle along some path always minimize the difference between its kinetic energy and its potential energy.  Mathematically, the motion of a particle always minimizes the Lagragian action functional.

In classical mechanics, the Least Action Principle states that the motion of a particle along some path always minimize the difference between its kinetic energy and its potential energy.  Mathematically, the motion of a particle always minimizes the Lagragian action functional.

In [[General Relativity]], the Least Action Principle states that the motion of a particle on a surface M must be a [[geodesic]] of M, so it must minimize the geodesic functional.

In [[General Relativity]], the Least Action Principle states that the motion of a particle on a surface M must be a [[geodesic]] of M, so it must minimize the geodesic functional. Additional, the Einstein field equations follow from applying the principle of least action to the action functional <math>\int R_{g}d\mathrm{vol}_g</math>, where the minimization takes place over the space of metrics <math>g</math> on spacetime, and <math>R_g</math> denotes the scalar curvature of the metric.

In [[Quantum Mechanics]], the amplitude for a particle start at point ''A'' and end at point ''B'' is given by the integral of <math>e^{iS(\mathrm{path})/\hbar}</math> over the space of all paths joining ''A'' and ''B''. Here <math>S(\mathrm{path})</math> denotes the action of the path. Thus, by the stationary phase approximation, the amplitude receives the greatest contribution from the path that minimizes the action. This is the [[Feynman path integral]] approach to quantum mechanics.

In [[Quantum Mechanics]], the amplitude for a particle start at point ''A'' and end at point ''B'' is given by the integral of <math>e^{iS(\mathrm{path})/\hbar}</math> over the space of all paths joining ''A'' and ''B''. Here <math>S(\mathrm{path})</math> denotes the action of the path. Thus, by the stationary phase approximation, the amplitude receives the greatest contribution from the path that minimizes the action. This is the [[Feynman path integral]] approach to quantum mechanics.