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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.

In [[Quantum Mechanics]], the Least Action Principle states that the motion of a subatomic particle from its initial state to its end state must minimize the [[Feynman path integral]], which is a sum over all possible path a particle can take from initial state to end state.  

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 ''S'' 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.

Recently, mathematician Alain Connes has formulated a ''spectral action principle'' that seeks to apply the Least Action Principle to the mathematical formulation of quantum gravity.

Recently, mathematician Alain Connes has formulated a ''spectral action principle'' that seeks to apply the Least Action Principle to the mathematical formulation of quantum gravity.

[[category: physics]]

[[category: physics]]