Difference between revisions of "Least Action Principle"
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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. Additionally, 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. | ||
Revision as of 19:02, July 5, 2008
Principle of Least Action was orginally formulated by Pierre Louis Maupertuis, who believe that nature must always operate in the most efficient way possible, so all laws of nature should be described as the minimization of a certain quantity. Pierre Louis Maupertuis considered the Principle of Least Action as a proof that God exists.
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. Additionally, the Einstein field equations follow from applying the principle of least action to the action functional 
, where the minimization takes place over the space of metrics 
on spacetime, and 
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 
over the space of all paths joining A and B. Here 
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.
