Difference between revisions of "Least Action"
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==Full Title== | ==Full Title== | ||
| + | [[Least Action]] is used with variational calculus to determine the least costly path to a given goal. | ||
| + | ==Context== | ||
| + | Typically [[Least Action]] is a metric to determine the least energetic path of a particle or wave in physics. | ||
| − | + | ===Brachistochrone=== | |
| + | the curve between two points that in the shortest time by a body moving under an external force without friction; the curve of quickest descent. | ||
| + | ===Hamilton's Principle=== | ||
| + | a general principle of least action for classical mechanics, which states that the dynamics of a physical system are determined by a variational problem for a functional based on its Lagrangian, which contains all the physical information concerning the system and the forces acting on it, which is the position and velocity of every component. It should be noted that this information does not include the angular motion of the component.<ref>L. D. Landau and E. M. Lifschitz, ''Mechanics'' Addison Wesley (1960 in English) p 2</ref> | ||
| − | == | + | ==The Principle== |
| − | + | The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics.<ref>Alberto Rojo and Anthony Bloch, ''The Principle of Least Action - History and Physics'' Cambridge UP (2018-04) ISBN: 9780521869027</ref> | |
| − | + | ||
| − | + | ==Optics== | |
| − | Maxwell's | + | In optics, Fermat proposed his principle that the path taken between two points by a ray of light is the path that can be traversed in the least time, such as a straight ray in a uniform medium, or the refraction of light passing through an interface between two media. |
| + | In this principle, the traversal time is the optimization object and the product of the refraction index and the optical path is regarded as the action. After Fermat, Maupertuis and Euler independently proposed Maupertuis’s principle for mechanics, which states that the path followed by a physical system is the one having the shortest length (with a suitable interpretation of path and length). | ||
| + | |||
| + | Light, however, is diffracted when passing near or through media and so the spot at which the light strikes the target is a pattern with a total probability of one. | ||
| + | |||
| + | ==Computation== | ||
| + | Landauers' Principle<ref name = bennett>Charles H. Bennett, ''Notes on Landauer's principle, Reversible Computation and Maxwell's Demon.'' Studies in History and Philosophy of Modern Physics volume=34 issue=3 pp. 501–510 (2003) http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf DOI 10.1016/S1355-2198(03)00039-X</ref> | ||
| + | |||
| + | ==Biology== | ||
| + | ==Quantum Mechanics== | ||
| + | Least action in [[Quantum Mechanics]] is a little different as [[All Possible Paths]] are considered with appropriate weights. | ||
| + | |||
| + | As noted above when a photon of energy or other atomic particle passes anywhere near another object, the full pattern of possibilities is open as the the location each particle strikes. | ||
| − | + | * See the wiki page on [[Feynman Least Action Thesis]] for Feynman's application of [[Least Action]] to [[Quantum Mechanics]] | |
| − | |||
==References== | ==References== | ||
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===Other Material=== | ===Other Material=== | ||
* See wiki page on the [[Eventful Universe]] | * See wiki page on the [[Eventful Universe]] | ||
| − | |||
[[Category: Physics]] | [[Category: Physics]] | ||
Latest revision as of 18:02, 10 January 2025
Contents
Full Title
Least Action is used with variational calculus to determine the least costly path to a given goal.
Context
Typically Least Action is a metric to determine the least energetic path of a particle or wave in physics.
Brachistochrone
the curve between two points that in the shortest time by a body moving under an external force without friction; the curve of quickest descent.
Hamilton's Principle
a general principle of least action for classical mechanics, which states that the dynamics of a physical system are determined by a variational problem for a functional based on its Lagrangian, which contains all the physical information concerning the system and the forces acting on it, which is the position and velocity of every component. It should be noted that this information does not include the angular motion of the component.[1]
The Principle
The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics.[2]
Optics
In optics, Fermat proposed his principle that the path taken between two points by a ray of light is the path that can be traversed in the least time, such as a straight ray in a uniform medium, or the refraction of light passing through an interface between two media. In this principle, the traversal time is the optimization object and the product of the refraction index and the optical path is regarded as the action. After Fermat, Maupertuis and Euler independently proposed Maupertuis’s principle for mechanics, which states that the path followed by a physical system is the one having the shortest length (with a suitable interpretation of path and length).
Light, however, is diffracted when passing near or through media and so the spot at which the light strikes the target is a pattern with a total probability of one.
Computation
Landauers' Principle[3]
Biology
Quantum Mechanics
Least action in Quantum Mechanics is a little different as All Possible Paths are considered with appropriate weights.
As noted above when a photon of energy or other atomic particle passes anywhere near another object, the full pattern of possibilities is open as the the location each particle strikes.
- See the wiki page on Feynman Least Action Thesis for Feynman's application of Least Action to Quantum Mechanics
References
- ↑ L. D. Landau and E. M. Lifschitz, Mechanics Addison Wesley (1960 in English) p 2
- ↑ Alberto Rojo and Anthony Bloch, The Principle of Least Action - History and Physics Cambridge UP (2018-04) ISBN: 9780521869027
- ↑ Charles H. Bennett, Notes on Landauer's principle, Reversible Computation and Maxwell's Demon. Studies in History and Philosophy of Modern Physics volume=34 issue=3 pp. 501–510 (2003) http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf DOI 10.1016/S1355-2198(03)00039-X
Other Material
- See wiki page on the Eventful Universe
