Principle of least action

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The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). The physicist Richard Feynman demonstrated how this principle can also be used in quantum calculations. [1] It was historically called "least" because its solution requires finding the path that has the least value.[2] Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.[3]

The principle remains central in modern physics and mathematics, being applied in thermodynamics,[4] fluid mechanics,[5] the theory of relativity, quantum mechanics,[6] particle physics, and string theory[7] and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in surveying and optics. Rope stretchers in ancient Egypt stretched corded ropes to measure the distance between two points. Ptolemy, in his Geography (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course".[citation needed] In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection.[citation needed] Hero of Alexandria later showed that this path was the shortest length and least time.[8]

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744[9] and 1746.[10] However, Leonhard Euler discussed the principle in 1744,[11] and evidence shows that Gottfried Leibniz preceded both by 39 years.[12][13][14]

In 1932, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes: for macroscopic systems, the dominant contribution to the apparent path is the classical path (the stationary, action-extremizing one), while any other path is possible in the quantum realm.[citation needed]

General statement[edit]




Mathematically the principle is[16][17][18]

where δ (Greek lowercase delta) means a small change. In words this reads:[15]


In applications the statement and definition of action are taken together:[19]


Origins, statements, and controversy[edit]


Main article: Fermat's principle



Main article: Maupertuis principle



— Pierre Louis Maupertuis[20]



Maupertuis' principle





Euler's principle


Disputed priority[edit]


Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752.[12] The claims of forgery were re-examined 150 years later, and archival work by C.I. Gerhardt in 1898[13] and W. Kabitz in 1913[14] uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.

Further development[edit]


Lagrange and Hamilton[edit]

Main article: Hamilton's principle

Much of the calculus of variations was stated by Joseph-Louis Lagrange in 1760[22][23] and he proceeded to apply this to problems in dynamics. In Méchanique Analytique (1788) Lagrange derived the general equations of motion of a mechanical body.[24] William Rowan Hamilton in 1834 and 1835[25] applied the variational principle to the classical Lagrangian function

to obtain the Euler–Lagrange equations in their present form.

Jacobi and Morse[edit]

In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary saddle points); most of his work focused on geodesics on two-dimensional surfaces.[26] The first clear general statements were given by Marston Morse in the 1920s and 1930s,[27] leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.

Gauss and Hertz[edit]


Disputes about possible teleological aspects[edit]




In particular, the fixing of the final state has been interpreted as giving the action principle a teleological character which has been controversial historically. However, according to W. Yourgrau and S. Mandelstam, the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess[28] In addition, some critics maintain this apparent teleology occurs because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be seen as a teleological explanation. Teleology can also be overcome if we consider the classical description as a limiting case of the quantum formalism of path integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths.[citation needed]

The short story Story of Your Life by the speculative fiction writer Ted Chiang contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension. Keith Devlin's The Math Instinct contains a chapter, "Elvis the Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in some animals as they solve the "least time" problem in actual situations.

More Fundamental Than Newton's 2nd Law[edit]

According to Richard Feynman, the principle of least action is mathematically more specific than Newton's 2nd law and more fundamental in theoretical physics because it explains a wider range of physical law. You can derive Newton's 2nd law from least action, but the converse is not true without also applying Newton's 1st and 3rd laws and disallowing non-conservative forces like friction. By being more specific and thereby explaining only conservative forces, the principle of least action is able to solve problems Newton's 2nd law can't, but the converse is not true. The principle of least action can be used to derive the conservation of momentum and energy if its symmetry in space and time are assumed (see Noether's theorem).[3] It correctly does not allow non-conservative potential fields, but Newton's 2nd law allows for them by allowing for non-conservative momenta and forces (such as friction) which are not fundamental forces.[29] The mathematical basis for the difference is that Newton's 2nd law (stated as F=dp/dt instead of F=ma) allows for momentums p(t)=q(t)+C where q(t) are conserved momenta allowed by least action and C is a constant that can be non-zero in Newton's 2nd law but not in least action. The constant allows for non-conservative momenta and therefore non-conservative forces and potentials in Newton's 2nd law. Newton's 2nd law explains conservation of energy and momentum and can be used to show equivalency with least action when forces are properly conserved, e.g. when forces are summed to zero in accordance with Newton's 1st and 3rd laws and when accounting for heat generated by friction. Derivations of Lagrangian and Hamiltonian methods do not begin with Newton's 2nd law, but with a more modern mathematical formulation of it that requires forces to be conservative.[citation needed]

See also[edit]

Notes and references[edit]

  1. ^ R. Feynman, Quantum Mechanics And Path Integrals, McGraw-Hill Companies (1965), ISBN 0070206503
  3. ^ a b "The Character of Physical Law" Richard Feynman
  4. ^ García-Morales, Vladimir; Pellicer, Julio; Manzanares, José A. (2008). "Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators". Annals of Physics. 323 (8): 1844–58. Bibcode:2008AnPhy.323.1844G. doi:10.1016/j.aop.2008.04.007. 
  5. ^
  6. ^ Feynman, Richard Phillips (1942). "The Principle of Least Action in Quantum Mechanics". Bibcode:1942PhDT.........5F. 
  7. ^[full citation needed]
  8. ^ Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 167–68. ISBN 0-19-501496-0. 
  20. ^ Chris Davis. Idle theory (1998)
  28. ^ Stöltzner, Michael (1994). Inside Versus Outside: Action Principles and Teleology. Springer. pp. 33–62. doi:10.1007/978-3-642-48647-0_3. ISBN 978-3-642-48649-4. 
  29. ^ "The Principle of Least Action" Richard Feynman

External links[edit]