Principle of least action
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| Principle of least action | |
|---|---|
| Name | Principle of least action |
| Field | Theoretical physics |
| Introduced | 18th century |
| Notable | Pierre-Louis Moreau de Maupertuis, Leonhard Euler, Joseph-Louis Lagrange, William Rowan Hamilton |
Principle of least action The Principle of least action is a variational statement used in theoretical physics, classical mechanics, and analytical dynamics that determines the trajectory of a system by extremizing an action functional. It underpins formulations by Leonhard Euler, Joseph-Louis Lagrange, William Rowan Hamilton, and influenced later developments by Sofia Kovalevskaya, Pierre-Simon Laplace, and Isaac Newton.
Definition and formulation
The principle asserts that the true path taken by a dynamical system between two states makes the action stationary, typically a minimum, of the integral of a Lagrangian over time; this was advocated by Pierre-Louis Moreau de Maupertuis and formalized by Leonhard Euler and Joseph-Louis Lagrange. The Lagrangian formalism connects to Hamiltonian mechanics developed by William Rowan Hamilton and to canonical transformations studied by Carl Gustav Jacobi and Adrien-Marie Legendre. In many physical problems the stationary-action condition yields Euler–Lagrange equations which are equivalent to Newtonian equations used by Isaac Newton and later generalized in celestial mechanics by Pierre-Simon Laplace and Joseph-Louis Lagrange.
Historical development
Early variational ideas appear in the work of Pierre de Fermat and his principle of least time, which influenced eighteenth-century debates culminating in the formulations by Maupertuis and disputes involving Voltaire and Émilie du Châtelet. Leonhard Euler provided systematic variational calculus, while Joseph-Louis Lagrange recast mechanics in his 1788 work, consolidating influences from Jean le Rond d'Alembert and Sadi Carnot's contemporaries. William Rowan Hamilton reformulated the principle in the 19th century, influencing later figures like Hermann von Helmholtz, James Clerk Maxwell, and Lord Kelvin; subsequent extensions informed research by Josiah Willard Gibbs and Paul Dirac.
Mathematical formulations
The standard action S = ∫ L dt uses a Lagrangian L(q, q̇, t) leading to Euler–Lagrange equations, a result central to analyses by Leonhard Euler and Joseph-Louis Lagrange. The Legendre transform relates Lagrangian and Hamiltonian formulations, developed by Adrien-Marie Legendre and used in canonical mechanics by William Rowan Hamilton and Carl Gustav Jacobi. Noether's theorem, proved by Emmy Noether, links continuous symmetries to conservation laws in the variational framework, a connection exploited in studies by Hendrik Lorentz, Albert Einstein, and Felix Klein.
Applications in physics
Variational methods based on the principle appear throughout mechanics and field theory: celestial mechanics problems treated by Pierre-Simon Laplace and Joseph-Louis Lagrange; continuum mechanics studied by Augustin-Louis Cauchy and Siméon Denis Poisson; electromagnetism formulated by James Clerk Maxwell uses action principles exploited by Oliver Heaviside and Heinrich Hertz. In thermodynamics and statistical mechanics the method influenced works by Ludwig Boltzmann and Josiah Willard Gibbs, while quantum field applications were developed by Paul Dirac, Richard Feynman, and Julian Schwinger. Modern applications include gauge theories associated with Murray Gell-Mann and Sheldon Glashow and general relativity by Albert Einstein.
Variational principles and connections
Related variational principles include Hamilton's principle, d'Alembert's principle from Jean le Rond d'Alembert, and Fermat's principle of least time by Pierre de Fermat; these are conceptually unified by the calculus of variations advanced by Leonhard Euler and Joseph-Louis Lagrange. The principle interfaces with symmetry analysis through Emmy Noether's theorem and with canonical quantization approaches pursued by Paul Dirac and path-integral methods by Richard Feynman. Gauge invariance and constraints are treated using techniques attributed to L. D. Landau, Evgeny Lifshitz, and methods developed in works connected to Lev Landau and Evgeny Lifshitz.
Quantum and relativistic extensions
Quantum extensions include Feynman's path integral formulation and operator approaches by Paul Dirac and Erwin Schrödinger; these recast classical action principles into quantum amplitude sums used in particle physics by Richard Feynman, Freeman Dyson, and Murray Gell-Mann. Relativistic generalizations of the action underpin Albert Einstein's general relativity and the Einstein–Hilbert action connected to David Hilbert; these influence studies in cosmology by Georges Lemaître and Alexander Friedmann. Quantum field theory and renormalization frameworks developed by Julian Schwinger and Kenneth Wilson rely on action-based formulations central to the Standard Model associated with Steven Weinberg and Sheldon Glashow.
Examples and solved problems
Classic solved problems include the brachistochrone and tautochrone posed to and solved by Jakob Bernoulli and Johann Bernoulli, and taut reflections of Fermat's principle considered by Pierre de Fermat and Christiaan Huygens. Celestial two-body and restricted three-body problems treated by Isaac Newton, Joseph-Louis Lagrange, and Henri Poincaré are elegantly addressed with action methods. Modern exemplar calculations include deriving Maxwell's equations from an electromagnetic action as in James Clerk Maxwell's framework and obtaining field equations in general relativity from the Einstein–Hilbert action as developed by Albert Einstein and David Hilbert.