Variational principle
In science, a variational principle is one that states a problem in terms of finding an unknown function that makes an integral take on an extremum (a maximum or a minimum). For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be described as a variational principle; in this case, the solution involves finding a function that minimizes the gravitational potential energy of the chain. These types of problems belong to the field of mathematics analysis called Calculus of variations
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Overview
Any physical law which can be expressed as a variational principle describes a self-adjoint operator.[1] These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
History
Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.
Examples
In mathematics
- The Rayleigh–Ritz method for solving boundary-value problems approximately
- Ekeland's variational principle in mathematical optimization
- The finite element method
In physics
- Fermat's principle in geometrical optics
- Maupertuis' principle in classical mechanics
- The principle of least action in mechanics, electromagnetic theory, and quantum mechanics
- The variational method in quantum mechanics
- Gauss's principle of least constraint and Hertz's principle of least curvature
- Hilbert's action principle in general relativity, leading to the Einstein field equations.
- Palatini variation
References
- Lanczos, Cornelius (1974) [1st published 1970, University of Toronto Press]. The Variational Principles of Mechanics (4th, paperback ed.). Dover. ISBN 0-8020-1743-6.
- Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.CS1 maint: ref=harv (link)
- S T Epstein 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
- R.P. Feynman, "The Principle of Least Action", an almost verbatim lecture transcript in Volume 2, Chapter 19 of The Feynman Lectures on Physics, Addison-Wesley, 1965. An introduction in Feynman's inimitable style.
- C Lanczos, The Variational Principles of Mechanics (Dover Publications)
- R K Nesbet 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
- S K Adhikari 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
- C G Gray, G Karl G and V A Novikov 1996, Ann. Phys. 251 1.
- C.G. Gray, G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles". 11 December 2003. physics/0312071 Classical Physics.
- Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
- Stephen Wolfram, A New Kind of Science (2002), p. 1052
- John Venables, "The Variational Principle and some applications". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
- Andrew James Williamson, "The Variational Principle -- Quantum monte carlo calculations of electronic excitations". Robinson College, Cambridge, Theory of Condensed Matter Group, Cavendish Laboratory. September 1996. (dissertation of Doctor of Philosophy)
- Kiyohisa Tokunaga, "Variational Principle for Electromagnetic Field". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI
- Komkov, Vadim (1986) Variational principles of continuum mechanics with engineering applications. Vol. 1. Critical points theory. Mathematics and its Applications, 24. D. Reidel Publishing Co., Dordrecht.
- Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.