Variational principle

In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding such functions which optimize the values of quantities that depend upon those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.

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
The variation principle relating topological entropy and Kolmogorov-Sinai entropy.

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.

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[1] Lanczos, Cornelius (1974) [1st published 1970, University of Toronto Press]. The Variational Principles of Mechanics (4th, paperback ed.). Dover. ISBN 0-8020-1743-6.