Variational principle

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v t e

In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using

The history of the variational principle in classical mechanics started with Maupertuis's principle in the 18th century.

Math

Felix Klein's 1872 Erlangen program attempted to identify invariants under a group of transformations.

Examples

In mathematics

• Ekeland's variational principle in mathematical optimization
• The finite element method
• The variation principle relating
  Kolmogorov-Sinai entropy
  .

In physics

• The Rayleigh–Ritz method for solving boundary-value problems in elasticity and wave propagation
• Fermat's principle in geometrical optics
• Hamilton's principle in classical mechanics
• Maupertuis' principle in classical mechanics
• The
  electromagnetic theory, and quantum mechanics
• The variational method in quantum mechanics
• Hellmann–Feynman theorem
• 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
• Hartree–Fock method
• Density functional theory
• Gibbons–Hawking–York boundary term
• Variational quantum eigensolver

References

External links

• The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action
• MR 0526967
  .
• S T Epstein 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
• 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
  .
• 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.