Variational principle

Short description: Scientific principles enabling the use of the calculus of variations

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Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

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 functions that optimize the values of quantities that depend on 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.

## Contents

* 1 History
* 1.1 Physics
* 1.2 Math
* 2 Examples
* 2.1 In mathematics
* 2.2 In physics
* 3 References
* 4 External links

## History

### Physics

Main page: Physics:History of variational principles in physics

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 topological entropy and 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
* Maupertuis' principle in classical mechanics
* The principle of least action in mechanics, 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
* 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.
* 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. https://archive.org/details/introductiontoel00grif_0.
* 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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