Variational principle

A variational principle is a mathematical procedure that renders a physical problem solvable by the 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.

History

Physics

The concept of a variational principle emerged from earlier work like Fermat's principle for optics in 1662. The first application of the variational technique, albeit as a special-case rather than a general principle, was James Bernoulli's solution of the brachistochrone problem in 1718.^[1]^: 68 Pierre Louis Maupertuis generalized Fermat's concept to mechanics,^[2]^: 97 in the form of a principle of least action. These principles were linked to a more general principle of least action by William Rowan Hamilton in 1831, showing that the motion of matter particles and the motion of light waves could be described in the same way. Hamilton's work was an important influence on the early 20th century research into wave-particle duality, culminating in the 1926 discovery of Schrodinger's equation.^[3]^: 119

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
Hamilton's principle in classical mechanics
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

^ Goldstine, Herman H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century. Studies in the History of Mathematics and Physical Sciences. Vol. 5. New York, NY: Springer New York. doi:10.1007/978-1-4613-8106-8. ISBN 978-1-4613-8108-2.
^ Whittaker, Edmund T. (1989). A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 (Repr ed.). New York: Dover Publ. ISBN 978-0-486-26126-3.
^ Yourgrau, Wolfgang; Mandelstam, Stanley (1979). Variational principles in dynamics and quantum theory. Dover books on physics and chemistry (Republ. of the 3rd ed., publ. in 1968 ed.). New York, NY: Dover Publ. ISBN 978-0-486-63773-0.

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. 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 Archived 2015-09-26 at the Wayback Machine". 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.

[Goldstine-1] Goldstine, Herman H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century. Studies in the History of Mathematics and Physical Sciences. Vol. 5. New York, NY: Springer New York. doi:10.1007/978-1-4613-8106-8. ISBN 978-1-4613-8108-2.

[Whittaker-2] Whittaker, Edmund T. (1989). A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 (Repr ed.). New York: Dover Publ. ISBN 978-0-486-26126-3.

[YourgrauMandelstam1979-3] Yourgrau, Wolfgang; Mandelstam, Stanley (1979). Variational principles in dynamics and quantum theory. Dover books on physics and chemistry (Republ. of the 3rd ed., publ. in 1968 ed.). New York, NY: Dover Publ. ISBN 978-0-486-63773-0.

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