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

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, 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 In physics and geometry, a catenary ( , ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, ...

—can be solved using

variational calculus The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, and in this case, the variational principle is the following: The solution is a function that minimizes the

gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy an object with mass has due to the gravitational potential of its position in a gravitational field. Mathematically, it is the minimum Work (physics), mechanical work t ...

of the chain.

History

Physics

The history of the variational principle in

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

started with

Maupertuis's principle In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis, 1698 – 1759) states that the path followed by a physical system is the one of least length (with a suitable interpretation of ''path'' and ''length''). It is a ...

in the 18th century.

Math

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

's 1872

Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...

attempted to identify invariants under a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of transformations.

Examples

In mathematics

Ekeland's variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems. Ekeland's principle can be used when the lower level set of a ...

in mathematical optimization * The

finite element method Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...

* The variation principle relating

topological entropy In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Th ...

and Kolmogorov-Sinai entropy.

In physics

* The

Rayleigh–Ritz method The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. In this method, an infinite-dimensiona ...

for solving boundary-value problems in elasticity and wave propagation *

Fermat's principle Fermat's principle, also known as the principle of least time, is the link between geometrical optics, ray optics and physical optics, wave optics. Fermat's principle states that the path taken by a Ray (optics), ray between two given ...

geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along ...

Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single funct ...

* Maupertuis' principle in

* The

principle of least action Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

electromagnetic theory In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interact ...

, and

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

* The

variational method The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

in quantum mechanics *

Hellmann–Feynman theorem In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem ...

Gauss's principle of least constraint The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the acceleration of a ...

and Hertz's principle of least curvature * Hilbert's action principle in general relativity, leading to the

Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...

. *

Palatini variation In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection. In fact, as is well known, the Einstein–Hilbert action for general relativity was first formulate ...

Hartree–Fock method In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The method is named after Douglas ...

Density functional theory Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...

Gibbons–Hawking–York boundary term In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most ele ...

* Variational quantum eigensolver

References

External links

The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action
* * 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. *{{cite book , author=Griffiths, David J. , title=Introduction to Quantum Mechanics (2nd ed.) , publisher=Prentice Hall , year=2004 , isbn=0-13-805326-X , url-access=registration , url=https://archive.org/details/introductiontoel00grif_0 * John Venables,

. Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics) * Andrew James Williamson,

-- 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,

. 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. Theoretical physics Articles containing proofs Principles