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 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

Main article: 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
• 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