Variational principle


In science, a variational principle is one that states a problem in terms of finding an unknown function that makes an integral take on an extremum. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be described as a variational principle; in this case, the solution involves finding a function that minimizes the gravitational potential energy of the chain. These types of problems belong to the field of mathematics analysis called Calculus of variations

Overview

Any physical law which can be expressed as a variational principle describes a self-adjoint operator. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.

History

's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.

Examples

In mathematics