Variational principle

A variational principle is a principle in physics which is expressed in terms of the calculus of variations.

According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.

Felix Klein'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 (what is now called a gauge group) for General Relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.

Examples

• Fermat's principle in geometrical optics
• The principle of least action in mechanics, electromagnetic theory, and quantum mechanics, where the dimension is action.
• The Einstein equation also involves a variational principle, according to Stephen Wolfram, (A New Kind of Science, p. 1052.), as a constraint on the Einstein-Hilbert Action.

Further readings

• Epstein S T 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
• Nesbet R K 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
• Adhikari S K 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
• Gray C G, Karl G and Novikov V A 1996 Ann. Phys. 251 1.

See also

• Action principle
• History of physics

External links and references

• Cornelius Lanczos, The Variational Principles of Mechanics
• Stephen Wolfram, A New Kind of Science p. 1052
• Gray, C.G., G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles". 11 Dec 2003. physics/0312071 Classical Physics.
• Venables, John, "The Variational Principle and some applications". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
• Williamson, Andrew James, "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)
• Tokunaga, Kiyohisa, "Variational Principle for Electromagnetic Field". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI