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| ⚫ | A '']'' is a principle in ] which | ||
| ⚫ | ] | ||
| ⚫ | is expressed in terms of the ]. | ||
| According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is ]. These expressions are also called ]. Such an expression describes an ] under a Hermitian transformation. | |||
| ]'s ] attempted to identify such invariants under a group of transformations. In what is referred to in physics as ], the ] of transformations (what is now called a ]) for ] defines symmetries under a group of transformations which depend on a variational principle, or ]. | |||
| ⚫ | A '' |
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| ⚫ | is expressed in terms of the ]. | ||
| ==Examples== | ==Examples== | ||
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| * ] | * ] | ||
| * ] | |||
| ==External links and references== | ==External links and references== | ||
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Revision as of 11:33, 13 February 2005
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
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