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A Variational Complex for Difference Equations

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Abstract

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler–Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler–Lagrange system).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7UX, England

    Peter E. Hydon

  2. Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, England

    Elizabeth L. Mansfield

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  1. Peter E. Hydon
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  2. Elizabeth L. Mansfield
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Correspondence to Peter E. Hydon or Elizabeth L. Mansfield.

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Hydon, P., Mansfield, E. A Variational Complex for Difference Equations. Found Comput Math 4, 187–217 (2004). https://doi.org/10.1007/s10208-002-0071-9

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  • DOI: https://doi.org/10.1007/s10208-002-0071-9

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