Communications in Mathematical Analysis

Discrete Calculus of Variations for Quadratic Lagrangians

P. Ryckelynck and L. Smoch

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Abstract

The intent of this paper is to develop a framework for discrete calculus of variations with action densities involving a new class of discretization operators. We introduce first the generalized scale derivatives, study their regularity and state some Leibniz formulas. Then, we deduce the discrete EulerLagrange equations for critical points of sampled actions that we compare to existing versions. Next, we investigate the case of general quadratic lagrangians and provide two examples of such lagrangians. At last, we find nontrivial properties for the discretization of a quadratic nulllagrangian.

Article information

Source
Commun. Math. Anal., Volume 15, Number 1 (2013), 44-60.

Dates
First available in Project Euclid: 18 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.cma/1374153494

Mathematical Reviews number (MathSciNet)
MR3082263

Zentralblatt MATH identifier
1275.49040

Subjects
Primary: 49K15: Problems involving ordinary differential equations 49K21: Problems involving relations other than differential equations 49M25: Discrete approximations 49N10: Linear-quadratic problems 65L03: Functional-differential equations 65L12: Finite difference methods

Keywords
Calculus of variation Functional equations Quadratic lagrangians Null Lagrangians

Citation

Ryckelynck, P.; Smoch, L. Discrete Calculus of Variations for Quadratic Lagrangians. Commun. Math. Anal. 15 (2013), no. 1, 44--60. https://projecteuclid.org/euclid.cma/1374153494


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