Communications in Mathematical Analysis

Discrete Calculus of Variations for Quadratic Lagrangians

P. Ryckelynck and L. Smoch

Full-text: Open access


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

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

First available in Project Euclid: 18 July 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Calculus of variation Functional equations Quadratic lagrangians Null Lagrangians


Ryckelynck, P.; Smoch, L. Discrete Calculus of Variations for Quadratic Lagrangians. Commun. Math. Anal. 15 (2013), no. 1, 44--60.

Export citation


  • F. M. Atici, D. C. Biles, A. Lebedinsky, An application of time scales to economics. Math. Comput. Modelling 43 (2006), no. 7-8, pp. 718–726.
  • R. Agarwal, M. Bohner, D. O'Regan, A. Peterson, Dynamic equations on time scales: a survey. J. Comp. App. Math. 141 (2002), pp. 1–26.
  • M. Bohner, Calculus of variations on time scales. Dynam. Systems Appl. 13 (2004), no. 3-4, pp. 339–349.
  • J. Cresson, Non-differentiable variational principles. J. Math. Anal. Appl. 307 (2005), no. 1, pp. 48–64.
  • J. Cresson, G. F. F. Frederico and D. F. M. Torres, Constants of Motion for Non-Differentiable Quantum Variational Problems. Topol. Methods Nonlinear Anal., 33 (2009), no. 2, pp. 217–232.
  • J. Cresson, A. B. Malinowska and D. F. M. Torres, Differential, integral, and variational delta-embeddings of Lagrangian systems. Comput. Math. Appl., 64 (2012), no. 7, pp. 2294–2301.
  • R. A. C. Ferreira and D. F. M. Torres, Remarks on the calculus of variations on time scales. Int. J. Ecol. Econ. Stat 9 (2007), no. F07, pp. 65–73.
  • E. Girejko, A. B. Malinowska, D. F. M. Torres, Delta-Nabla optimal control problems. J. Vib. Control 17 (2011), no.11, pp. 1634–1643.
  • E. Girejko, A. B. Malinowska, D. F. M. Torres, The contingent epiderivative and the calculus of variations on time scales. Optimization 61 (2012), no.3, pp. 251-264.
  • R. Hilscher and V. Zeidan, Calculus of variations on time scales: weak local piecewise $\mathcal{C}^1_{rd}$ solutions with variable endpoints. J. Math. Anal. Appl. 289 (2004), no. 1, pp. 143–166.
  • N. Martins and D. F. M. Torres, Calculus of variations on time scales with nabla derivatives. Nonlinear Anal. Theor. Meth. App. 71 (2009), no. 12, pp. 763–773.
  • J. E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001), pp. 357–514.
  • S. Ober-Blöbaum, O. Junge, J. E. Marsden, Discrete Mechanics and Optimal Control: an Analysis. ESAIM Control Optim. Calc. Var. 17 (2011), pp. 322–352.
  • B. Schmidt, S. Leyendecker, M. Ortiz, $\Gamma$-convergence of variational integrators for constrained systems, J. Nonlinear Sci., 19 (2009), no. 2, pp. 153–177.