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.
References
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. MR2218315 1187.91125 10.1016/j.mcm.2005.08.014 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. MR2218315 1187.91125 10.1016/j.mcm.2005.08.014
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. MR1908825 1020.39008 10.1016/S0377-0427(01)00432-0 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. MR1908825 1020.39008 10.1016/S0377-0427(01)00432-0
M. Bohner, Calculus of variations on time scales. Dynam. Systems Appl. 13 (2004), no. 3-4, pp. 339–349. MR2106410 1069.39019 M. Bohner, Calculus of variations on time scales. Dynam. Systems Appl. 13 (2004), no. 3-4, pp. 339–349. MR2106410 1069.39019
J. Cresson, Non-differentiable variational principles. J. Math. Anal. Appl. 307 (2005), no. 1, pp. 48–64. MR2138974 1077.49033 10.1016/j.jmaa.2004.10.006 J. Cresson, Non-differentiable variational principles. J. Math. Anal. Appl. 307 (2005), no. 1, pp. 48–64. MR2138974 1077.49033 10.1016/j.jmaa.2004.10.006
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. MR2549615 1188.49023 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. MR2549615 1188.49023
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. MR2966865 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. MR2966865
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. MR2405376 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. MR2405376
E. Girejko, A. B. Malinowska, D. F. M. Torres, Delta-Nabla optimal control problems. J. Vib. Control 17 (2011), no.11, pp. 1634–1643. MR2885049 10.1177/1077546310381271 E. Girejko, A. B. Malinowska, D. F. M. Torres, Delta-Nabla optimal control problems. J. Vib. Control 17 (2011), no.11, pp. 1634–1643. MR2885049 10.1177/1077546310381271
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. MR2879335 10.1080/02331934.2010.506615 1239.49025 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. MR2879335 10.1080/02331934.2010.506615 1239.49025
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. MR2020533 1043.49004 10.1016/j.jmaa.2003.09.031 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. MR2020533 1043.49004 10.1016/j.jmaa.2003.09.031
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. MR2671876 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. MR2671876
J. E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001), pp. 357–514. MR2009697 1123.37327 10.1017/S096249290100006X J. E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001), pp. 357–514. MR2009697 1123.37327 10.1017/S096249290100006X
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. MR2801322 10.1051/cocv/2010012 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. MR2801322 10.1051/cocv/2010012
B. Schmidt, S. Leyendecker, M. Ortiz, $\Gamma$-convergence of variational integrators for constrained systems, J. Nonlinear Sci., 19 (2009), no. 2, pp. 153–177. MR2495892 1179.37112 10.1007/s00332-008-9030-1 B. Schmidt, S. Leyendecker, M. Ortiz, $\Gamma$-convergence of variational integrators for constrained systems, J. Nonlinear Sci., 19 (2009), no. 2, pp. 153–177. MR2495892 1179.37112 10.1007/s00332-008-9030-1
