×

Local truncation error of low-order fractional variational integrators. (English) Zbl 1458.37088

Nielsen, Frank (ed.) et al., Geometric science of information. 4th international conference, GSI 2019, Toulouse, France, August 27–29, 2019. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 11712, 541-548 (2019).
Summary: We study the local truncation error of the so-called fractional variational integrators, recently developed by the authors in [A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings. 6th IFAC LHMNC-2018 Proceedings, IFAC-PapersOnLine 51, No. 3, 50–55 (2018); Fractional damping through restricted calculus of variations. arXiv:1905.05608 (2019)] based on previous work by F. Riewe [Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, No. 2, 1890–1899 (1996)] and J. Cresson and P. Inizan [J. Math. Anal. Appl. 385, No. 2, 975–997 (2012; Zbl 1250.49024)]. These integrators are obtained through two main elements: the enlarging of the usual mechanical Lagrangian state space by the introduction of the fractional derivatives of the dynamical curves; and a discrete restricted variational principle, in the spirit of discrete mechanics and variational integrators [J. E. Marsden and M. West, Acta Numerica 10, 357–514 (2001; Zbl 1123.37327)]]. The fractional variational integrators are designed for modelling fractional dissipative systems, which, in particular cases, reduce to mechanical systems with linear damping. All these elements are introduced in the paper. In addition, as original result, we prove (Sect. 3, Theorem 2) the order of local truncation error of the fractional variational integrators with respect to the dynamics of mechanical systems with linear damping.
For the entire collection see [Zbl 1428.94016].

MSC:

37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
26A33 Fractional derivatives and integrals
49K05 Optimality conditions for free problems in one independent variable
49M25 Discrete approximations in optimal control
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
PDFBibTeX XMLCite
Full Text: DOI Link