Local Truncation Error of Low-Order Fractional Variational Integrators

We study the local truncation error of the so-called fractional variational integrators, recently developed in [1,2] based on previous work by Riewe and Cresson [3,4]. 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 [5]. 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, Theorem2) the order of local truncation error of the fractional variational integrators with respect to the dynamics of mechanical systems with linear damping.