Variational integrators for fractional Birkhoffian systems

In this paper, we generalize the Pfaff–Birkhoff principle to the case of containing fractional derivatives and obtain the so-called fractional Pfaff–Birkhoff–d’Alembert principle. The fractional Birkhoff equations in the sense of Riemann–Liouville fractional derivative are derived. Under the framework of variational integrators, we develop the discrete fractional Birkhoff equations by approximating the Riemann–Liouville fractional derivative with the shifted Grünwald–Letnikov fractional derivative. The resulting algebraic equations can be served as an algorithm to numerically solve the fractional Birkhoff equations. A numerical example is demonstrated to show the validity and applicability of the presented methodology.