On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative

TL;DR

This paper explores fractional mechanics by analyzing fractional Lagrangians and deriving fractional Euler-Lagrange and Hamilton equations, extending classical results through fractional derivatives and providing concrete examples.

Contribution

It introduces a fractional generalization of total time derivatives and derives corresponding Euler-Lagrange and Hamilton equations using Riemann-Liouville derivatives.

Findings

Fractional Lagrangians differing by fractional derivatives are analyzed.

Classical results are recovered as a special case.

Two detailed examples illustrate the fractional equations.

Abstract

Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the equivalent Lagrangians play an important role because they admit the same Euler-Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant the Lagrangian we obtain the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann-Liouville fractional derivatives. As a consequence of applying this procedure the classical results are reobtained as a special case. The fractional generalization of $Fa\grave{a}$ di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler-Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated and two examples are analyzed in details.