We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

The calculus with fractional derivatives and integrals of noninteger order started more than three centuries ago when Leibniz proposed a derivative of order

The fractional calculus of variation was introduced in the context of classical mechanics. Riewe [

Despite that the Riewe approach has been successfully applied to study open and/or nonconservative linear systems, it cannot be directly applied to nonlinear open systems. The limitation follows from the fact that, in order to obtain a final equation of motion containing only integer order derivatives, the Lagrangian should contain only quadratic terms depending on fractional derivatives. In the present work we formulated a generalization of Riewe fractional action principle by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz’s rules.

As examples, we applied our generalized fractional variational principle to some nonlinear chaotic third-order dynamical systems, so-called jerk dynamical systems because the derivative of the acceleration with respect to time is referred to as the jerk [

The fractional calculus of derivative and integration of noninteger orders started more than three centuries ago with l’Hôpital and Leibniz when the derivative of order

Despite haing many different approaches to fractional calculus, several known formulations are somehow connected with the analytical continuation of Cauchy formula for

Let

For integer

The integration operators

The left and the right Riemann-Liouville fractional derivatives of order

On the other hand, the Caputo fractional derivatives are defined by inverting the order between derivatives and integrations.

The left and the right Caputo fractional derivatives of order

An important consequence of definitions (

It is important to remark, for the purpose of this work, that the fractional derivatives (

In addition to the definitions (

Let

It is important to notice that the formulas of integration by parts (

Finally, in order to obtain the equation of motion for our examples, we are going to use the following two relations:

In classical calculus of variations it is of no conceptual and practical importance to deal with Lagrangian functions depending on derivatives of nonlinear functions of the unknown function

Our main result is the following theorem.

Let

In order to develop the necessary conditions for the extremum of the action (

It is important to notice that our theorem can be easily extended for Lagrangians depending on left Caputo derivatives, and Riemann-Liouville fractional derivatives. Actually, it is also easy to generalize in order to include a nonlinear function

For

As an example for application of our generalized Euler-Lagrange equation (

The Lagrangian (

In the present work we obtained an Euler-Lagrange equation for Lagrangians depending on fractional derivatives of nonlinear functions of the unknown function

The author is grateful to the Brazilian foundations FAPERGS, CNPq, and Capes for financial support.