Volterra series fractional mechanics

Abstract

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. That is, we start with an action for our system. The equations of motion are derived by requiring the perturbation of the action to vanish. We seek to be able to include, e.g., dissipation terms in our equations through this method. Of particular interest to us is including fractional derivative operators in our system's equation. These operators lie 'between' the ordinary integer derivatives. One of their main attributes is that they are non-local operators. By successfully including fractional derivatives in our formalism, we are able to model many nonconservative systems. Previous work in this area has been largely unsatisfactory. One of our main ideas is to treat actions as Volterra series, which are a generalization of power series to functionals. The kernels in this series are what give rise to the fractional derivatives in our equations of motion. The Volterra series concept is a convenient setting for modelling actions. In particular, it explicitly separates the derivative operators from the system's position function. Fractional derivatives commonly raise many difficulties when they are used. Our work is no exception to this. The fundamental difficulty for us is that fractional derivatives come in both advanced and retarded forms. This results in a pair of equations when we use the standard variation technique: an advanced and a retarded equation of motion. This causes us to reexamine many of our assumptions when using a variational principle to model systems.