Nonconservative Lagrangian mechanics: a generalized function approach

Abstract

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.

Nonconservative Lagrangian Mechanics: Purely Causal Equations of Motion

“…Neither Riewe [9,10] nor Dreisigmeyer and Young [6] were able to remove RFDs completely. Riewe suggested replacing RFDs with LFDs in the resulting equations of motion.…”

“…In examining Riewe's approach, Dreisigmeyer and Young showed that his procedure of replacing anti-causal with causal operations may not be a wise idea [6]. While still employing fractional derivatives, Dreisigmeyer and Young instead allowed for a causal equation and an anti-causal dual equation to arise.…”

“…In [6] the idea of treating the actions and Lagrangians as Volterra series was introduced. (Volterra series are the generalization to functionals of the power series concept.)…”

“…(A fuller discussion of this material can be found in [6,7,8].) First, define the generalized functions…”

An equation with left and right fractional derivatives

“…Many results have been published with concrete problems solved in classical spaces of functions and in the spaces of generalized functions. We cite only some of them, recently published or with a new approach: [2]- [4], [7], [8], [13], [15], [17], [19], [20], [22], [23] and with Karamata's regularly varying functions: [11], [24]. In this paper we treat such an equation with the boundary condition y(0) = y(b) = 0 in the space L 1 (0, b) and in a subspace of tempered distributions constructed for this problem.…”