Integrability and the variational formulation of non-conservative mechanical systems

TL;DR

This paper introduces a novel variational approach for deriving equations of motion in non-conservative mechanical systems using a first variation functional, extending classical methods to include damping and other non-conservative effects.

Contribution

It develops a new framework that derives dynamical equations from a first variation functional, applicable to non-conservative systems, and relates integrability to the Spencer operator.

Findings

Successfully models damped oscillators within the new framework.

Shows that the dynamical equations can be obtained without an action functional.

Links integrability conditions to the Spencer operator's properties.

Abstract

It is shown that one can obtain canonically-defined dynamical equations for non-conservative mechanical systems by starting with a first variation functional, instead of an action functional, and finding their zeroes. The kernel of the first variation functional, as an integral functional, is a 1-form on the manifold of kinematical states, which then represents the dynamical state of the system. If the 1-form is exact then the first variation functional is associated with the first variation of an action functional in the usual manner. The dynamical equations then follow from the vanishing of the dual of the Spencer operator that acts on the dynamical state. This operator, in turn, relates to the integrability of the kinematical states. The method is applied to the modeling of damped oscillators.