The mixed convolved action

TL;DR

This paper introduces the concept of mixed convolved action, a unified variational principle using fractional calculus that applies to both conservative and non-conservative dynamical systems with discrete and continuous spatial representations.

Contribution

It develops a novel variational framework based on mixed convolved action, incorporating fractional derivatives to handle a broad class of dynamical systems, including non-conservative processes.

Findings

Unified variational principle for diverse dynamical systems

Inclusion of fractional derivatives extends applicability

Derivation of governing equations from a single scalar functional

Abstract

A series of stationary principles are developed for dynamical systems by formulating the concept of mixed convolved action, which is written in terms of mixed variables, using temporal convolutions and fractional derivatives. Dynamical systems with discrete and continuous spatial representations are considered as initial applications. In each case, a single scalar functional provides the governing differential equations, along with all the pertinent initial and boundary conditions, as the Euler-Lagrange equations emanating from the stationarity of this mixed convolved action. Both conservative and non-conservative processes can be considered within a common framework, thus resolving a long-standing limitation of variational approaches for dynamical systems. Several results in fractional calculus also are developed.