A Novel Approach to Setting the Problem of Lagrange for Dynamical Systems and Nonlinear Elastodynamics

Abstract

The classical Lagrange problem for dynamical systems introduces a Lagrangian action functional defined for any dynamical process that is envisioned to take place over a fixed interval of time with its state at each time lying on an unknown, but prescribed, configuration between two given end points in an \(n\) -dimensional state space \(\mathbb{R}^{n}\) . It is proposed that the fundamental dynamical field equation that characterizes the dynamical process and determines the precise motion between the two given end points is the Euler–Lagrange equation related to the stationarity of the Lagrangian action functional, expressed as the integral of a particularly formulated action density over the fixed time interval, among all admissible configurations that span the two given end points. Thus stated, this variational calculus problem introduces variations of a configuration that carries a dynamical process, and emphasizes the novelty and need to express explicitly how the configuration influences the state of that process. At each time during a dynamical process the state is subjected to an extrinsic force (classically taken to be conservative) which must be transmitted to the configuration that carries the process and, by action-reaction the configuration responds with a configuration contact force on the state of equal magnitude but opposite direction. This allows the Lagrangian action functional for a dynamical process to be interpreted as the difference between the average kinetic energy of the dynamical process that is carried by that configuration and the average configurational work done by the configuration contact force on the moving state as the state traverses that configuration during the fixed time interval. The aim in the Problem of Lagrange is to extremize this difference over all admissible configurations. The implication is that given a time interval and initial and final end points in the space of all states, the dynamical process of physical interest must follow a configuration that optimizes the gap between the average expended kinetic energy and the average expended configurational work. When the optimal condition is met and the dynamical process is so restricted, the difference between these average expenditures of energy and work will be at a local maximum, a local minimum, or a saddle point known as a condition of “least action”.

Herein, we investigate the optimization implications of this novel interpretation of the action functional for the Problem of Lagrange for dynamical systems for a general, possibly non-conservative, state-dependent extrinsic force field. We show that only a conservative state-dependent extrinsic force field is allowable within the statement of the problem and, thus, reaffirm the predominant classical hypothesis of restricting attention to conservative extrinsic force fields.

We close with a section which covers an analogous investigation of the Problem of Lagrange for nonlinear elastodynamics.