Classical mechanics with inequality constraints

TL;DR

This paper explores how inequality constraints can be incorporated into classical mechanics by modifying the action, illustrating the approach with a harmonic oscillator model that exhibits relativistic-like behavior at large amplitudes.

Contribution

It introduces a method to include inequality constraints in classical mechanics through action modification and demonstrates its effectiveness with a harmonic oscillator example.

Findings

Modified action approach effectively incorporates inequality constraints.

Large amplitude oscillators exhibit relativistic-like behavior.

Higher derivative inequality constraints are briefly discussed.

Abstract

In this paper we discuss mechanical systems with inequality constraints. We demonstrate how such constraints can be taken into account by proper modification of the action which describes the original unconstrained dynamics. To illustrate this approach we consider a harmonic oscillator in the model with limiting velocity. We compare the behavior of such an oscillator with the behavior of a relativistic oscillator and demonstrated that when an amplitude of the oscillator is large the properties of both type of oscillators are quite similar. We also briefly discuss inequality constraints which contain higher derivatives.