Functional Equations and Poincare Invariant Mechanical Systems

TL;DR

This paper investigates a functional equation linked to Poincare-invariant mechanical systems, developing new methods to find general solutions and discovering novel solutions within a specific function class.

Contribution

It introduces new techniques for solving a key functional equation in Poincare-invariant mechanics and provides the first comprehensive solutions within a defined function class.

Findings

New solution methods for the functional equation.

Discovery of novel solutions within a specific function class.

Enhanced understanding of Poincare-invariant mechanical systems.

Abstract

We study the following functional equation that has arisen in the context of mechanical systems invariant under the Poincare algebra: \sum\limits_{i=1}^{n+1}\dfrac{\partial}{\partial x_{i}}\prod\limits_{j\neq i}f(x_{i}-x_{j}) =0,\qquad n \geq 2. New techniques are developed and the general solution within a certain class of functions is given. New solutions are found.