Decomposition and characterization of curl forces for all space dimensions

TL;DR

This paper presents a PDE-free, geometric framework for decomposing forces in any dimension, generalizing curl forces beyond three dimensions and characterizing non-conservative dynamics without PDEs.

Contribution

Introduces a novel PDE-free, geometric decomposition method for forces in arbitrary dimensions, generalizing curl forces and analyzing non-conservative dynamics using differential forms and the Frobenius theorem.

Findings

Decomposition into exact and antiexact components using homotopy operator.

Generalization of curl forces to all space dimensions.

Characterization of non-conservative dynamics via integrability analysis.

Abstract

This paper introduces a PDE-free algorithmic framework for the local decomposition of classical forces in arbitrary dimensions. By representing a force field as a differential $1$-form (work form), we employ the homotopy operator on a star-shaped domain to achieve a geometric decomposition into exact (gradient) and antiexact components. The antiexact part serves as a formal generalization of the curl force - or circulatory force - outside of three-dimensional Euclidean space. To further characterize the non-conservative dynamics, we apply the Frobenius theorem to the antiexact component, resolving it into integrable terms associated with generalized potentials and a path-dependent 'core' representing fundamental obstructions to integrability. Unlike the Darboux-based classification, this constructive approach bypasses the requirement for solving partial differential equations, offering a practical tool for analyzing non-autonomous influences and scaling effects in complex physical systems.