Decomposition and characterization of curl forces for all space dimensions

2025/07/13 by R. A. Kycia, Kycia, Radosław Antoni
Engineering · #Classical Physics (physics.class-ph) #Differential Geometry (math.DG) #Elasticity and Wave Propagation #FOS: Mathematics #FOS: Physical sciences #Geotechnical and Geomechanical Engineering #Mathematical Physics (math-ph) #Mechanics and Biomechanics Studies

paper · pdf · doi:10.48550/arxiv.2507.09817

openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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.

Citations

• The Darboux Classification of Curl Forces
• Nonlinear Cauchy Elasticity
• Inverting covariant exterior derivative
• Inverting covariant exterior derivative
• Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates
• The Poincare lemma for codifferential, anticoexact forms, and applications to physics
• On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists

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