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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.09817 |
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Table of Contents:
- 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.