Decomposition and characterization of curl forces for all space dimensions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908940301762560 |
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| author | Kycia, Radosław Antoni |
| author_facet | Kycia, Radosław Antoni |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposition and characterization of curl forces for all space dimensions Kycia, Radosław Antoni Mathematical Physics Differential Geometry Classical Physics 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. |
| title | Decomposition and characterization of curl forces for all space dimensions |
| topic | Mathematical Physics Differential Geometry Classical Physics |
| url | https://arxiv.org/abs/2507.09817 |