An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908654927609856 |
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| author | Colombo, Leonardo de Lucas, Javier Rivas, Xavier Zawora, Bartosz M. |
| author_facet | Colombo, Leonardo de Lucas, Javier Rivas, Xavier Zawora, Bartosz M. |
| contents | This work presents a comprehensive review of the $k$-polysymplectic Marsden-Weinstein reduction theory, rectifying prior errors and inaccuracies in the literature while introducing novel findings. It also emphasises the genuine practical significance of seemingly minor technical details. On this basis, we introduce a novel $k$-polysymplectic energy-momentum method, new related stability analysis techniques, and apply them to Hamiltonian systems of ordinary differential equations relative to a $k$-polysymplectic manifold. We provide detailed examples of both physical and mathematical significance, including the study of complex Schwarz equations related to the Schwarz derivative, a series of isotropic oscillators, integrable Hamiltonian systems, quantum oscillators with dissipation, affine systems of differential equations, and polynomial dynamical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_15035 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold Colombo, Leonardo de Lucas, Javier Rivas, Xavier Zawora, Bartosz M. Mathematical Physics Classical Analysis and ODEs Differential Geometry Exactly Solvable and Integrable Systems 34A26, 34D20, 37J39 (primary), 53B50, 53C15 (secondary) This work presents a comprehensive review of the $k$-polysymplectic Marsden-Weinstein reduction theory, rectifying prior errors and inaccuracies in the literature while introducing novel findings. It also emphasises the genuine practical significance of seemingly minor technical details. On this basis, we introduce a novel $k$-polysymplectic energy-momentum method, new related stability analysis techniques, and apply them to Hamiltonian systems of ordinary differential equations relative to a $k$-polysymplectic manifold. We provide detailed examples of both physical and mathematical significance, including the study of complex Schwarz equations related to the Schwarz derivative, a series of isotropic oscillators, integrable Hamiltonian systems, quantum oscillators with dissipation, affine systems of differential equations, and polynomial dynamical systems. |
| title | An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold |
| topic | Mathematical Physics Classical Analysis and ODEs Differential Geometry Exactly Solvable and Integrable Systems 34A26, 34D20, 37J39 (primary), 53B50, 53C15 (secondary) |
| url | https://arxiv.org/abs/2311.15035 |