Remarks on structures and preservation in forced discrete mechanical systems of Routh type
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| Format: | Preprint |
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2024
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| _version_ | 1866908823629856768 |
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| author | Caruso, Matías I. Fernández, Javier Tori, Cora Zuccalli, Marcela |
| author_facet | Caruso, Matías I. Fernández, Javier Tori, Cora Zuccalli, Marcela |
| contents | We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete Legendre transform of the canonical symplectic structure on $T^*Q$ modified by a "magnetic term". One example of this type of system is provided by the Lagrangian reduction of a symmetric (unforced) discrete mechanical system in the Routh style. In this particular case, we do not reduce by the full symmetry group but, rather, by an appropriate isotropy subgroup. In this context, the preserved symplectic structure can be alternatively seen as the Marsden-Weinstein reduction of the canonical symplectic structure $ω_{L_d}$ on $Q\times Q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05305 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on structures and preservation in forced discrete mechanical systems of Routh type Caruso, Matías I. Fernández, Javier Tori, Cora Zuccalli, Marcela Differential Geometry Dynamical Systems 37J06, 70H33 (Primary) 70G75 (Secondary) We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete Legendre transform of the canonical symplectic structure on $T^*Q$ modified by a "magnetic term". One example of this type of system is provided by the Lagrangian reduction of a symmetric (unforced) discrete mechanical system in the Routh style. In this particular case, we do not reduce by the full symmetry group but, rather, by an appropriate isotropy subgroup. In this context, the preserved symplectic structure can be alternatively seen as the Marsden-Weinstein reduction of the canonical symplectic structure $ω_{L_d}$ on $Q\times Q$. |
| title | Remarks on structures and preservation in forced discrete mechanical systems of Routh type |
| topic | Differential Geometry Dynamical Systems 37J06, 70H33 (Primary) 70G75 (Secondary) |
| url | https://arxiv.org/abs/2403.05305 |