Contactifications: a Lagrangian description of compact Hamiltonian systems
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913569254146048 |
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| author | Grabowska, Katarzyna Grabowski, Janusz Kuś, Marek Marmo, Giuseppe |
| author_facet | Grabowska, Katarzyna Grabowski, Janusz Kuś, Marek Marmo, Giuseppe |
| contents | If $η$ is a contact form on a manifold $M$ such that the orbits of the Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the presymplectic 2-form $dη$ on $M$ induces a symplectic structure $ω$ on the quotient manifold $N=M/\mathcal{F}$. We call $(M,η)$ a $\textit contactification$ of the symplectic manifold $(N,ω)$. First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space $\mathbb{C}P^{n-1}$, being the unit sphere $S^{2n-1}$ in $\mathbb{C}^{n}$, and equipped with the restriction of the Liouville 1-form on $\mathbb{C}^n$. Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden-Weinstein-Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds $(N,ω)$, on which symplectic forms never admit a `vector potential'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contactifications: a Lagrangian description of compact Hamiltonian systems Grabowska, Katarzyna Grabowski, Janusz Kuś, Marek Marmo, Giuseppe Symplectic Geometry Mathematical Physics 53D05, 53D10 (Primary) 53D35, 70Hxx, 70H25, 70S05 (Secondary) If $η$ is a contact form on a manifold $M$ such that the orbits of the Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the presymplectic 2-form $dη$ on $M$ induces a symplectic structure $ω$ on the quotient manifold $N=M/\mathcal{F}$. We call $(M,η)$ a $\textit contactification$ of the symplectic manifold $(N,ω)$. First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space $\mathbb{C}P^{n-1}$, being the unit sphere $S^{2n-1}$ in $\mathbb{C}^{n}$, and equipped with the restriction of the Liouville 1-form on $\mathbb{C}^n$. Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden-Weinstein-Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds $(N,ω)$, on which symplectic forms never admit a `vector potential'. |
| title | Contactifications: a Lagrangian description of compact Hamiltonian systems |
| topic | Symplectic Geometry Mathematical Physics 53D05, 53D10 (Primary) 53D35, 70Hxx, 70H25, 70S05 (Secondary) |
| url | https://arxiv.org/abs/2404.19560 |