Symplectization of certain Hamiltonian systems in fibered almost-symplectic manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913613433798656 |
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| author | Fassò, Francesco Sansonetto, Nicola |
| author_facet | Fassò, Francesco Sansonetto, Nicola |
| contents | There is an important difference between Hamiltonian-like vector fields in an almost-symplectic manifold $(M,σ)$, compared to the standard case of a symplectic manifold: in the almost-symplectic case, a vector field such that the contraction iXσ is closed need not be a symmetry of $σ$. We thus call partially-Hamiltonian those vector fields which have the former property and fully-Hamiltonian those which have both properties. We consider 2n-dimensional almost symplectic manifolds with a fibration $π: M \longrightarrow B$ by Lagrangian tori. Trivially, all vertical partially-Hamiltonian vector fields are fully-Hamiltonian. We investigate the existence and the properties of non-vertical fully-Hamiltonian vector fields. We show that this class is non-empty, but under certain genericity conditions that involve the Fourier spectrum of their Hamiltonian, these vector fields reduce, under a (possibly only semi-globally defined, if $n \ge 4$) torus action, to families of standard symplectic-Hamiltonian vector fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11265 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symplectization of certain Hamiltonian systems in fibered almost-symplectic manifolds Fassò, Francesco Sansonetto, Nicola Symplectic Geometry Differential Geometry Dynamical Systems Exactly Solvable and Integrable Systems 53D15, 37J40, 70H08 There is an important difference between Hamiltonian-like vector fields in an almost-symplectic manifold $(M,σ)$, compared to the standard case of a symplectic manifold: in the almost-symplectic case, a vector field such that the contraction iXσ is closed need not be a symmetry of $σ$. We thus call partially-Hamiltonian those vector fields which have the former property and fully-Hamiltonian those which have both properties. We consider 2n-dimensional almost symplectic manifolds with a fibration $π: M \longrightarrow B$ by Lagrangian tori. Trivially, all vertical partially-Hamiltonian vector fields are fully-Hamiltonian. We investigate the existence and the properties of non-vertical fully-Hamiltonian vector fields. We show that this class is non-empty, but under certain genericity conditions that involve the Fourier spectrum of their Hamiltonian, these vector fields reduce, under a (possibly only semi-globally defined, if $n \ge 4$) torus action, to families of standard symplectic-Hamiltonian vector fields. |
| title | Symplectization of certain Hamiltonian systems in fibered almost-symplectic manifolds |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems Exactly Solvable and Integrable Systems 53D15, 37J40, 70H08 |
| url | https://arxiv.org/abs/2412.11265 |