Geometry of bi-Hamiltonian systems. (Géométrie des systèmes bihamiltoniens.) (French) Zbl 0793.58015
It is well known that the various notions of compatibility between geometrical structures play a very important role in differential geometry. For instance, the compatibility between complex structures and Riemannian structures lead us naturally to the Kähler geometry. The compatibility between Riemannian structures and foliated structures determines the study of Riemannian foliations and so on.
In the paper under review the authors prove that the compatibility of two symplectic structures gives rise to a very rich geometry which contains as particular cases some old results due to Darboux and Lie.
The material is organized as follows. In §1 are described the nondegenerate bi-Hamiltonian mechanical systems which are situated in a neighborhood of an invariant torus. The bi-Lagrangian fibrations make the object of the following two sections. Finally in the last section all the above results are exemplified in the particular case of a 4-dimensional compact manifold, connected and without boundary.
In the paper under review the authors prove that the compatibility of two symplectic structures gives rise to a very rich geometry which contains as particular cases some old results due to Darboux and Lie.
The material is organized as follows. In §1 are described the nondegenerate bi-Hamiltonian mechanical systems which are situated in a neighborhood of an invariant torus. The bi-Lagrangian fibrations make the object of the following two sections. Finally in the last section all the above results are exemplified in the particular case of a 4-dimensional compact manifold, connected and without boundary.
Reviewer: M.Puta (Timişoara)
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
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