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Second order Hamiltonian vector fields on tangent bundles. (English) Zbl 0822.58021
Let $$M$$ be a differentiable manifold. A vector field $$\Gamma$$ of $$TM$$ which corresponds to a system of second order ordinary differential equations on $$M$$ is called a second order Hamiltonian vector field if it is the Hamiltonian field of a function $$F \in C^ \infty(TM)$$ with respect to a Poisson structure $$P$$ of $$TM$$. We formulate the direct problem as that of finding $$\Gamma$$ if $$P$$ is given, and the inverse problem as that of finding $$P$$ if $$\Gamma$$ is given. We show the solution of the direct problem if $$P$$ is a symplectic structure such that the fibers of $$TM$$ are Lagrangian submanifolds. For the inverse problem we generalize Henneaux’ method of looking for a solution by studying a corresponding system of linear algebraic restrictions.
Reviewer: I.Vaisman (Haifa)

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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##### References:
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