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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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