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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)

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI
[1] Coste, A.; Dazord, P.; Weinstein, A., Groupoïdes symplectiques, Dept. math. univ. Lyon 2/A, 1-62, (1987) · Zbl 0668.58017
[2] Dazord, P.; Lichnerowicz, A.; Marle, Ch.-M., Structures locale des variétés de Jacobi, J. math. pures et appl., 70, 101-152, (1991) · Zbl 0659.53033
[3] Henneaux, M., On the inverse problem of the calculus of variations, J. physics A, 15, L93-98, (1992)
[4] Kosmann-Schwarzbach, Y.; Magri, F., Poisson-Nijenhuis structures, Ann. inst. H. Poincaré série A (physique théorique), 53, 35-81, (1990) · Zbl 0707.58048
[5] Lichnerowicz, A., LES variétés de Poisson et leurs algèbres de Lie associées, J. diff. geometry, 12, 253-300, (1977) · Zbl 0405.53024
[6] MacKenzie, K., Lie groupoids and Lie algebroids in differential geometry, () · Zbl 0683.53029
[7] Morandi, G.; Ferrario, C.; Lo Vecchio, G.; Marmo, G.; Rubano, C., The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics reports, 188, 3-4, (1990) · Zbl 1211.58008
[8] Vaisman, I., Cohomology and differential forms, (1973), M. Dekker New York
[9] Vaisman, I., The Poisson-Nijenhuis manifolds revisited, Rendiconti sem. mat. Torino, 52, 377-394, (1994) · Zbl 0852.58042
[10] Weinstein, A., The local structure of Poisson manifolds, J. diff. geometry, 18, 523-557, (1983) · Zbl 0524.58011
[11] Weinstein, A., Lagrangian mechanics and groupoids, (1992), Univ. of Calif Berkeley, Preprint
[12] Yano, K.; Ishihara, S., Tangent and cotangent bundles, (1973), M. Dekker New York · Zbl 0262.53024
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