Geodesic
Hamiltonian structures of the~first variation equations and symplectic connections
Sbornik. Mathematics, Tome 191 (2000) no. 4, pp. 477-502

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Necessary and sufficient conditions in terms of symplectic connections, ensuring that the first variation equation of a Hamiltonian system along a fixed invariant symplectic submanifold is also a Hamiltonian system with respect to some admissible symplectic structure are obtained. The class of admissible symplectic structures is distinguished by means of the natural condition of compatibility with the symplectic 2-form in the ambient space. Possible obstructions to the existence of a Hamiltonian structure on the first variation equation are investigated. 
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     author = {Yu. M. Vorob'ev},
     title = {Hamiltonian structures of the~first variation equations and symplectic connections},
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Yu. M. Vorob'ev. Hamiltonian structures of the~first variation equations and symplectic connections. Sbornik. Mathematics, Tome 191 (2000) no. 4, pp. 477-502. http://geodesic.mathdoc.fr/item/SM_2000_191_4_a0/