Gutkin, D. Variétés bi-structurées et opérateurs de récursion. (Manifolds with double structure and recursion operators). (French) Zbl 0587.58015 Ann. Inst. Henri Poincaré, Phys. Théor. 43, 349-357 (1985). We study the implications of the existence of a Poisson structure and a presymplectic structure on a manifold, for the complete integrability in Liouville’s sense of certain Hamiltonian vector fields. Under certain conditions, the recursion operator defined by these two structures generates first integrals in involution. Cited in 2 ReviewsCited in 4 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:Poisson structure and a presymplectic structure on a manifold; complete integrability; first integrals × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] F. Magri , A geometrical approach to the nonlinear solvable equations , Lecture Notes in Physics , t. 120 , Springer-Verlag , 1980 , p. 233 - 263 . MR 581899 [2] F. Magri , C. Morosi , A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds , preprint Milano Univ. , 1984 . [3] F. Magri , C. Morosi , O. Ragnisco , Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications . Communications in Mathematical Physics , t. 99 , 1985 , p. 115 - 140 . Article | MR 791643 | Zbl 0602.58017 · Zbl 0602.58017 · doi:10.1007/BF01466596 [4] G. Marmo , A geometrical characterization of completely integrable systems , in Proceedings of the meeting Geometry and Physics , Florence , 1982 , p. 257 - 262 . MR 760848 | Zbl 0548.58018 · Zbl 0548.58018 [5] S. De Filippo , G. Marmo , M. Salerno , G. Vilasi , A new characterization of completely integrable systems , preprint Salerno Univ. , 1983 . [6] De Filippo , M. Salerno , G. Vilasi . A geometrical approach to the integrability of soliton equations . Letters in Mathematical Physics , t. 9 , n^\circ 2 , 1985 , p. 85 - 91 . MR 785860 | Zbl 0586.35087 · Zbl 0586.35087 · doi:10.1007/BF00400704 [7] B. Fuchssteiner , The Lie algebra structure of degenerate Hamiltonian and bihamiltonian systems , Progress of Theoretical Physics , t. 68 , n^\circ 4 , 1982 , p. 1082 - 1104 . MR 688120 | Zbl 01662564 · Zbl 1098.37540 · doi:10.1143/PTP.68.1082 [8] A. Lichnerowicz . Les variétés de Poisson et leurs algèbres de Lie associées , Journal of Differential Geometry , t. 12 , 1977 , p. 253 - 300 . MR 501133 | Zbl 0405.53024 · Zbl 0405.53024 [9] C.M. Marle , Poisson manifolds in mechanics, in Bifurcation Theory, Mechanics and Physics , Reidel Publishing Company , 1983 , p. 47 - 76 . MR 726243 | Zbl 0525.58019 · Zbl 0525.58019 [10] Y. Kosmann-Schwarzbach , Cours de 3e cycle , Université de Lille , 1984 . [11] A. Weinstein , Symplectic manifolds and their Lagrangian submanifolds , Advances in Mathematics , t. 6 , 1971 , p. 329 - 346 . MR 286137 | Zbl 0213.48203 · Zbl 0213.48203 · doi:10.1016/0001-8708(71)90020-X [12] V.I. Arnold , Mathematical methods of classical mechanics , Springer-Verlag , New York , 1978 . MR 690288 | Zbl 0386.70001 · Zbl 0386.70001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.