Lacomba, Ernesto; Losco, Lucette Caractérisation variationnelle globale des flots canoniques et de contact dans leurs groupes de difféomorphismes. (Global variational characterization of canonical and contact flows through their diffeomorphism groups). (French) Zbl 0605.58021 Ann. Inst. Henri Poincaré, Phys. Théor. 45, 99-116 (1986). The main theorem generalizes V. I. Arnol’d’s result [Ann. Inst. Fourier 16, 319-361 (1966; Zbl 0148.453)] for Euler equations in hydrodynamics to arbitrary Riemannian manifolds M with boundary. The standard variational principle (integral of action over M) is shown to be equivalent to a ”global” variational principle (integral over the group of diffeomorphisms). In terms of the latter, a characterization of Hamiltonian and contact vector fields is given. Reviewer: S.V.Duzhin MSC: 58E30 Variational principles in infinite-dimensional spaces 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 76B99 Incompressible inviscid fluids Keywords:variational principles; Euler equations; hydrodynamics; Riemannian manifolds; group of diffeomorphisms; Hamiltonian and contact vector fields Citations:Zbl 0148.453 PDFBibTeX XMLCite \textit{E. Lacomba} and \textit{L. Losco}, Ann. Inst. Henri Poincaré, Phys. Théor. 45, 99--116 (1986; Zbl 0605.58021) Full Text: Numdam EuDML References: [1] R. Abraham , J. Marsden , Foundations of Mechanics , 2 nd. Ed. Benjamin-Cummings , New York , 1978 . MR 515141 | Zbl 0393.70001 · Zbl 0393.70001 [2] V. Arnold , Ann. Inst. Fourier ( Grenoble ), t. 16 , 1966 , p. 319 - 361 . Numdam | Zbl 0148.45301 · Zbl 0148.45301 · doi:10.5802/aif.233 [3] V. Arnold , Journal de Mécanique , t. 5 , n^\circ 1 , 1966 , p. 29 - 43 . Zbl 0161.22903 · Zbl 0161.22903 [4] V. Arnold , Mathematical Methods of Classical Mechanics . Springer-Verlag , New York , 1978 . MR 690288 | Zbl 0386.70001 · Zbl 0386.70001 [5] D. Ebin , J. Marsden , Ann. of Math. , t. 92 , 1970 , p. 102 - 163 . MR 271984 | Zbl 0211.57401 · Zbl 0211.57401 · doi:10.2307/1970699 [6] C. Godbillon , Géométrie Différentielle et Mécanique Analytique . Hermann , Paris , 1969 . MR 242081 | Zbl 0174.24602 · Zbl 0174.24602 [7] E. Lacomba , Symplectic Geometry , Crumeyrolle, Grifone, Eds., Pitman Books , Londres , 1982 , p. 66 - 75 . MR 712162 [8] E. Lacomba , L. Losco , Group Theoretical Methods in Physics , K. B. Wolf, Ed., Lecture Notes in Physics , t. 135 , Springer Verlag , New York , 1980 , p. 122 - 128 . MR 651523 · Zbl 0446.58007 [9] E. Lacomba , L. Losco , Physica , t. 114 A, 1982 , p. 124 - 128 . MR 678393 | Zbl 0512.58017 · Zbl 0512.58017 · doi:10.1016/0378-4371(82)90270-9 [10] J. Leslie , Topology , t. 6 , 1967 , p. 263 - 271 . MR 210147 | Zbl 0147.23601 · Zbl 0147.23601 · doi:10.1016/0040-9383(67)90038-9 [11] L. Losco , C. R. Acad. Sci. Paris , t. 278 A, 1974 , p. 1641 - 1643 . MR 350777 | Zbl 0286.58008 · Zbl 0286.58008 [12] J. Marsden , R. Abraham , Proc. Sympos. Pure Math. , t. 16 , Amer. Math. Soc. , Providence, R. I. , 1970 , p. 237 - 243 . Zbl 0211.57402 · Zbl 0211.57402 [13] H. Omori , Proc. Sympos. Pure Math. , t. 15 , Amer. Math. Soc. , Providence, R. I ., 1970 , p. 167 - 184 . MR 271983 | Zbl 0214.48805 · Zbl 0214.48805 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.