×

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

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

Citations:

Zbl 0148.453
PDFBibTeX XMLCite
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.