On the geometric approach to the motion of inertial mechanical systems

Journal of Physics A: Mathematical and General

Volumn 35, Issue 32, 2002, Pages

  Constantin, Adrian ^a   Kolev, Boris ^b  

^a LUND UNIVERSITY   (Sweden)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042279206     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/35/32/201     Document Type: Review

References (42)

1. Arnold V 1966 Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits Ann. Inst. Fourier (Grenoble) 16 319-61
2. Arnold V 1989 Mathematical Methods of Classical Mechanics (New York: Springer)
3. Arnold V and Khesin B 1998 Topological Methods in Hydrodynamics (New York: Springer)
4. Barenblatt G I 1996 Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge: Cambridge University Press)
5. Beals R, Sattinger D and Szmigielski J 1999 Multipeakons and a theorem of Stieltjes Inverse Probl. 15 1-4
6. Brezis H and Bourguignon J P 1974 Remarks on the Euler equation J. Funct. Anal. 15 341-63
7. Burgers J 1948 A mathematical model illustrating the theory of turbulence Adv. Appl. Mech. 1 171-99
8. Camassa R and Holm D 1993 An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661-4
9. Camassa R, Holm D and Hyman J 1994 A new integrable shallow water equation Adv. Appl. Mech. 311-33
10. Chen S, Foias C, Olson E, Titi E and Wynne S 1999 A connection bewteen the Camassa-Holm equations and turbulent flows in channels and pipes Phys. Fluids 11 2343-53
11. Cioranescu D and Girault V 1996 Solutions variationnelles et classiques d'une famille de fluides de grade deux C. R. Acad. Sci., Paris 322 1163-8
12. Constantin A 2000 Existence of permanent and breaking waves for a shallow water equation: a geometric approach Ann. Inst. Fourier (Grenoble) 50 321-62
13. Constantin A and Escher J 1998 Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation Commun. Pure Appl. Math. 51 475-504
14. Constantin A and Escher J 2000 On the blow-up rate and the blow-up set of breaking waves for a shallow water equation Math. Z. 233 75-91
15. Constantin A and McKean H P 1999 A shallow water equation on the circle Commun. Pure Appl. Math. 52 949-82
16. Constantin A and Molinet L 2000 Global weak solutions for a shallow water equation Commun. Math. Phys. 211 45-61
17. Constantin A and Strauss W 2000 Stability of peakons Commun. Pure Appl. Math. 53 603-10
18. Dai H-H 1998 Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod Acta Mech. 127 293-308
19. Dieudonné J 1969 Foundations of Modern Analysis (New York: Academic)
20. Drazin P and Johnson R 1992 Solitons: An Introduction (Cambridge: Cambridge University Press)
21. Dunn J and Fosdick R 1974 Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade Arch. Ration. Mech. Anal. 56 191-252
22. Ebin D and Marsden J E 1970 Groups of diffeomorphisms and the notion of an incompressible fluid Ann. Math. 92 102-63
23. Ebin D, Marsden J E and Fischer A 1972 Diffeomorphism groups, hydrodynamics and relativity and the notion of an incompressible fluid Proc. 13th Biennal Seminar of Canad. Math. Congress (Montreal) pp 135-279
24. Fokas A S and Fuchssteiner B 1981 Symplectic structures, their Bäcklund transformation and hereditary symmetries Physica D 4 47-66
25. Gilson C and Pickering A 1995 Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations J. Phys. A: Math. Gen. 28 2871-88
26. Hamilton R 1982 The inverse function theorem of Nash and Moser Bull. Am. Math. Soc. 7 65-222
27. Holm D, Marsden J and Ratiu T 1998 The Euler-Poincari equations and semidirect products with applications to continuum theories Adv. Math. 137 1-81
28. Hörmander L 1997 Lectures on Nonlinear Hyperbolic Differential Equations (Berlin: Springer)
29. Johnson R 2002 Camassa-Holm, Korteweg-de Vries and related models for water waves J. Fluid Mech. 455 63-82
30. Kato T 1975 The Cauchy problem for quasi-linear symmetric hyperbolic systems Arch. Ration. Mech. Anal. 58 181-205
31. Kato T 1975 Quasi-linear equations of evolution, with applications to partial differential equations Spectral Theory and Differential Equations (Springer Lecture Notes in Mathematics vol 448) (Berlin: Springer) pp 25-70
32. Lang S 1999 Fundamentals of Differential Geometry (New York: Springer)
33. McKean H P 1979 Integrable systems and algebraic curves Global Analysis (Springer Lecture Notes in Mathematics vol 755) (Berlin: Springer) pp 83-200
34. McKean H P 1998 Breakdown of a shallow water equation Asian J. Math. 2 203-8
35. Milnor J 1984 Remarks on infinite-dimensional Lie groups Relativity, Groups and Topology (Les Houches, 1983) (Amsterdam: North-Holland) pp 1009-57
36. Misiolek G 1998 A shallow water equation as a geodesic flow on the Bott-Virasoro group J. Geom. Phys. 24 203-8
37. Olver P 1993 Applications of Lie Groups to Differential Equations (New York: Springer)
38. Ono T 1995 Riemannian geometry of the motion of an ideal incompressible magnetohydrodynamical fluid Phys. D 81 207-20
39. Palais R 1997 The symmetries of solitons Bull. Am. Math. Soc. (N.S.) 34 339-403
40. Shnirelman A 1987 The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid Matem. Sb. 128 79-105
41. Talman R 2000 Geometric Mechanics (New York: Wiley)
42. Zeitlin V and Pasmanter R 1994 On the differential geometry approach to the geophysical flows Phys. Lett. A 189 59-63