Fractional embedding of differential operators and Lagrangian systems

Journal of Mathematical Physics

Volumn 48, Issue 3, 2007, Pages

  Cresson, Jacky ^a,b  

^a UNIVERSITÉ DE PAU ET DES PAYS DE L'ADOUR   (France)

^b INSTITUT DES HAUTES ÉTUDES SCIENTIFIQUES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34047097242     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2483292     Document Type: Article

References (49)

1. Agrawal, O. P., "Formulation of Euler-Lagrange equations for fractional variational problems," J. Math. Anal. Appl. 272, 368-379 (2002).
2. Andler, M. and Leray, J. Proc. Am. Philos. Soc. 144 (2000).
3. Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, New York, 1989).
4. Avez, A., Calcul Différentiel (Masson, Paris, 1983).
5. Bateman, H., "On dissipative systems and related variational problems," Phys. Rev. 38, 815-819 (1931).
6. Bauer, P. S., Proc. Natl. Acad. Sci. U.S.A. 17, 311 (1931).
7. Ben Adda, F. and Cresson, J., "About non-differentiable functions," Appl. Math. Comput. 263, 721-737 (2001).
8. Ben Adda, F. and Cresson, J., "Fractional differential equations and the Schrödinger equation," Acta Math. Acad. Sci. Hung. 161, 323-345 (2005).
9. Ben Adda, F. and Cresson, J., "Quantum derivatives and the Schrödinger equation," Chaos, Solitons Fractals 19, 1323-1334 (2004).
10. Carpinteri, A. and Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (Springer-Verlag, Wien, 1997).
11. Carreras, B. A., Lynch, V. E., and Zaslavsky, G. M., "Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models," Phys. Plasmas 8, 5096-5103 (2001).
12. Comte, F., "Opérateurs fractionnaires en Econométrie et en Finance" (unpublished).
13. Compte, A., "Stochastic foundations of fractional dynamics," Phys. Rev. E 53, 4191-4193 (1996).
14. Cresson, J., "Non-differentiable variational principles," J. Math. Anal. Appl. 307, 48-64 (2005).
15. Cresson, J., "Scale calculus and the Schrödinger equation," J. Math. Phys. 44, 4907-4938 (2003).
16. Cresson, J., "Théories de plongeaient des systèmes dynamiques - Un programme," p. 21 (2005).
17. Cresson, J., "Quantum embedding and partial differential equations" (unpublished).
18. Cresson, J. and Darses, S., "Plongement stochastique des systèmes dynamiques," C. R. Acad. Sci., Ser. I: Math. 342, 333-336 (2006).
19. Cresson, J. and Darses, S., "Stochastic embedding of dynamical systems," preprint IHES 06/27, p. 87 (2006).
20. Cresson, J. and Darses, S., "Stochastic perturbation theory" (unpublished).
21. Cresson, J. and Darses, S., "Théorème de Noether stochastique," in C. R. Acad. Sci., Ser. I: Math, (to be published).
22. Darses, S. and Nourdin, I., "Stochastic derivatives for fractional diffusions," Ann. Probab. (to be published).
23. Erwin, V. J. and Roop, J. P., "Variational formulation for the stationary fractional advection dispersion equation," Numer. Methods Partial Differ. Equ. 22, 558-576 (2006).
24. Frederico, G. and Torres, D., "A formulation of Noether's theorem for fractional problems of the calculus of variations," Report, 2005 (unpublished).
25. Gelfand, I. M. and Fomin, S. V., Calculus of Variations (Dover, New York, 2000).
26. Gorenflo, R. and Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (Ref. 10), pp. 223-276.
27. Henry, B. I. and Wearne, S. L., "Fractional reaction-diffusion, " Physica A 276, 448-455 (2000).
28. Kosmann-Scwarzback, Y., Les Théorèmes de Noether; invariance et lois de conservation au XX ème siécle (Ecole Polytechnique, Paris, 2004).
29. Leray, J., "Sur le mouvement d'un liquide visqueux emplissant l'espace," Acta Math. 63, 193-248 (1934).
30. Marmi, S., "Chaotic behavior in the Solar system (following J. Laskar)" (unpublished).
31. Le Mehaute, A., Nigmatullin, R. R., and Nivanen, R., Flèches du temps et géométries/raciales (Hermes, Paris, 1998).
32. Miller, K. S. and Ross, B., An Introduction to Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993).
33. Morrison, P. J., "A paradigm for joined Hamiltonian and dissipative systems," Physica D 18, 410-419 (1986).
34. Muslih, S. I. and Baleanu, D., "Formulation of Hamiltonian equations for fractional variational problems," e-print math-ph/0510029.
35. Muslih, S. I. and Baleanu, D., "Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives," J. Math. Anal. Appl. 304, 599-606 (2005).
36. Nottale, L., Fractal Space-Time and Microphysics (World Scientific, Singapore, 1993).
37. Oldham, K. B. and Spanier, J., The Fractional Calculus (Academic, New York, 1974).
38. Podlubny, I., Fractional Differential Equations (Academic, New York, 1999).
39. Riewe, F., "Mechanics with fractional derivatives," Phys. Rev. E 55, 3581-3592 (1997).
40. Riewe, F., "Nonconservative Lagrangian and Hamiltonian mechanics," Phys. Rev. E 53, 1890-1899 (1996).
41. Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1998).
42. Schneider, W. R. and Wyss, W., "Fractional diffusion and wave equations," J. Math. Phys. 30, 134-144 (1989).
43. Shlesinger, M. F., West, B. J., and Klafter, J., "Lévy dynamics of enhanced diffusion: Application to turbulence," Phys. Rev. Lett. 58, 1100-1103 (1987).
44. Stanislavsky, A. A., "Hamiltonian formalism of fractional systems," Eur. Phys. J. B 49, 93-101 (2006).
45. Tolman, R. C., Statistical Mechanics (Chemical Catalog, New York, 1927).
46. Widder, D. V., Advanced Calculus, 2nd ed. (Dover, New York, 1961).
47. Wyss, W. "The fractional diffusion equation," J. Math. Phys. 27, 2782-2785 (1986).
48. Zaslavsky, G. M., "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371, 461-580 (2002).
49. Zaslavsky, G. M., Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, New York, 2005).