Fractional embedding of differential operators and Lagrangian systems

This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the stochastic embedding theory developed with Darses [C. R. Acad. Sci. Ser. I: Math 342, 333 (2006); (preprint IHES 06/27, p. 87, 2006)], we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provides a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e., the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski...

[1]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[2]  J. Cresson,et al.  Théorème de Noether stochastique , 2007 .

[3]  Delfim F. M. Torres,et al.  A formulation of Noether's theorem for fractional problems of the calculus of variations , 2007, Journal of Mathematical Analysis and Applications.

[4]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[5]  Sébastien Darses,et al.  Stochastic derivatives for fractional diffusions. , 2006, math/0604315.

[6]  Aleksander Stanislavsky,et al.  Hamiltonian formalism of fractional systems , 2006, 1111.3064.

[7]  J. Cresson,et al.  Plongement stochastique des systèmes lagrangiens , 2005, math/0510655.

[8]  D. Baleanu,et al.  Formulation of Hamiltonian Equations for Fractional Variational Problems , 2005, math-ph/0510029.

[9]  Sébastien Darses,et al.  Stochastic embedding of dynamical systems , 2005, math/0509713.

[10]  M. Combescure.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[11]  Dumitru Baleanu,et al.  Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives , 2005 .

[12]  Jacky Cresson,et al.  Fractional differential equations and the Schrödinger equation , 2005, Appl. Math. Comput..

[13]  J. Cresson.  Non-differentiable variational principles , 2004, math/0410377.

[14]  Jacky Cresson,et al.  Quantum derivatives and the Schrödinger equation , 2004 .

[15]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[16]  J. Cresson.  Scale calculus and the Schrödinger equation , 2002, math/0211071.

[17]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[18]  John Dauns,et al.  COHERENCE , 2002 .

[19]  Jacky Cresson,et al.  About Non-differentiable Functions , 2001 .

[20]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[21]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[22]  Frederick E. Riewe,et al.  Mechanics with fractional derivatives , 1997 .

[23]  Compte,et al.  Stochastic foundations of fractional dynamics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Riewe,et al.  Nonconservative Lagrangian and Hamiltonian mechanics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[26]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[27]  Laurent Nottale,et al.  Fractal Space-Time And Microphysics: Towards A Theory Of Scale Relativity , 1993 .

[28]  G. W. Stewart,et al.  Stochastic Perturbation Theory , 1990, SIAM Rev..

[29]  J. Wisdom.  Chaotic behavior in the solar system , 1987 .

[30]  Bruce J. West,et al.  Lévy dynamics of enhanced diffusion: Application to turbulence. , 1987, Physical review letters.

[31]  W. Wyss.  The fractional diffusion equation , 1986 .

[32]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[33]  H. Bateman.  On Dissipative Systems and Related Variational Principles , 1931 .

[34]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[35]  G. Serra.  Les théorèmes de Noether , 2009 .

[36]  G. M. Zaslavskii.  Hamiltonian chaos and fractional dynamics , 2005 .

[37]  Clovis Darrigan,et al.  Université de Pau et des Pays de l'Adour , 1998 .

[38]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[39]  L. Nottale.  Fractal space-time and microphysics , 1993 .

[40]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[41]  Philip J. Morrison,et al.  A paradigm for jointed Hamiltonian and dissipative systems , 1986 .

[42]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[43]  A. Weil.  The extension problem , 1971 .

[44]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.