Jacky Cresson

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...

Non-differentiable variational principles

Abstract We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler–Lagrange equation. We finally prove that solutions of the Schrodinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamiltons principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space–time.

Stochastic embedding of dynamical systems

Most physical systems are modeled by an ordinary or a partial differential equation, like the n-body problem in celestial mechanics. In some cases, for example, when studying the long term behavior of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modeled or even known. One way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic. In this paper, we develop a theory for the stochastic embedding of ordinary differential equations. We apply this method to Lagrangian systems. In this particular case, we extend many results of classical mechanics, namely, the least action principle, the Euler-Lagrange equations, and Noether’s theorem. We also obtain a Hamiltonian formulation for our stochastic Lagrangian systems. Many applications are discussed at the end of the paper.

Constants of motion for non-differentiable quantum variational problems

We extend the DuBois-Reymond necessary optimality condition and Noethers symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noethers theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrodinger equation.

Scale calculus and the Schrödinger equation

This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,e) such that f(t,e)→f(t) when e goes to zero. This led to several new notions as representations: fractal functions and e-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schrodinger equation v...

Fractional differential equations and the Schrödinger equation

In a previous paper, we defined, following a previous work of Kolvankar and Gangal, a notion of @a-derivative, 0<@a<1. In this article, we study @a-differential equations associated to our fractional calculus. We then discuss a fundamental problem concerning the Schrodinger equation in the framework of Nottales scale relativity theory.

Quantum derivatives and the Schrödinger equation

Abstract We define a scale derivative for non-differentiable functions. It is constructed via quantum derivatives which take into account non-differentiability and the existence of a minimal resolution for mean representation. This justify heuristic computations made by Nottale in scale-relativity. In particular, the Schrodinger equation is derived via the scale-relativity principle and Newton’s fundamental equation of dynamics.

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolictorus intersec t transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometricodynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolicperiodicorbits near the given torus. The relevanc e of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolicsets. r 2002 Elsevier Science (USA). All rights reserved.

Time scale differential, integral, and variational embeddings of Lagrangian systems

We introduce differential, integral, and variational delta embeddings. We prove that the integral delta embedding of the Euler-Lagrange equations and the variational delta embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least-action principle is obtained.

Scale relativity theory for one-dimensional non-differentiable manifolds

Abstract We discuss a rigorous foundation of the pure scale relativity theory for a one-dimensional space variable. We define several notions as “representation” of a continuous function, scale law and minimal resolution. We define precisely the meaning of a scale reference system and space reference system for non-differentiable one-dimensional manifolds.