Searcharxiv⌕ Search

arXiv subjects

Jacky Cresson

Publications and source records attributed to Jacky Cresson.

54 records · Page 3Linked to original sources
arXiv2008

Fractional embeddings and stochastic time

As a model problem for the study of chaotic Hamiltonian systems, we look for the effects of a long-tail distribution of recurrence times on a fixed Hamiltonian dynamics. We follow Stanislavsky's approach of Hamiltonian formalism for fractional systems. We prove that his formalism can be retrieved from the fractional embedding theory. We deduce that the fractional Hamiltonian systems of Stanislavsky stem from a particular least action principle, said causal. In this case, the fractional embedding becomes coherent.

math.DS↗
arXiv2008

Constants of Motion for Non-Differentiable Quantum Variational Problems

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's 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.

math-ph↗
arXiv2008

Mould Calculus for Hamiltonian Vector Fields

We present the general framework of Écalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then Écalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then prove a Kolmogorov theorem on Hamiltonian vector fields near a diophantine torus in action-angle coordinates using moulds techniques.

math.DS↗
arXiv2007

On the first integral conjecture of Rene Thom

More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word \textquotedblleft generically\textquotedblright, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class $C^{k}$ with $k\geq d,$ where $d$ is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the $C^{1}$ sense.

math.DS↗
arXiv2007

Phénomènes de symétrie dans des formes linéaires en polyzêtas

We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $ζ$ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric type, which can be written as linear forms in some specific multiple zeta values. The proof makes use of the regularization procedure for multiple zeta values with logarithmic divergence.

math.NT↗
arXiv2006

Séries hypergéométriques multiples et polyzêtas

We describe a theoretical and effective algorithm which enables us to prove that rather general hypergeometric series and integrals can be decomposed as linear combinations of multiple zeta values, with rational coefficients.

math.NT↗
arXiv2006

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 S. Darses, 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 provide 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. that the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of F. Riewe, we obtain a fractional Ostrogradski formalism which allows us to derive non-conservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equation and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss as well as the fractional diffusion equation.

math.DS↗
arXiv2006

About the Trimmed and the Poincare-Dulac normal form of diffeomorphisms

We study two particular continuous prenormal forms as defined by Jean Ecalle and Bruno Vallet for local analytic diffeomorphism: the Trimmed form and the Poincare-Dulac normal form. We first give a self-contain introduction to the mould formalism of Jean Ecalle. We provide a dictionary between moulds and the classical Lie algebraic formalism using non-commutative formal power series. We then give full proofs and details for results announced by J. Ecalle and B. Vallet about the Trimmed form of diffeomorphisms. We then discuss a mould approach to the classical Poincare-Dulac normal form of diffeomorphisms. We discuss the universal character of moulds taking place in normalization problems.

math.DS↗
arXiv2005

Plongement stochastique des systèmes lagrangiens

We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding lagrangian systems, we obtain a stochastic Euler-Lagrange equation which, in the case of natural lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation.

math.PR↗
arXiv2005

Lemme de coherence et théorème de Noether stochastique

The stochastic embedding procedure associates a stochastic Euler-Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral.

math.PR↗
arXiv2005

Stochastic embedding of dynamical systems

Most physical systems are modelled 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 behaviour of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modelled 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.

math.PR↗
arXiv2005

Dynamique des nombres et physique des oscillateurs

We consider the superheterodyning system discovered by Armstrong and Schottky in 1924. This system is the basic piece of any communication system. We prove that the frequency spectrum of this system is governed by arithmetical rule. We provide a number theoritical framework which allows us to predict all the particular features of the experimental frequency spectrum. We also introduce a new natural dynamical system on numbers and study its first properties.

math-ph↗
arXiv2005

About analytic non integrability

We prove several general results on non existence of analytic first integrals for analytic diffeomorphisms possessing a hyperbolic fixed point.

math.DS↗
arXiv2005

Calcul Moulien

This paper is an introduction to mould calculus as introduced by Jean Ecalle. We give a precise definition of moulds and describe there main properties. We translate mould symmetries (alterna(e)l and symetra(e)l) using non commutative formal power series in two given bialgebras. We apply this formalism to the problem of normal forms for vector fields and diffeomrophisms.

math.DS↗
arXiv2004

Non-differentiable variational principles

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 Schrödinger equation can be obtained as extremals of a non differentiable variational principle, leading to an extended Hamilton's 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.

math.GM↗
arXiv2002

Scale calculus and the Schrodinger equation

We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal resolution, and quantum representation of a non differentiable function. We then define a scale quantization procedure for classical Lagrangian systems inspired by the Scale relativity theory developped by Nottale. We prove that the scale quantization of Newtionian mechanics is a non linear Schrodinger equation. Under some specific assumptions, we obtain the classical linear Schrodinger equation.

math.GM↗