Generalized Euler-Lagrange equations for variational problems with scale derivatives

We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.Comment: Submitted on 03-Aug-2009; accepted for publication 16-March-2010; in "Letters in Mathematical Physics"

About analytic non integrability

We prove several general results on non existence of analytic first integrals for analytic diffeomorphisms possessing a hyperbolic fixed point.Comment: 14 page

Multiscale functions, Scale dynamics and Applications to partial differential equations

Modeling phenomena from experimental data, always begin with a \emph{choice of hypothesis} on the observed dynamics such as \emph{determinism}, \emph{randomness}, \emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \emph{"With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \emph{multi-scale functions}, \emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale calculus} (see \cite{bohn}). These definitions will be illustrated on the \emph{multi-scale Okamoto's functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \emph{scale regimes} whereas the equation is \emph{scale invariant}. A typical example of such an equation, is the \emph{Euler-Lagrange equation} and particularly the \emph{Newton's equation} which will be discussed. Notably, we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton's equation} and also the \emph{non-linear Schr\"odinger equation} via the \emph{scale Newton's equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schr\"odinger equation}

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.Comment: 11 page

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.Comment: 112 page

Hyperboliity versus partial-hyperbolicity and the transversality-torsion phenomenon

In this paper, we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrres of freedom Hamiltonian systems: the transversality-torsion phenomenon.Comment: 10 page

Irreversibility, least action principle and causality

The least action principle, through its variational formulation, possesses a finalist aspect. It explicitly appears in the fractional calculus framework, where Euler-Lagrange equations obtained so far violate the causality principle. In order to clarify the relation between those two principles, we firstly remark that the derivatives used to described causal physical phenomena are in fact left ones. This leads to a formal approach of irreversible dynamics, where forward and backward temporal evolutions are decoupled. This formalism is then integrated to the Lagrangian systems, through a particular embedding procedure. In this set-up, the application of the least action principle leads to distinguishing trajectories and variations dynamical status. More precisely, when trajectories and variations time arrows are opposed, we prove that the least action principle provides causal Euler-Lagrange equations, even in the fractional case. Furthermore, the embedding developped is coherent.Comment: 14 page