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

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

Fractal traps and fractional dynamics

Anomalous diffusion may arise in typical chaotic Hamiltonian systems. According to G.M. Zaslavsky's analysis, a description can be done by means of fractional kinetics equations. However, the dynamical origin of those fractional derivatives is still unclear. In this talk we study a general Hamiltonian dynamics restricted to a subset of the phase space. Starting from R. Hilfer's work, an expression for the average infinitesimal evolution of trajectories sets is given by using Poincar\'{e} recurrence times. The fractal traps within the phase space which are described by G.M. Zaslavsky are then taken into account, and it is shown that in this case, the derivative associated to this evolution may become fractional, with order equal to the transport exponent of the diffusion process

Homogeneous fractional embeddings

Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retain homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.Comment: 14 page

Dynamics of wrinkling in ultrathin elastic sheets

The wrinkling of thin elastic objects provides a means of generating regular patterning at small scales in applications ranging from photovoltaics to microfluidic devices. Static wrinkle patterns are known to be governed by an energetic balance between the object's bending stiffness and an effective substrate stiffness, which may originate from a true substrate stiffness or from tension and curvature along the wrinkles. Here we investigate dynamic wrinkling, induced by the impact of a solid sphere onto an ultra-thin polymer sheet floating on water. The vertical deflection of the sheet's centre induced by impact draws material radially inwards, resulting in an azimuthal compression that is relieved by the wrinkling of the entire sheet. We show that this wrinkling is truly dynamic, exhibiting features that are qualitatively different to those seen in quasi-static wrinkling experiments. Moreover, we show that the wrinkles coarsen dynamically because of the inhibiting effect of the fluid inertia. This dynamic coarsening can be understood heuristically as the result of a dynamic stiffness, which dominates the static stiffnesses reported thus far, and allows new controls of wrinkle wavelength.Comment: 8 pages, 4 figures. Please see published version for supplementary movies and SI Appendi

Variational integrators of fractional Lagrangian systems in the framework of discrete embeddings

International audienceThis paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a variational structure called Lagrangian structure. We are specially interested in the conservation at the discrete level of this Lagrangian structure by discrete embeddings. We then replace in this framework the variational integrators developed in [10, Chapter VI.6] and in [12]. Finally, we extend the notion of discrete embeddings and variational integrators to fractional Lagrangian systems

Variational integrator for fractional Euler–Lagrange equations

International audienceWe extend the notion of variational integrator for classical Euler-Lagrange equations to the fractional ones. As in the classical case, we prove that the variational integrator allows to preserve Noether-type results at the discrete level

La piscine de Pantin (1935-1937), une réalisation architecturale et sociale d’envergure

En 1935, à l’occasion du projet de construction d’une usine élévatoire et de traitement des eaux par la C.G.E, la mairie de Pantin décide d’édifier à proximité une piscine municipale. Elle serait alimentée par les eaux chaudes extraites d’un des puits avoisinants. Le maire de l’époque, Charles Auray, passe commande à son fils, jeune architecte de vingt-quatre ans et lui adjoint un ingénieur plus expérimenté, Jean Molinié. Les deux bâtiments possèdent une parenté architecturale certaine, paradoxalement accentuée par les dernières modifications effectuées sur les façades de l’usine. Tout en récapitulant les traits propres aux piscines de l’époque, la piscine de Pantin emprunte sans complexe aux réalisations d’une des personnalités les plus marquantes du mouvement moderne néerlandais, l’architecte Willem-Marinus DudokIn 1935, while the construction of a sewage pumping and treatment plant was planned by the administration of the French water utility, the town council of Pantin decided to have a municipal swimming pool built nearby. It would be supplied with warm waters drawn from one of the neighbouring wells. The mayor of the time, Charles Auray, commissioned his son, as a twenty-four-year-old architect, and appointed Jean Molinié, as a more experienced engineer, to him. The two buildings were definitely architecturally related, which would be paradoxically emphasized by the eventual alterations of the factory façades. Combining all the features characteristic of the contemporary swimming pools, the swimming pool of Pantin freely borrows from the works of one of the most outstanding personalities of the Dutch Modern Movement, the architect Willem-Marinus Dudok.Die Compagnie Générale des Eaux (Gesellschaft für Wasserversorgung) plant 1935 die Errichtung einer Pumpen- und Abwasserreinigungsanlage in Pantin. Sogleich entschließt sich die Stadtverwaltung in deren Nähe ein städtisches Schwimmbad zu bauen, welches mit Warmwasser durch eine nahe artesische Quelle versorgt werden soll. Der damalige Bürgermeister Charles Auray beauftragt damit seinen Sohn, einen 24 jährigen jungen Architekten, und den erfahrenen Bauingenieur Jean Molinié als Mitarbeiter. Die Kläranlage und das Schwimmbad führen eine gewisse architektonische Verwandschaft vor Augen, die besonders deutlich wird nach den Veränderungen an den Fassaden der Abwasserreinigungsanlage. An dem Schwimmbad von Pantin lassen sich zwar alle Charakterzüge der zeitgenössischen Schwimmanstalten erkennen, aber auch der Einfluss der Werke des Architekten Willem-Marinus Dudok, einer bedeutenden Figur der niederländischen modernen Architektu