The tiered Aubry set for autonomous Lagrangian functions

Marie-Claude Arnaud

Displaying similar documents to “The tiered Aubry set for autonomous Lagrangian functions”

Generic Nekhoroshev theory without small divisors

Abed Bounemoura, Laurent Niederman (2012)

Annales de l’institut Fourier

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In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new...

Formal geometric quantization

Paul-Émile Paradan (2009)

Annales de l’institut Fourier

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Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M, Ω)$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $Φ$ is proper so that the reduced space $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ is compact for all $μ$ . Then, we can define the “formal geometric quantization” of $M$ as $𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .$ The aim of this article is to study the functorial properties of the assignment $(M, K) \to 𝒬_{K}^{- \infty} (M)$ .

Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Dmitri V. Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, Fabrizio Pugliese (2012)

Annales de l’institut Fourier

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We study the geometry of multidimensional scalar $2^{n d}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $ℰ$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2 n + 1)$ -dimensional contact manifold $(M, 𝒞)$ . We develop the theory of characteristics of $ℰ$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $ℰ$ . After specializing such results to general...

Periodic conservative solutions of the Camassa–Holm equation

Helge Holden, Xavier Raynaud (2008)

Annales de l’institut Fourier

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We show that the periodic Camassa–Holm equation $u_{t} - u_{x x t} + 3 u u_{x} - 2 u_{x} u_{x x} - u u_{x x x} = 0$ possesses a global continuous semigroup of weak conservative solutions for initial data ${u |}_{t = 0}$ in $H_{per}^{1}$ . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $μ$ with $μ_{ac} = (u^{2} + u_{x}^{2}) d x$ . The total energy is preserved by the solution.

Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

Frédéric Faure (2007)

Annales de l’institut Fourier

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We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\hat{M}$ . The usual semi-classical Trace formula expresses $T r ({\hat{M}}^{t})$ for finite time $t$ , in the limit $ℏ \to 0$ , in terms of periodic orbits of $M$ of period $t$ . Recent work reach time $t ≪ t_{E} / 6$ where $t_{E} = log (1 / ℏ) / λ$ is the Ehrenfest time, and $λ$ is the Lyapounov coefficient. Using a semi-classical...