Editors’ preface for the topical issue “Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics”
Effective Hamiltonians and Quantum States
We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function solving the eikonal equation aėȧnd a probability measure solving a related transport equation.We present some elementary formal identities relating certain quantum states and . We show also how to build out of an approximate...
Ehresmann Connection in the Geometry of Nonholonomic Systems
Elastic Problems and Optimal Control: Integrable Systems
Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy
This note deals with Lagrangian fibrations of elliptic K3 surfaces and the associated Hamiltonian monodromy. The fibration is constructed through the Weierstraß normal form of elliptic surfaces. There is given an example of K3 dynamical models with the identity monodromy matrix around 12 elementary singular loci.
Engouffrement symplectique et intersections lagrangiennes.
Entropy and complexity of a path in sub-riemannian geometry
We characterize the geometry of a path in a sub-riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset of a metric space is the minimum number of balls of a given radius needed to cover . It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-riemannian manifold as the infimum of the lengths of all trajectories contained in an -neighborhood of the path,...
Entropy and complexity of a path in sub-Riemannian geometry
We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ε-neighborhood of the path,...
Equivalence of Lagrangian germs in the presence of a surface
Equivalence problem for Lagrangians
Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems
Ergodic averages and free actions
If the ergodic transformations S, T generate a free action on a finite non-atomic measure space (X,S,µ) then for any there exists a measurable function f on X for which and -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.
Ergodic geodesic flows on product manifolds with low dimensional factors.
Errata : «Classification géométrique des structures internes des systèmes dynamiques à 2 spins»
Erratum for "Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature".
A mistake was found in the reasoning leading to a Lagrangian which we considered as equivalent from the formula for the action S(γ) below the classical mechanical problem (3) on "Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature", page 271.
Erratum : “On the geometry of subbundles and Lagrange foliations”
Espaces de Krein et index des systèmes hamiltoniens
Estimates for the energy of a symplectic map.
Estimation de l'entropie des systèmes lagrangiens sans points conjugués
Exemples de hamiltoniens non intégrables en mécanique analytique réelle