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Editors’ preface for the topical issue “Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics”

Andreas Čap, Vladimir Matveev, Karin Melnick, Galliano Valent (2012)

Open Mathematics

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000/2001)

Séminaire Équations aux dérivées partielles

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 $u$ solving the eikonal equation aėȧnd a probability measure $σ$ solving a related transport equation.We present some elementary formal identities relating certain quantum states $ψ$ and $u, σ$ . We show also how to build out of $u, σ$ an approximate...

Ehresmann Connection in the Geometry of Nonholonomic Systems

Aleksandar Bakša (2012)

Publications de l'Institut Mathématique

Elastic Problems and Optimal Control: Integrable Systems

Velimir Jurdjević (2013)

Zbornik Radova

Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy

Daisuke Tarama (2012)

Open Mathematics

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.

Francois Laudenbach (1995)

Commentarii mathematici Helvetici

Entropy and complexity of a path in sub-riemannian geometry

Frédéric Jean (2003)

ESAIM: Control, Optimisation and Calculus of Variations

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,...

Entropy and complexity of a path in sub-Riemannian geometry

Frédéric Jean (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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

Wojciech Domitrz, Stanisław Janeczko (1997)

Banach Center Publications

Equivalence problem for Lagrangians

Alois Švec (1988)

Czechoslovak Mathematical Journal

Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems

Eva Miranda, Nguyen Tien Zung (2004)

Annales scientifiques de l'École Normale Supérieure

Ergodic averages and free $ℤ^{2}$ actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

If the ergodic transformations S, T generate a free $ℤ^{2}$ action on a finite non-atomic measure space (X,S,µ) then for any $c_{1}, c_{2} \in ℝ$ there exists a measurable function f on X for which ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (S^{j} x) \to c_{1}$ and ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (T^{j} x) \to c_{2} µ$ -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.

Keith Burns, Marlies Gerber (1994)

Journal für die reine und angewandte Mathematik

Errata : «Classification géométrique des structures internes des systèmes dynamiques à 2 spins»

P. Iglesias (1983)

Annales de l'I.H.P. Physique théorique

Erratum for "Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature".

Ernesto A. Lacomba, J. Guadalupe Reyes (2000)

Publicacions Matemàtiques

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.

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