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A geometric analysis of dynamical systems with singular Lagrangians

Monika Havelková (2011)

Communications in Mathematics

We study dynamics of singular Lagrangian systems described by implicit differential equations from a geometric point of view using the exterior differential systems approach. We analyze a concrete Lagrangian previously studied by other authors by methods of Dirac’s constraint theory, and find its complete dynamics.

First steps in stable Hamiltonian topology

Kai Cieliebak, Evgeny Volkov (2015)

Journal of the European Mathematical Society

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on S 3 is homotopic to a stable Hamiltonian structure which cannot be embedded in 4 . Moreover, we derive a structure theorem for stable...

Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action

Frédéric Bourgeois, Alexandru Oancea (2010)

Journal of the European Mathematical Society

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the L 2 -gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic S 1 -invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define S 1 -equivariant...

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space ( W , Ω ¯ ) , having chosen a given reference...

Generalized PN manifolds and separation of variables

Fernand Pelletier, Patrick Cabau (2008)

Banach Center Publications

The notion of generalized PN manifold is a framework which allows one to get properties of first integrals of the associated bihamiltonian system: conditions of existence of a bi-abelian subalgebra obtained from the momentum map and characterization of such an algebra linked with the problem of separation of variables.

Geometric integrators for piecewise smooth Hamiltonian systems

Philippe Chartier, Erwan Faou (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we consider C1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that...

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

We show that if ( M , ω ) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of  ( M , ω ) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in  M × M when M satisfies...

Isomorphisms of Poisson and Jacobi brackets

Janusz Grabowski (2000)

Banach Center Publications

We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.

Length minimizing Hamiltonian paths for symplectically aspherical manifolds

Ely Kerman, François Lalonde (2003)

Annales de l’institut Fourier

In this note we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of Polterovich and Schwarz, we study the role, in the Floer complex of the generating Hamiltonian, of the global extrema which remain fixed as the time varies. Our main result determines a natural condition which implies that the corresponding path minimizes the positive Hofer length. We use this to prove that a quasi-autonomous Hamiltonian...

Maximal Hamiltonian tori for polygon spaces

Jean-Claude Hausmann, Susan Tolman (2003)

Annales de l’institut Fourier

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.

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