Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows

Ghoussoub, Nassif; Tzou, Leo
August 2006
Calculus of Variations & Partial Differential Equations;Aug2006, Vol. 26 Issue 4, p511
Academic Journal
Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex�concave Hamiltonians $${\cal H}$$ on a�possibly infinite dimensional�symplectic space H 2, the existence of a solution for the Hamiltonian system $$-J\dot u (t)\in \partial {\cal H}(u(t))$$ that connects in a given time T > 0, two Lagrangian submanifolds. Another application deals with the construction of multiparameter flows, including those generated by vector fields that represent superpositions of skew-adjoint operators with gradients of convex potentials. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [5�7].


Related Articles

  • Generic Dynamics of 4-Dimensional C 2 Hamiltonian Systems. Bessa, M�rio; Dias, Jo�o Lopes // Communications in Mathematical Physics;Jul2008, Vol. 281 Issue 3, p597 

    We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C 2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov...

  • HEEGAARD GRADIENT OF SEIFERT FIBERED 3-MANIFOLDS. KAZUHIRO ICHIHARA // Bulletin of the London Mathematical Society;Jul2004, Vol. 36 Issue 4, p537 

    The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach towards proving the well-known virtually Haken conjecture. As instructive examples, Seifert fibered 3-manifolds are considered in this paper. The author shows that a compact orientable Seifert...

  • Determining quasidiabatic coupled electronic state Hamiltonians using derivative couplings: A normal equations based method. Papas, Brian N.; Schuurman, Michael S.; Yarkony, David R. // Journal of Chemical Physics;9/28/2008, Vol. 129 Issue 12, p124104 

    A self-consistent procedure for constructing a quasidiabatic Hamiltonian representing Nstate coupled electronic states in the vicinity of an arbitrary point in nuclear coordinate space is described. The matrix elements of the Hamiltonian are polynomials of arbitrary order. Employing a crude...

  • Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation. Ahmad, Tahir; Ken, Tan Lit // Journal of Mathematics & Statistics;2011, Vol. 7 Issue 3, p207 

    Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows...

  • Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants. Prykarpats'kyi, Ya. A. // Ukrainian Mathematical Journal;Mar2008, Vol. 60 Issue 3, p441 

    We study the differential-geometric properties of Hamiltonian connections on symplectic manifolds for adiabatically perturbed Hamiltonian systems. In particular, an associated Hamiltonian connection is constructed on the principal fibration. Its description is given in terms of covariant...

  • Limit Cycles Bifurcated from Some Z4-Equivariant Quintic Near-Hamiltonian Systems. Simin Qu; Cangxin Tang; Fengli Huang; Xianbo Sun // Abstract & Applied Analysis;2014, p1 

    We study the number and distribution of limit cycles of some planar Z4-equivariant quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the perturbed system can have 24 limit cycles with some new distributions. The configurations of limit...

  • On the Linearization of Hamiltonian Systems on Poisson Manifolds. Vorob'ev, Yu. M. // Mathematical Notes;Sep/Oct2005, Vol. 78 Issue 3/4, p297 

    The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there...

  • Hamiltonian Systems Admitting a Runge�Lenz Vector and an Optimal Extension of Bertrand�s Theorem to Curved Manifolds. Ballesteros, �ngel; Enciso, Alberto; Herranz, Francisco; Ragnisco, Orlando // Communications in Mathematical Physics;Sep2009, Vol. 290 Issue 3, p1033 

    Bertrand�s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result...

  • Fusion of Symmetric D-Branes and Verlinde Rings. Carey, Alan; Bai-Ling Wang // Communications in Mathematical Physics;Jan2008, Vol. 277 Issue 3, p577 

    We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken�s quasi-Hamiltonian G-spaces. The...


Read the Article


Sign out of this library

Other Topics