Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.Mathematics Subject Classification (2002): Primary: 58E10, 83C10; Secondary: 53D12     

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.     

Morse homology for generating functions of Lagrangian submanifolds

The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional' symplectic invariants constructed via generating functions to the ``infinite dimensional' ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional. I, J. Diff. Geom. 46 (1997), 499-577.

     

On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

Maslov P-Index Theory for a Symplectic Path with Applications

Abstract The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered. *Project supported by the National Natural Science Foundation of China (No.10531050) and FANEDD.     

Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rⁿ, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.     

Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.     

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

Multiple periodic points of the Poincaré map of Lagrangian systems on tori

In this paper, suppose , A is positive definite and symmetric, and both A and V are and 1-periodic in all of their variables. We prove that the Poincaré map (i.e. the time-1-solution map) of the Lagrangian system possesses infinitely many periodic points on produced by contractible integer periodic solutions. Received July 23, 1997; in final form December 17, 1998     

Maslov P-Index Theory for a Symplectic Path with Applications

The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.     

Localization for Involutions in Floer Cohomology

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.     

Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems

Clarke has shown that the problem of findingT-periodic solutions for a convex Hamiltonian system is equivalent to the problem of finding critical points to a certain functional, dual to the classical action functional. In this paper, we relate the Morse index of the critical point to the minimal period of the correspondingT-periodic solution. In particular, we show that if the critical point is obtained by the Ambrosetti-Rabinowitz mountain-pass theorem the corresponding solution has minimal periodT, that is, it cannot beT/k-periodic withk integer,k2. As a consequence, we prove that if the Hamiltonian is flat near an equilibrium and superquadratic near infinity, then for anyT>0, the corresponding Hamiltonian system has a periodic solution with minimal periodT.     

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.     

Relative asymptotic behavior of pseudoholomorphic half‐cylinders

In this article we study the asymptotic behavior of pseudoholomorphic half‐cylinders that converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, and embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference” of two such maps, generalizing results from [6, 7, 12, 15]. Using this result with a technique from [14], we then show that a finite collection of pseudoholomorphic half‐cylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to solutions to a linear equation. © 2007 Wiley Periodicals, Inc.     

Index computations in Rabinowitz Floer homology

In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a nondegenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one moves up the orbit cylinder. We also provide an example to show that, even above the Ma?é critical value, the periods may be increasing thus producing a jump in the Morse index of the free time action functional in relation to the Morse index of the fixed time action functional.     

A Morse Index Theorem for Elliptic Operators on Bounded Domains

Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ? ? ⁿ is deformed. For a family of domains {Ω _t } _{t∈[a, b]} we prove that the Morse index of L on Ω _a differs from the Morse index of L on Ω _b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω _a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the “original” problem (on Ω _b ) and the “simplified” problem (on Ω _a ). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.     

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ, and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in ℝ×Y, with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces.?This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology. Received November 2, 2000 / final version received December 16, 2001?Published online November 19, 2002     

Perturbed closed geodesics are periodic orbits: Index and transversality

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of equals minus its Conley-Zehnder index when viewed as a critical point of and if is trivial. Otherwise a correction term +1 appears. Received: 21 May 2001; in final form: 10 October 2001 / Published online: 4 April 2002