A Generalized Sturm Theorem

Edwards, Harold M.

doi:10.2307/1970490

Cited by 47 publications

(36 citation statements)

References 5 publications

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“…This result is yet another version of the comparison theorem in (symplectic) Sturm theory, similar to those established in, e.g., [Ar1,Bo,Ed]. The proposition can be easily verified by combining the construction of the generalized Maslov index, [RS], with the Arnold comparison theorem, [Ar1], and utilizing (2.2).…”

Section: The Salamon-zehnder Invariant ∆supporting

confidence: 71%

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Ginzburg¹,

Gürel²

2009

Comment. Math. Helv.

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Abstract. In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.

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Section: The Salamon-zehnder Invariant ∆supporting

confidence: 71%

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Ginzburg¹,

Gürel²

2009

Comment. Math. Helv.

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“…The structure of U -manifolds and the EM-index. In this paragraph we will briefly recall some useful facts about the U -manifold and we will define the EM-index , by generalizing the intersection theory proposed by Edwards in [3]. Notation 2.3 A (real-valued) Hermitian form on a complex vector space V is a real valued function Q on V which satisfies:…”

Section: Linear Preliminariesmentioning

confidence: 99%

“…We use the variational approach to (1.3) as described in [3] and we will stick to the notations of that paper. Given the complex n-dimensional Hermitian space (C n , ·, · ), for any m ∈ N let …”

Section: Variational Settingmentioning

confidence: 99%

“…where p i is a smooth path of matrices on the complex n-dimensional vector space C n and p 2m is the symmetry represented by the diagonal block matrix diag (I n−ν , −I ν ) for some integer 0 ≤ ν ≤ n, by using the reformulation in terms of calculus of variations as in [3]. However, we observe in this respect, that the situation we are dealing with is completely different and a new approach is needed, since both the hand-sides of the equation (1.2) are meaningless.…”

Section: Introductionmentioning

confidence: 99%

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On a Generalized Sturm Theorem

Порталури

2010

Advanced Nonlinear Studies

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Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm theorem for indefinite systems with Dirichlet boundary conditions of the formwhere pi is a smooth path of matrices on the complex n-dimensional vector space C n and p2m is the symmetry represented by diag (In−ν, −Iν) for some integer 0 ≤ ν ≤ n.

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“…By minor changes, the Morse Index Theorem is also valid in the case of Lorentzian metrics g, i.e., metrics having index one, provided that one considers causal, i.e., timelike or lightlike, geodesics and that one restricts the index form I to vector fields that are pointwise orthogonal to the geodesic (see [3]). Subsequent results have extended the Index Theorem to the case of solutions with non fixed initial and/or final endpoint (see [15,23]) and to the case of ordinary differential operators of even order (see [9]). …”

Section: Introductionmentioning

confidence: 99%

An Index Theorem for Non-Periodic Solutions of Hamiltonian Systems

Piccione

Tausk

2001

Proceedings of the London Mathematical Society

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ABSTRACT. We consider a Hamiltonian setup (M, ω, H, L, Γ, P), where (M, ω) is a symplectic manifold, L is a distribution of Lagrangian subspaces in M, P a Lagrangian submanifold of M, H is a smooth time dependent Hamiltonian function on M and Γ : [a, b] → M is an integral curve of the Hamiltonian flow H starting at P. We do not require any convexity property of the Hamiltonian function H. Under the assumption that Γ(b) is not P-focal it is introduced the Maslov index i maslov (Γ) of Γ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances i maslov (Γ) is computed as a sort of algebraic count of the P-focal points along Γ. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of Γ is equal to the sum of i maslov (Γ) and a convexity term of the Hamiltonian H relative to the submanifold P. When the result is applied to the case of the cotangent bundle M = T M * of a semi-Riemannian manifold (M, g) and to the geodesic Hamiltonian H(q, p) = 1 2 g −1 (p, p), we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.

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A Generalized Sturm Theorem

Cited by 47 publications

References 5 publications

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

On a Generalized Sturm Theorem

An Index Theorem for Non-Periodic Solutions of Hamiltonian Systems

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