On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Gosson, Maurice de; Gosson, Serge de; Piccione, Paolo

doi:10.1007/s10455-008-9106-z

Cited by 19 publications

(30 citation statements)

References 25 publications

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“…We observe that this map coincides, up to a sign with, the one defined in [9, Equation2.3] or with the local chart ϕ L 0 ,L 1 (L) given by authors in[8, Section 2] or in[15, Section 2]. However our choice is coherent with the crossing forms defined through Q in[29, Section 1],[34, Equation 2 & Remark 3.1] with[20, Section 3] and[6].…”

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confidence: 62%

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Sturm theory with applications in geometry and classical mechanics

et al. 2021

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Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

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confidence: 62%

“…We emphasize that, for curves having degenerate endpoints with respect to there are several conventions for how the endpoints contribute to the Maslov index. For other different choices we refer the interested reader to [8,20,29] and references therein.…”

Section: Iteration Inequalities For Periodic Boundary Conditionsmentioning

confidence: 99%

Sturm theory with applications in geometry and classical mechanics

et al. 2021

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“…Similarly, B(β j ) is in Sp ψ (2). With these further notations established, we may now state the aforementioned theorem, which combines Corollary 3.7 and Lemma 3.3 from [DDP08].…”

Section: Proof Of Lemma 55mentioning

confidence: 97%

“…Explicitly, the Conley-Zehnder index on a 2n-dimensional manifold is n minus the Robbin-Salamon index. The index of a path of shear matrices is an easy and well-known computation, and so the bulk of the work is relating the index of the concatenation to the indices of the concatenated paths, for which we use results from [DDP08]. We begin by recalling some old notation and introducing some new notation.…”

Section: The Conley-zehnder Indexmentioning

confidence: 99%

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

et al. 2019

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Given a symplectic surface (Σ, ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ, ω), dH), where dH is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. * This paper was the outcome of the authors' work in the computational symplectic topology graduate student team-based research program held Hofer's geometryLet (M, ω) be a symplectic manifold. Given a smooth function H :where H t (p) := H(t, p). Let ϕ t H : M → M be the flow of the ODEẋ(t) = X t (x(t)), making sufficient assumptions to ensure that the flow is globally defined on the time interval [0, 1] (for example, we could take M to be compact). Inside the group Symp(M, ω) = {φ ∈ Diff(M ) : φ * ω = ω} of symplectomorphisms we have the subgroup of Hamiltonian diffeomorphisms Ham(M, ω), which consists of the time-one maps ϕ 1 H : M → M of flows as above. The group Ham(M, ω) is equipped with a geometrically meaningful bi-invariant metric introduced by Hofer. The resulting metric group is an important object of study in symplectic geometry. For φ ∈ Ham(M, ω), we define the Hofer norm φ H = inf H 1 0

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“…In this section we will prove an iteration formula for the Maslov index of a periodic solution of a Hamiltonian system, using a similar formula proved in [19] for the Conley-Zehnder index, and a formula relating the two indices via the Hörmander index. The reader should note that in the literature there are several definitions Maslov index for a continuous Lagrangian path; these definitions differ by a boundary term when the path has endpoints in the Maslov cycle.…”

Section: On the Maslov Index And Iteration Formulasmentioning

confidence: 99%