On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

Frauenfelder, Urs; Moreno, Agustín Vivas

doi:10.48550/arxiv.2109.09147

Cited by 5 publications

(11 citation statements)

References 12 publications

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“…In the examples above, where the spaces consists of matrices, the transition from to the geometric quotient to the GIT quotient basically means, in practice, to ignore Jordan factors, replacing them with diagonal blocks. The resulting matrices, while not necessarily equivalent in the original quotient, become so in the GIT one, see [10,Appendix A]. In [10], the cases n = 1 and n = 2 (for instance, relevant for the planar and the spatial three-body problems, respectively) are studied in detail.…”

Section: Remark 12 (Git Quotient)mentioning

confidence: 99%

“…a mass parameter). For this purpose, we will illustrate the use of global topological methods, via the GIT sequence introduced in [10] by the first and third authors. This is a sequence of three spaces ("top", "middle" and "base") consisting of equivalence classes of symplectic matrices, and concrete maps between them, which are also explicitly computable.…”

Section: Introductionmentioning

confidence: 99%

“…B-signature for symmetric orbits. We now explain the notion of B-sign, introduced in [10]. For the case n = 2, the following construction will assign a pair = ( 1 , 2 ), the B-signature, where i = ± is a "plus" or a "minus" label for i = 1, 2, to the eigenvalues of the 2 × 2-block A of M = M A,B,C , assuming that they are real and different.…”

Section: Introductionmentioning

confidence: 99%

“…However, in order to refine the data proved by the point p, we can still appeal to classical Krein theory, as proposed by Krein [18, 19] [20, 21] and rediscovered by M öser [22], which associates a sign only to the elliptic eigenvalues. It turns out that, in the elliptic case, the notion of B-signature coincides with the more classical one of Krein signature; see [10]. Before introducing this notion, first, recall that the reduced monodromy matrix of any orbit at any point is a symplectic matrix M in Sp(4), and changing the base point only changes the matrix up to symplectic conjugation.…”

Section: Introductionmentioning

confidence: 99%

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Symplectic methods in the numerical search of orbits in real-life planetary systems

Frauenfelder¹,

Koh²,

Moreno³

2022

Preprint

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The intention of this article is to illustrate the use of invariants coming from Floer-type theories, as well as global topological methods, for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). In this paper, we illustrate the use of the GIT sequence introduced in [10] by the first and third authors, consisting of a sequence of spaces and maps between them. Roughly speaking, closed orbits of an arbitrary Hamiltonian system induce points in these spaces, and so their topology imposes restrictions on the existence of regular orbit cylinders, as well as encodes information on all types of bifurcations. We also consider the notion of the SFT-Euler characteristic, as the Euler characteristic of the local Floer homology groups associated to a closed orbit of a Hamiltonian system. This number stays invariant before and after a bifurcation, and can be computed in explicit terms from the eigenvalues of the reduced monodromy matrix. This makes it useful for practical applications, e.g. for predicting the existence of orbits for optimization and space mission design. For the case of symmetric orbits (i.e. preserved by an antisymplectic involution), which may be thought both as open or closed strings, we further consider the notion of the real Euler characteristic, as the Euler characteristic of the corresponding Lagrangian Floer homology group. We illustrate the practical use of these invariants, together with the GIT sequence, via numerical work based on the cell-mapping method as described in [16], for the Jupiter-Europa and the Saturn-Enceladus systems. These are currently systems of interest, as they fall in the agenda of space agencies like NASA, as these icy moons are considered candidates for harbouring conditions suitable for extraterrestrial life. CONTENTS 1. Introduction 2. The SFT-Euler characteristic of a periodic orbit 3. The real Euler characteristic of a symmetric orbit 4. Non-symmetric orbits and branching structure 5. Symmetric subtle division 6. Numerics Appendix A. Reduced vs nonreduced monodromy matrices Appendix B. Planar orbits and spatial vs. planar bifurcations Appendix C. Basis changes References

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Section: Remark 12 (Git Quotient)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symplectic methods in the numerical search of orbits in real-life planetary systems

Frauenfelder¹,

Koh²,

Moreno³

2022

Preprint

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“…See also [BFK19,Theorem 6.1]. It should also be interesting to study bifurcations via SFT-Euler characteristic, see [FM21].…”

Section: Introductionmentioning

confidence: 99%

The Limit Set of a Family of Periodic Orbits

Yannis¹

2021

Preprint

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On doubly symmetric periodic orbits

Frauenfelder

Moreno

2023

Celest Mech Dyn Astron

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In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e., those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) All covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory (Eliashberg et al. Geom Funct Anal Special Volume Part II:560–673, 2000); (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in Frauenfelder et al. in Symplectic methods in the numerical search of orbits in real-life planetary systems. Preprint arXiv:2206.00627). The above results follow from: (5) A symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in Frauenfelder and Moreno 2021) differ.

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On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

Cited by 5 publications

References 12 publications

Symplectic methods in the numerical search of orbits in real-life planetary systems

Symplectic methods in the numerical search of orbits in real-life planetary systems

The Limit Set of a Family of Periodic Orbits

On doubly symmetric periodic orbits

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