Helium and Hamiltonian delay equations

Frauenfelder, Urs

doi:10.1007/s11856-021-2242-x

Cited by 7 publications

(6 citation statements)

References 13 publications

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“…Proof: That q 1 is constant in the case r = 1 is proved in [5,Lemma 3.1]. It suffices therefore to consider the case r < 1.…”

Section: Proof Of the Main Resultsmentioning

confidence: 95%

“…In order to get a more tractable system the author replaced in [5] the instantaneous interaction of the two electrons by a mean interaction. It turns out that for mean interaction between the electrons the outer electron is really frozen, i.e., it becomes stationary.…”

Section: Introductionmentioning

confidence: 99%

“…It turns out that for mean interaction between the electrons the outer electron is really frozen, i.e., it becomes stationary. One obtains a delay equation for the inner electron and it is shown analytically in [5] that there exists a unique nondegenerate solution.…”

Section: Introductionmentioning

confidence: 99%

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A compactness theorem for frozen planets

Frauenfelder¹

2020

Preprint

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“…Proof: That q 1 is constant in the case r = 1 is proved in [5,Lemma 3.1]. It suffices therefore to consider the case r < 1.…”

Section: Proof Of the Main Resultsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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A compactness theorem for frozen planets

Frauenfelder¹

2020

Preprint

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“…where µ > 1. This functional describes two electrons attracted by a nucleus of charge µ and interacting with each other by their mean position, see [5].…”

Section: Examplementioning

confidence: 99%

“…The purpose of this note is to introduce a class of Hamiltonian delay equations for which such a uniform bound on the nullity still holds although the problems are in general not local. The author discovered this kind of equations in connection with his study on Helium [5]. The idea is to study particles which interact with each other just through their orbits.…”

Section: Introductionmentioning

confidence: 99%

Nullity bounds for certain Hamiltonian delay equations

Frauenfelder¹

2020

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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Helium and Hamiltonian delay equations

Cited by 7 publications

References 13 publications

A compactness theorem for frozen planets

A compactness theorem for frozen planets

Nullity bounds for certain Hamiltonian delay equations

On doubly symmetric periodic orbits

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