Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses

Lewis, Mark L.; Offin, Daniel C.; Buono, Pietro‐Luciano; Kovacic, Mitchell

doi:10.3934/dcds.2013.33.1137

Cited by 6 publications

(2 citation statements)

References 19 publications

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“…These symmetries may be useful in finding periodic orbits via global calculus of variations, and perhaps find their stability using symplectic techniques [29], [68].…”

Section: Discrete Symmetriesmentioning

confidence: 99%

About simple variational splines from the Hamiltonian viewpoint

Balseiro¹,

Stuchi²,

Cabrera³

et al. 2017

Journal of Geometric Mechanics

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In this paper, we study simple splines on a Riemannian manifold Q from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case Q is the 2-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

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“…These symmetries may be useful in finding periodic orbits via global calculus of variations, and perhaps find their stability using symplectic techniques [29], [68].…”

Section: Discrete Symmetriesmentioning

confidence: 99%

About simple variational splines from the Hamiltonian viewpoint

Balseiro¹,

Stuchi²,

Cabrera³

et al. 2017

Journal of Geometric Mechanics

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show abstract

“…At a linear level, this problem has been treated by several authors, mainly motivated by the dynamics of the so-called Aubry-Mather-Mañé sets (see, e.g. [16,29], [33, section 2.6], [4]), and problems from celestial mechanics ( [23,26]). Maslov-index arguments have also been used in this setting, see, e.g., [45].…”

Section: Introductionmentioning

confidence: 99%

Instability of closed orbits obtained by minimization*

Ureña

2022

Nonlinearity

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We study the dynamics around closed orbits of autonomous Lagrangian systems. When the configuration space is two-dimensional and orientable we show that every closed orbit minimizing the free-period action functional is orbitally unstable. This result applies even when the minimizers are degenerate or nonisolated, but a particularly strong form of instability holds in the isolated case. Under some symmetry assumptions, free-period action minimizers are unstable also in the higher-dimensional case. Applications to geodesics and celestial mechanics are given.

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