First integrals for the Kepler problem with linear drag

Margheri, Alessandro; Ortega, Rafael; Rebelo, Carlota

doi:10.1007/s10569-016-9715-y

Cited by 8 publications

(27 citation statements)

References 11 publications

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“…However some formal calculations suggest the existence of special orbits with an asymptotically spiraling convergence to the center. We conclude that section by a few remarks and observations about the main result of [11,12] on asymptotically spiraling solutions. In Section 5, we show that equation (1.3) does have solutions for which |u(t)| is an exact decreasing exponential Ce −ηt if, and only if α = δ or α = 3δ 2 .…”

Section: 2)mentioning

confidence: 79%

“…It is quite natural to introduce this equation which was used systematically in [11,12] since this procedure is standard in the case δ = 0, c.f. eg [10].…”

Section: A Global Existence Resultsmentioning

confidence: 99%

“…Concerning (1.2), it turns out that all solutions have a bounded displacement |u(t)| and converge asymptotically to the center as was established by R. Ortega and his co-authors in [11,12]. In those works, the existence of a conserved quantity was used to establish that some trajectories behave as e −2δt v(t) for t large, where v(t) is confined to an ellipse.…”

Section: 2)mentioning

confidence: 86%

“…In section 2 we recall, to make subsequent comparisons easier, the Rutherford-Bohr atomic model with its circular orbits. In Section 3, we give direct and elementary proofs of some results from [11,12] together with some refinements concerning uniform bounds on u(t). In Section 4, we prove that (1.2) does not have any solution for which |u(t)| is an exact decreasing exponential Ce −ηt .…”

Section: 2)mentioning

confidence: 99%

“…It was established in [11] that all solutions u = u(t) of (1.2) are bounded. However the velocity u is generally unbounded.…”

Section: Boundedness Properties Of Solutionsmentioning

confidence: 99%

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On Some Damped 2 Body Problems

Haraux¹

2021

Preprint

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The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to $0$ for $t$ large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology

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Section: 2)mentioning

confidence: 79%

“…It is quite natural to introduce this equation which was used systematically in [11,12] since this procedure is standard in the case δ = 0, c.f. eg [10].…”

Section: A Global Existence Resultsmentioning

confidence: 99%

Section: 2)mentioning

confidence: 86%

Section: 2)mentioning

confidence: 99%

“…It was established in [11] that all solutions u = u(t) of (1.2) are bounded. However the velocity u is generally unbounded.…”

Section: Boundedness Properties Of Solutionsmentioning

confidence: 99%

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On Some Damped 2 Body Problems

Haraux¹

2021

Preprint

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A dissipative Kepler problem with a family of singular drags

Margheri

Misquero

2020

Celest Mech Dyn Astr

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Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

Kristiansen

2024

Nonlinearity

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In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in Margheri et al (2017 Celest. Mech. Dyn. Astron. 127 35–48) it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. We believe that our approach and results will have general interest in problems with blowup dynamics, including the Kepler problem with generalized nonlinear drag.

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First integrals for the Kepler problem with linear drag

Cited by 8 publications

References 11 publications

On Some Damped 2 Body Problems

On Some Damped 2 Body Problems

A dissipative Kepler problem with a family of singular drags

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

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