Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

Maciej J. Capiński; Shane Kepley; J. D. Mireles James

doi:10.1016/j.jde.2023.03.053

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

Maciej J. Capiński, Shane Kepley, J. D. Mireles James^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L_4,5), families of periodic orbits passing through collision, and orbits connecting L₄ to ejection or collision.

Original language	English
Pages (from-to)	132-191
Number of pages	60
Journal	Journal of Differential Equations
Volume	366
Early online date	18 Apr 2023
DOIs	https://doi.org/10.1016/j.jde.2023.03.053
Publication status	Published - 5 Sept 2023

Bibliographical note

Funding Information:
M.C. was partially supported by the NCN grants 2019/35/B/ST1/00655 and 2021/41/B/ST1/00407.J.D.M.J. was partially supported by NSF Grant DMS 1813501.

Publisher Copyright:
© 2023 Elsevier Inc.

Funding

M.C. was partially supported by the NCN grants 2019/35/B/ST1/00655 and 2021/41/B/ST1/00407.J.D.M.J. was partially supported by NSF Grant DMS 1813501.

Funders	Funder number
National Science Foundation	DMS 1813501
National Science Foundation
Narodowe Centrum Nauki	2021/41/B/ST1/00407, 2019/35/B/ST1/00655
Narodowe Centrum Nauki

Keywords

Celestial mechanics
Collisions
Computer assisted proofs
Transverse homoclinic

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

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10.1016/j.jde.2023.03.053

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title = "Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem",

abstract = "This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Str{\"o}mgren's asymptotic periodic orbits (transverse homoclinics for L4,5), families of periodic orbits passing through collision, and orbits connecting L4 to ejection or collision.",

keywords = "Celestial mechanics, Collisions, Computer assisted proofs, Transverse homoclinic",

author = "Capi{\'n}ski, {Maciej J.} and Shane Kepley and {Mireles James}, {J. D.}",

note = "Funding Information: M.C. was partially supported by the NCN grants 2019/35/B/ST1/00655 and 2021/41/B/ST1/00407.J.D.M.J. was partially supported by NSF Grant DMS 1813501. Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

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AU - Kepley, Shane

AU - Mireles James, J. D.

N1 - Funding Information: M.C. was partially supported by the NCN grants 2019/35/B/ST1/00655 and 2021/41/B/ST1/00407.J.D.M.J. was partially supported by NSF Grant DMS 1813501. Publisher Copyright: © 2023 Elsevier Inc.

PY - 2023/9/5

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N2 - This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L4,5), families of periodic orbits passing through collision, and orbits connecting L4 to ejection or collision.

AB - This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L4,5), families of periodic orbits passing through collision, and orbits connecting L4 to ejection or collision.

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Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

Abstract

Bibliographical note

Funding

Keywords

UN SDGs

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