Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle

Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle Mitsuru Shibayama, Kaito Miyazaki This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4115964/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract There are various solutions to the circular restricted three-body problem, which is used as a model to describe the movement of spacecraft. One of the most famous solutions is the periodic orbits and quasi-periodic solutions that are located near the equilibrium point on the same line as the main and secondary celestial bodies. These solutions are advantageous as they can help reduce fuel consumption while maintaining the spacecraft's orbit. It is expected that these solutions will be applied in many missions. In this paper, we will first explain a functional that uses Percival's variational principle to apply an invariant torus to the circular restricted three-body problem. We will then use a method to obtain an approximate quasi-periodic solution by minimizing this functional using the steepest descent method. Next, we will compare the results obtained from the proposed method with the ones obtained by numerically integrating the circular restricted three-body problem. Finally, we will discuss the prospects of these solutions. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4115964","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":281725452,"identity":"175e774d-d535-4c46-b126-1129a3f70830","order_by":0,"name":"Mitsuru Shibayama","email":"data:image/png;base64,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","orcid":"","institution":"Kyoto University","correspondingAuthor":true,"prefix":"","firstName":"Mitsuru","middleName":"","lastName":"Shibayama","suffix":""},{"id":281725453,"identity":"027dcd40-3318-4c08-9390-b69e3542f171","order_by":1,"name":"Kaito Miyazaki","email":"","orcid":"","institution":"Kyoto University","correspondingAuthor":false,"prefix":"","firstName":"Kaito","middleName":"","lastName":"Miyazaki","suffix":""}],"badges":[],"createdAt":"2024-03-17 07:59:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4115964/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4115964/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":70919507,"identity":"f0779078-de5e-479d-8980-c9765834c2f0","added_by":"auto","created_at":"2024-12-09 08:32:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":241220,"visible":true,"origin":"","legend":"","description":"","filename":"version1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4115964/v1_covered_a99d8920-7bef-41e7-9c3c-1d13524579e7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4115964/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4115964/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThere are various solutions to the circular restricted three-body problem,\u0026nbsp; which is used as a model to describe the movement of spacecraft. One of\u0026nbsp; the most famous solutions is the periodic orbits and quasi-periodic solutions\u0026nbsp; that are located near the equilibrium point on the same line as the main and\u0026nbsp; secondary celestial bodies. These solutions are advantageous as they can help\u0026nbsp; reduce fuel consumption while maintaining the spacecraft's orbit.\u0026nbsp; It is expected that these solutions will be applied in many missions.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn this paper, we will first explain a functional that uses Percival's variational principle\u0026nbsp; to apply an invariant torus to the circular restricted three-body problem. We will then use\u0026nbsp; a method to obtain an approximate quasi-periodic solution by minimizing this functional using\u0026nbsp; the steepest descent method. 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