[2209.07097] Proof of a conjecture by H. Dullin and R. Montgomery Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Dynamical Systems arXiv:2209.07097 (math) [Submitted on 15 Sep 2022 ( v1 ), last revised 29 Apr 2026 (this version, v4)] Title: Proof of a conjecture by H. Dullin and R. Montgomery Authors: Gabriella Pinzari View a PDF of the paper titled Proof of a conjecture by H. Dullin and R. Montgomery, by Gabriella Pinzari View PDF HTML (experimental) Abstract: In the framework of the planar Euler problem in the quasi--periodic regime, the formulae of the periods available in the literature are simple only on one side of their singularity. In this paper, we complement such formulae with others, which result simpler on the other side. The derivation of such new formulae uses the Keplerian limit and complex analysis tools. As an application, we prove a conjecture by H. Dullin and R. Montgomery, which states that such periods, as well as their ratio, the {\it rotation number}, are monotone functions of their non--trivial first integral, at any fixed energy level. Comments: 26 pages, 3 Figures Subjects: Dynamical Systems (math.DS) ; Mathematical Physics (math-ph) MSC classes: 34C20, 34C25, 70F05, 70F10, 37J35, 70K43 Cite as: arXiv:2209.07097 [math.DS] (or arXiv:2209.07097v4 [math.DS] for this version) https://doi.org/10.48550/arXiv.2209.07097 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Annali Scuola Normale Superiore, Classe Scienze, 2024 Related DOI : https://doi.org/10.2422/2036-2145.202402_016 Focus to learn more DOI(s) linking to related resources Submission history From: Gabriella Pinzari [ view email ] [v1] Thu, 15 Sep 2022 07:26:35 UTC (31 KB) [v2] Wed, 21 Feb 2024 11:11:31 UTC (27 KB) [v3] Wed, 26 Jun 2024 16:50:04 UTC (25 KB) [v4] Wed, 29 Apr 2026 05:05:00 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled Proof of a conjecture by H. Dullin and R. Montgomery, by Gabriella Pinzari View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2022-09 Change to browse by: math math-ph math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from