Instabilities and chaos in the classical three-body and three-rotor problems by Himalaya Senapati A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics to Chennai Mathematical Institute Submitted: April 2020 Defended: July 23, 2020 arXiv:2008.02670v1 [nlin.CD] 6 Aug 2020 Plot H1, SIPCOT IT Park, Siruseri, Kelambakkam, Tamil Nadu 603103, India Advisor: Prof. Govind S. Krishnaswami, Chennai Mathematical Institute. Doctoral Committee Members: 1. Prof. K. G. Arun, Chennai Mathematical Institute. 2. Prof. Arul Lakshminarayan, Indian Institute of Technology Madras. Declaration This thesis is a presentation of my original research work, carried out under the guidance of Prof. Govind S. Krishnaswami at the Chennai Mathematical Institute. This work has not formed the basis for the award of any degree, diploma, associateship, fellowship or other titles in Chennai Mathematical Institute or any other university or institution of higher education. Himalaya Senapati April, 2020 In my capacity as the supervisor of the candidate's thesis, I certify that the above statements are true to the best of my knowledge. Prof. Govind S. Krishnaswami April, 2020 iii Acknowledgments I would like to start by thanking my supervisor, Professor Govind S. Krishnaswami for his support and guidance, it has been a pleasure to be his student and learn from him. I am grateful to him for suggesting interesting research directions while giving me necessary space to pursue my own reasoning. His office was always open and he was willing to carefully listen to arguments, even when it was a different line of thought than his own, and offer corrections and insights. His persistent effort to improve the presentation of our results so that they are self-contained and appeal to a wide audience is something to strive for. Aside from research, he has also been very kind in other aspects of life, ranging from helping preparing applications for workshops and conferences to taking care of administrative matters. I thank Prof. K. G. Arun who is on my doctoral committee for his encouragement and for organizing departmental seminars with Prof. Alok Laddha where Research Scholars could present their work. I thank Prof. Arul Lakshminarayan for his insightful comments and references to the literature during our doctoral committee meetings. I also extend my thanks to Prof. Athanase Papadopoulos for encouraging me to contribute articles on non-Euclidean geometries to a book and for inviting me to conferences at MFO, Oberwolfach and BHU, Varanasi. I thank Prof. Sudhir Jain for valuable discussions. I would also like to thank Professors S G Rajeev and MS Santhanam for carefully reading this thesis and for their questions and suggestions. I am indebted to Prof. Swadheenananda Pattanayak and Prof. Chandra Kishore Moha- patra for instilling in me a love for mathematical sciences from a young age via Olympiad training camps at the state level. I am grateful to Banamali Mishra, Gokulananda Das and Ramachandra Hota for their teaching. I also thank Sandip Dasverma for his support. I thank Kedar Kolekar, Sonakshi Sachdev and T R Vishnu for many coffee time discus- sions. I also thank my officemates Abhishek Bharadwaj, Dharm Veer and Sarjick Bakshi as well as A Manu, Anbu Arjunan, Aneesh P B, Anudhyan Boral, Athira P V, Gaurav Patil, Keerthan Ravi, Krishnendu N V, Miheer Dewaskar, Naveen K, Pratik Roy, Praveen Roy, Priyanka Rao, Rajit Datta, Ramadas N, Sachin Phatak, Shanmugapriya P, Shraddha Sri- vastava and Swati Gupta for their help and friendship. A special thanks to Apolline Louvet for hosting me in France for a part of my stay during both my visits to Europe. I thank the faculty at the Chennai Mathematical Institute including Professors H S Mani, G Rajasekaran, N D Hari Dass, R Jagannathan, A Laddha, K Narayan, V V Sreedhar, R iv v Parthasarathy, P B Chakraborty, G Date and A Virmani for their time and help. Thanks are also due to the administrative staff at CMI including S Sripathy, Rajeshwari Nair, G Ranjini and V Vijayalakshmi as well as the mess, security and housekeeping staff. I gratefully acknowledge support from the Science and Engineering Research Board, Govt. of India in the form of an International Travel Support grant to attend the Berlin Mathe- matical Summer School in 2017, for sponsoring a school on nonlinear dynamics at SPPU, Pune in 2018 and for travel support to attend a CIMPA school at BHU, Varanasi and CNSD 2019 at IIT Kanpur. I also acknowledge MFO, Oberwolfach for awarding me an Oberwolfach Leibniz Graduate Students travel grant to attend a conference held there. I thank CMI for my research fellowship and for supporting my travel to attend schools and workshops at ICTS Bengaluru, MFO Oberwolfach, RKMVERI Belur Math, IISER Tirupati and IIT Kanpur and to meet Professors A Chenciner and L Zdeborova in Paris. I am also grateful to the Infosys Foundation and J N Tata trust for financial support. Finally, I would like to thank my parents Niranjan Senapati and Annapurna Senapati and brother Meghasan Senapati for their love and support. Abstract This thesis studies instabilities and singularities in a geometrical approach to the planar three-body problem as well as instabilities, chaos and ergodicity in the three-rotor problem. Trajectories of the three-body problem are expressed as geodesics of the Jacobi-Maupertuis (JM) metric on the configuration space. Translation, rotation and scaling isometries lead to reduced dynamics on quotients of the configuration space, which encode information on the full dynamics. Riemannian submersions are used to find quotient metrics and to show that the geodesic formulation regularizes collisions for the 1=r2 , but not for the 1=r potential. Extending work of Montgomery, we show the negativity of the scalar curvature on the center of mass configuration space and certain quotients for equal masses and zero energy. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. In the three-rotor problem, three equal masses move on a circle subject to attractive cosine inter-particle potentials. This problem arises as the classical limit of a model of coupled Josephson junctions. The energy serves as a control parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendula and breathers at all energies and choreographies up to moderate energies. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy with several manifestations: (a) a dramatic rise in the fraction of Poincar´esurfaces occupied by chaotic sections, (b) spontaneous breaking of discrete symmetries, (c) a geometric cascade of stability transitions in pendula and (d) a change in sign of the JM curvature. Poincar´esections indicate global chaos in a band of energies slightly above this transition where we provide numerical evidence for ergodicity and mixing with respect to the Liouville measure and study the statistics of recurrence times. vi List of publications This thesis is based on the following publications. 1. G. S. Krishnaswami and H. Senapati, Curvature and geodesic instabilities in a geomet- rical approach to the planar three-body problem, J. Math. Phys., 57, 102901 (2016). arXiv:1606.05091. [Featured Article] 2. G. S. Krishnaswami and H. Senapati, An introduction to the classical three-body prob- lem: From periodic solutions to instabilities and chaos, Resonance, 24, 87-114 (2019). arXiv:1901.07289. 3. G. S. Krishnaswami and H. Senapati, Stability and chaos in the classical three rotor problem, Indian Academy of Sciences Conference Series, 2 (1), 139 (2019). arXiv:1810.01317. 4. G. S. Krishnaswami and H. Senapati, Classical three rotor problem: periodic solutions, stability and chaos, Chaos, 29 (12), 123121 (2019). arXiv:1811.05807. [Editor's pick, Featured article] 5. G. S. Krishnaswami and H. Senapati, Ergodicity, mixing and recurrence in the three rotor problem, Chaos, 30 (4), 043112 (2020). arXiv:1910.04455. [Editor's pick] vii Contents Acknowledgments iv Abstract vi List of publications vii 1 Introduction 1 1.1 Geometrical approach to the planar three-body problem . .1 1.2 Classical three-rotor problem . .4 2 Instabilities in the planar three-body problem9 2.1 Trajectories as geodesics of the Jacobi-Maupertuis metric . .9 2.2 Planar three-body problem with inverse-square potential . 11 2.2.1 Jacobi-Maupertuis metric on the configuration space . 11 2.2.1.1 Isometries and Riemannian submersions . 12 2 3 3 2 2.2.1.2 Hopf coordinates on C and quotient spaces R , S and S .... 13 2.2.1.3 Lagrange, Euler, collinear and collision configurations . 16 3 2.2.2 Quotient JM metrics on shape space, S and the shape sphere . 17 2 3 2.2.2.1 Submersion from C to shape space R ................ 17 3 2 2.2.2.2 Submersion from R to the shape sphere S ............. 17 2 3 2 2.2.2.3 Submersion from C to S and then to S .............. 17 2.2.3 JM metric in the near-collision limit and its completeness . 18 2.2.3.1 Geometry near pairwise collisions . 18 3 2 2.2.3.2 Geometry on R and C near triple collisions . 20 2.2.4 Scalar curvature for equal masses and zero energy . 22 2 2.2.4.1 Scalar curvature on the shape sphere S ............... 23 2 2.2.4.2 Scalar curvature on the center-of-mass configuration space C ... 23 viii CONTENTS ix 3 3 2.2.4.3 Scalar curvatures on shape space R and on S ........... 24 2.2.5 Sectional curvature for three equal masses . 25 2.2.6 Stability tensor and linear stability of geodesics . 27 2.2.6.1 Rotational Lagrange solutions in Newtonian potential .