A New Compactification for Celestial Mechanics
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Graduate Theses, Dissertations, and Problem Reports 2017 A New Compactification for Celestial Mechanics Daniel Solomon Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Solomon, Daniel, "A New Compactification for Celestial Mechanics" (2017). Graduate Theses, Dissertations, and Problem Reports. 6689. https://researchrepository.wvu.edu/etd/6689 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. 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A New Compactification for Celestial Mechanics Daniel Solomon Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Harry Gingold, Ph.D., Chair Harvey Diamond, Ph.D. Leonard Golubovic, Ph.D. Harumi Hattori, Ph.D. Dening Li, Ph.D. Department of Mathematics Morgantown, West Virginia 2017 Keywords: escape in celestial mechanics, compactification, singular points of differential equations, critical points at infinity, Lorenz system, completeness, asymptotic analysis, degenerate critical points, expanding gravitational systems Copyright 2017 Daniel Solomon ABSTRACT A New Compactification for Celestial Mechanics Daniel Solomon This is an exposition of a new compactification of Euclidean space enabling the study of trajectories at and approaching spatial infinity, as well as results obtained for polynomial differential systems and Celestial Mechanics. The Lorenz system has an attractor for all real values of its parameters, and almost all of the complete quadratic systems share an interesting feature with the Lorenz system. Informed by these results, the main theorem is established via a contraction mapping arising from an integral equation derived from the Celestial Mechanics equations of motion. We establish the existence of an open set of initial conditions through which pass solutions without singularities, to Newton’s gravitational equations in R3 on a semi-infinite interval in forward time, for which every pair of particles separates like At; A > 0, as t ! 1 . The solutions are constructible as series with rapid uniform convergence and their asymptotic behavior to any order is prescribed. We show that this family of solutions depends on 6N parameters subject to certain constraints. This confirms the logical converse of Chazy’s 100-year old result assuming solutions exist for all time, they take the form given by Bohlin: solutions of Bohlin’s form do exist for all time. An easy consequence not found elsewhere is that the asymptotic directions of many configurations exiting the universe depend solely on the initial velocities and not on their initial positions. The N-body problem is fundamental to astrodynamics, since it is an idealization to point masses of the general problem of the motion of gravitating bodies, such as spacecraft motion within the Solar System. These new trajectories model paths of real particles escaping to infinity. A particle escaping its primary on a hyperbolic trajectory in the Kepler problem is the simplest example. This work may have relevance to new interplanetary trajectories or insight into known trajectories for potential space missions. I know that the tide is not an independent force, but merely the submission of the water to the movement of the moon in its orbit. And this in turn is subject to other orbits which are mightier far than it. And so the whole universe is held fast in the clinging grip of strong hands, the forces of Earth and Sun, planets and comets, and galaxies, blindly erupting forces ceaselessly stirring in ripples of silence to the very depth of black space. — Amos Oz DEDICATION This work is dedicated to my children and grandchildren, without whom it would have been finished many years earlier; I am proud of them every day. It is also dedicated to the memory of Charles Conley, a pioneer in the use of cohomology methods for dynamical systems [21]. I am grateful that he interested me in dynamical systems. In one sense this dissertation is the completion of a 41-year effort, since I started grad school in 1976. On the other hand, I come from a family of late bloomers. My father left aerospace engineering and was ordained at the age of 59. He held a pulpit before retiring again in his 70s and along the way he earned an MA in Theology at 63. Now at 86, he continues to audit advanced courses in math and physics at a nearby university. While raising five sons my mother finished college at 47 and then law school at 58, in just four years while working full time. After retiring from her job clerking for an administrative law judge at 60, she opened her own legal practice until retiring again at 72. My stepmother left music education at 59, and at 80 continues to publish on the subject, teach workshops in the US and around the world, direct a choir, and perform piano concerts in New York. My oldest brother was an Army officer when he began medical school, from which he graduated at 45. He became a neurologist and was recently promoted to full Colonel in the Medical Corps. Our middle brother has a successful career in the banking industry, but he cannot escape the need to play the blues on a slide guitar, which he started at 45. My next brother is an Israeli tour guide and educator based in Jerusalem. While not a late bloomer, he has one of the coolest jobs in one of the most special places in the world, and he lives a few hundred meters from where our grandfather was born. My youngest brother (also a lawyer) built an airplane in his garage. After ten years of work on the plane, it was certified for flight when he was 44. At almost 64, I’m right on time. If I had not lived until I was 90, I would not have been able to write this book. ... It could not have been done even when I was ten years younger. I wasn’t ready. God knows what other potentials lurk in other people, if we could only keep them alive well into their 90s.1 — Harry Bernstein 1Quoted in Senior Wonders [81], a sixtieth birthday present from my wife. Mr. Bernstein was awarded a Guggenheim Fellowship in 2008 to further pursue his writing (at 98!) after the publication of two well-received autobiographical novels, The Invisible Wall (2007) and The Dream (2008). It worked, too; his final novel The Golden Willow was published in 2009. iii ACKNOWLEDGEMENTS Much of this work was supported by NASA’s academic program. NASA is very generous with training, and it has been very generous with me! In addition, I would like to express my gratitude to many people for a wide variety of help and support: • A very special acknowledgement goes to my wife, Daya. I am beyond grateful for the life and love that we share. • My parents, who wakened my interests in math and science. • Three teachers from Moorestown Junior High School that encouraged me: Charles Barnosky, Flo- rence Kerr, and Andrea Rittenhouse. • Adam Bincer, from whom I learned to solve hard problems. He gave incredibly difficult homework problems in his courses in Quantum Mechanics, Quantum Electrodynamics, and Relativistic Quan- tum Electrodynamics. Since they were homework, one knew (or at least assumed) that they could be solved, and this was enough incentive for me to keep at it until the problems were solved. • Bernice and Randy Durand, who took me in a little bit while we studied topological methods in high energy physics together. • I thank my bosses for committed support: Carl Adams, Tupper Hyde, Juan Roman, Karen Flynn and Dennis Andrucyk. I still owe Karen a dollar. • Harry Gingold – enormous thanks is due to my long-suffering advisor. I am grateful for the opportunity to study under and work with an expert in asymptotic methods and singular problems. • A book that I’ve read more often than any other is Ursula K LeGuin’s The Dispossessed, which I read whenever I feel in need of inspiration. In addition, I treasure a post card she sent me in response to my fan mail. iv Contents ABSTRACT DEDICATION iii ACKNOWLEDGEMENTS iv List of Figures vii Chapter 1. Introduction 1 1.1. Celestial Mechanics, Singularities, and Infinity 3 1.2. Compactifications, Old and New 6 1.3. The Flow on the Boundary and at 1 9 1.4. Notation Glossary and Key Definitions 12 Chapter 2. Applications to the Lorenz and Other Polynomial Systems 15 2.1. The Lorenz System has a Global Repeller at Infinity 15 2.1.1. Compactification and Behavior at Infinity 16 2.1.2. Proof of Repulsion 18 2.2. Completeness of Quadratic Systems 20 2.2.1. Completeness and Structure of Lorenz-like Systems 21 2.2.2. Some incomplete systems 23 2.3. Fields of Lorenz-like Systems Near 1 25 Chapter 3. Celestial Mechanics at 1 via Compactification 31 3.1. Compactifying the N-Body Problem 31 3.1.1. Time derivatives 34 3.1.2. Including infinity 36 3.1.3. Seeking a series expansion 39 3.2.