The Kepler Problem

Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.

While much has been written on the ramifications of Newton's dynamics, until now the details of Newton's solution were available only to the physics expert. The Key to Newton's Dynamics clearly explains the surprisingly simple analytical structure that underlies the determination of the force necessary to maintain ideal planetary motion. J. Bruce Brackenridge sets the problem in historical and conceptual perspective, showing the physicist's debt to the works of both Descartes and Galileo. He tracks Newton's work on the Kepler problem from its early stages at Cambridge before 1669, through the revival of his interest ten years later, to its fruition in the first three sections of the first edition of the Principia.

This book is based on the Colloquium Lectures presented by Shlomo Sternberg in 1990. The authors delve into the mysterious role that groups, especially Lie groups, play in revealing the laws of nature by focusing on the familiar example of Kepler motion: the motion of a planet under the attraction of the sun according to Kepler's laws. Newton realized that Kepler's second law--that equal areas are swept out in equal times--has to do with the fact that the force is directed radially to the sun. Kepler's second law is really the assertion of the conservation of angular momentum, reflecting the rotational symmetry of the system about the origin of the force. In today's language, we would say that the group $O(3)$ (the orthogonal group in three dimensions) is responsible for Kepler's second law. By the end of the nineteenth century, the inverse square law of attraction was seen to have $O(4)$ symmetry (where $O(4)$ acts on a portion of the six-dimensional phase space of the planet). Even larger groups have since been found to be involved in Kepler motion. In quantum mechanics, the example of Kepler motion manifests itself as the hydrogen atom. Exploring this circle of ideas, the first part of the book was written with the general mathematical reader in mind. The remainder of the book is aimed at specialists. It begins with a demonstration that the Kepler problem and the hydrogen atom exhibit $O(4)$ symmetry and that the form of this symmetry determines the inverse square law in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. The space of regularized elliptical motions of the Kepler problem (also known as the Kepler manifold) plays a central role in this book. The last portion of the book studies the various cosmological models in this same conformal class (and having varying isometry groups) from the viewpoint of projective geometry. The computation of the hydrogen spectrum provides an illustration of the principle that enlarging the phase space can simplify the equations of motion in the classical setting and aid in the quantization problem in the quantum setting. The authors provide a short summary of the homological quantization of constraints and a list of recent applications to many interesting finite-dimensional settings. The book closes with an outline of Kostant's theory, in which a unitary representation is associated to the minimal nilpotent orbit of $SO(4,4)$ and in which electromagnetism and gravitation are unified in a Kaluza-Klein-type theory in six dimensions.

The book consists of articles based on the XXXVII Białowieża Workshop on Geometric Methods in Physics, 2018. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. This edition of the workshop featured a special session dedicated to Professor Daniel Sternheimer on the occasion of his 80th birthday. The previously unpublished papers present cutting-edge current research, typically grounded in geometry and analysis, with applications to classical and quantum physics. For the past seven years, the Białowieża Workshops have been complemented by a School on Geometry and Physics comprising a series of advanced lectures for graduate students and early-career researchers. The book also includes abstracts of the five lecture series that were given at the seventh school.

This book derives and analyzes all solutions to the Kepler problem with dark energy (DE), presenting significant results such as: (a) all radial infinite motions obey Hubble’s law at large times; (b) all orbital infinite motions are asymptotically radial and obey Hubble’s law; (c) infinite orbital motions strongly dominate the finite ones. This clearly shows the effect of repulsive DE: In the classical Kepler problem, all orbital motions are finite for negative energies and infinite in the opposite case. Another DE effect is spatial localization of bounded orbits: mostly, they are within the equilibrium sphere, where the attractive Newtonian force outbalances the repulsive force of DE. This problem is of particular current interest due to recent studies of the local flows of galaxies showing domination of DE in their dynamics; the book discusses this observation in detail.

The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.