The Kepler Problem

Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure. This book contains a comprehensive treatment of the Kepler problem, i.e., the two body problem. It is divided into four parts.

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Author: Bruno Cordani

Publisher: Birkhäuser

ISBN: 9783034880510

Category: Mathematics

Page: 442

View: 118

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.

Kepler Problem in the Presence of Dark Energy and the Cosmic Local Flow

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 ...

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Author: Alexander Silbergleit

Publisher: Springer

ISBN: 3030367517

Category: Science

Page: 68

View: 241

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 Key to Newton s Dynamics

J. Bruce Brackenridge sets the problem in historical and conceptual perspective, showing the physicist's debt to the works of both Descartes and Galileo.

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Author: J. Bruce Brackenridge

Publisher: Univ of California Press

ISBN: 9780520916852

Category: Science

Page: 330

View: 282

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.

Qualitative Analysis of the Anisotropic Kepler Problem

This paper gives a qualitative analysis of the flow of the anisotropic Kepler problem described by a Hamiltonian system as introduced by Gutzwiller and later studied by Devaney.

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Author: Josefina Casasayas

Publisher: American Mathematical Soc.

ISBN: 9780821823095

Category: Mathematics

Page: 115

View: 390

Variations on a Theme by Kepler

This book is based on the Colloquium Lectures presented by Shlomo Sternberg in 1990.

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Author: Victor Guillemin

Publisher: American Mathematical Soc.

ISBN: 9780821841846

Category: Mathematics

Page: 88

View: 904

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 h 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 set

Lesson 22

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Author: ADvTECH.

Publisher:

ISBN: OCLC:732355692

Category: Supervision

Page:

View: 805

Kepler Problem in the Presence of Dark Energy and the Cosmic Local Flow

This alone demonstrates the fundamental role that the Kepler problem with DE plays in the local cosmological studies; of course, the two-body problem is the ...

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Author: Alexander Silbergleit

Publisher: Springer Nature

ISBN: 9783030367527

Category: Science

Page: 68

View: 571

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.

Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature

Let us consider the Kepler problem on a sphere using the gnomonic coordinates. The advantage of this projection stems from the fact that free particle ...

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Author: T.G. Vozmischeva

Publisher: Springer Science & Business Media

ISBN: 9789401703031

Category: Science

Page: 184

View: 372

Introd uction The problem of integrability or nonintegrability of dynamical systems is one of the central problems of mathematics and mechanics. Integrable cases are of considerable interest, since, by examining them, one can study general laws of behavior for the solutions of these systems. The classical approach to studying dynamical systems assumes a search for explicit formulas for the solutions of motion equations and then their analysis. This approach stimulated the development of new areas in mathematics, such as the al gebraic integration and the theory of elliptic and theta functions. In spite of this, the qualitative methods of studying dynamical systems are much actual. It was Poincare who founded the qualitative theory of differential equa tions. Poincare, working out qualitative methods, studied the problems of celestial mechanics and cosmology in which it is especially important to understand the behavior of trajectories of motion, i.e., the solutions of differential equations at infinite time. Namely, beginning from Poincare systems of equations (in connection with the study of the problems of ce lestial mechanics), the right-hand parts of which don't depend explicitly on the independent variable of time, i.e., dynamical systems, are studied.

The Key to Newton s Dynamics

J. Bruce Brackenridge sets the problem in historical and conceptual perspective, showing the physicist's debt to the works of both Descartes and Galileo.

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Author: J. Bruce Brackenridge

Publisher:

ISBN: 0520200659

Category: Science

Page: 299

View: 234

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. 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.

Notes on Dynamical Systems

The SO ( 4 ) Symmetry of the Kepler Problem ( a ) The Kepler problem in R3 . This problem deals with the Hamiltonian ( 1.76 ) H = ž \ pl - 191-1 where p ...

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Author: Jürgen Moser

Publisher: American Mathematical Soc.

ISBN: 0821883526

Category: Science

Page: 256

View: 826

This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial N-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. --Publisher description.