The Kepler Problem

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The Kepler Problem

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Author : Bruno Cordani
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

The Kepler Problem written by Bruno Cordani and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

The Key To Newton S Dynamics

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Author : J. Bruce Brackenridge
language : en
Publisher: Univ of California Press
Release Date : 1996-02-29

The Key To Newton S Dynamics written by J. Bruce Brackenridge and has been published by Univ of California Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-02-29 with Science categories.

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.

Transformation Of The Kepler Problem

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Author : Marcel Reginatto
language : en
Publisher:
Release Date : 1992

Transformation Of The Kepler Problem written by Marcel Reginatto and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with categories.

Kepler Problem In The Presence Of Dark Energy And The Cosmic Local Flow

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Author : Alexander Silbergleit
language : en
Publisher: Springer Nature
Release Date : 2019-11-29

Kepler Problem In The Presence Of Dark Energy And The Cosmic Local Flow written by Alexander Silbergleit and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-29 with Science categories.

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.

Variations On A Theme By Kepler

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Author : Victor Guillemin
language : en
Publisher: American Mathematical Soc.
Release Date : 1990

Variations On A Theme By Kepler written by Victor Guillemin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Mathematics categories.

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.

Hidden Symmetries Of The Kepler Problem

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Author : Julia Kathryn Sheffler
language : en
Publisher:
Release Date : 2022

Hidden Symmetries Of The Kepler Problem written by Julia Kathryn Sheffler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Energy conservation categories.

The orbits of planets can be described by solving Kepler's problem which considers the motion due to by gravity (or any inverse square force law). The solutions to Kepler's problem, for energies less then 0, are ellipses, with a few conserved quantities: energy, angular momentum and the Laplace-Runge-Lenz (LRL) vector. Each conserved quantity corresponds to symmetries of the system via N öther's theorem. Energy conservation relates to time translations and angular momentum to three dimensional rotations. The symmetry related to the LRL vector is more difficult to visualize since it lives in phase space rather than configuration space. To understand the symmetry corresponding to the LRL vector, I use tools from Hamiltonian Mechanics, including the Poisson bracket, flow parameters, and action angle variables to make a visualization of the effect of the symmetry corresponding to the LRL vector. In particular the LRL vector corresponds to four-dimensional rotations in phase space. Though it is beyond the scope of this project I hope to use the solidified understanding of the relationship between conserved quantities and symmetries to simplify the derivation of the probability distribution of semi-major axis given a single direct image of an exoplanet.

Solving Kepler S Equation Over Three Centuries

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Author : Peter Colwell
language : en
Publisher:
Release Date : 1993

Solving Kepler S Equation Over Three Centuries written by Peter Colwell and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Science categories.

The Kepler Problem Half Century After Fock

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Author : Bruno Cordani
language : en
Publisher:
Release Date : 1988

The Kepler Problem Half Century After Fock written by Bruno Cordani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with categories.

Qualitative Analysis Of The Anisotropic Kepler Problem

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Author : Josefina Casasayas
language : en
Publisher: American Mathematical Soc.
Release Date : 1984

Qualitative Analysis Of The Anisotropic Kepler Problem written by Josefina Casasayas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.

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.

Some Geometric And Algebraic Aspects Of The Kepler Problem

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Author : Abdellatif El-Homani
language : en
Publisher:
Release Date : 2004

Some Geometric And Algebraic Aspects Of The Kepler Problem written by Abdellatif El-Homani and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with categories.