The Resource Relative equilibria of the curved Nbody problem, Florin Diacu
Relative equilibria of the curved Nbody problem, Florin Diacu
Resource Information
The item Relative equilibria of the curved Nbody problem, Florin Diacu represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Missouri Libraries.This item is available to borrow from 1 library branch.
Resource Information
The item Relative equilibria of the curved Nbody problem, Florin Diacu represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Missouri Libraries.
This item is available to borrow from 1 library branch.
 Summary
 The guiding light of this monograph is a question easy to understand but difficult to answer: {What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a circle, as between Paris and New York, or take some other path, and if so, what would that path look like? If you accept that the model proposed here, which assumes a gravitational law extended to a universe of constant curvature, is a good approximation of the physical reality (and I will later outline a few arguments in this direction), then we can answer the above question for distances comparable to those of our solar system. More precisely, this monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. Physicists take the flatness of space for granted in regions of that size. But it is good to finally have a mathematical confirmation in this sense. Our main goals, however, are mathematical. We will shed some light on the dynamics of N point masses that move in spaces of nonzero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics: classical dynamics, nonEuclidean geometry, geometric topology, Lie groups, and the theory of polytopes
 Language
 eng
 Extent
 1 online resource
 Contents

 Relative Equilibria (RE)
 Fixed Points (FP)
 pt. 3.
 Criteria and Qualitative Behavior
 Existence Criteria
 Qualitative Behavior
 pt. 4.
 Examples
 Positive Elliptic RE
 Positive EllipticElliptic RE
 Introduction
 Negative RE
 pt. 5.
 The 2dimensional case
 Polygonal RE
 Lagrangian and Eulerian RE
 Saari's Conjecture
 pt. 1.
 Background and Equations of Motion
 Preliminary Developments
 Equations of motion
 pt. 2.
 Isometries and Relative Equilibria
 Isometric Rotations
 Isbn
 9789491216688
 Label
 Relative equilibria of the curved Nbody problem
 Title
 Relative equilibria of the curved Nbody problem
 Statement of responsibility
 Florin Diacu
 Subject

 Celestial mechanics
 Celestial mechanics
 Differentiable dynamical systems.
 Differential Equations.
 Differential equations
 Differential equations
 Differential equations
 Dynamical Systems and Ergodic Theory.
 Gravitation
 Manybody problem  Numerical solutions
 Manybody problem  Numerical solutions
 Manybody problem  Numerical solutions
 Mathematical Concepts
 Mathematics
 Mathematics, general.
 Mathematics.
 Mechanics
 Ordinary Differential Equations.
 Physics  methods
 SCIENCE  Waves & Wave Mechanics
 Celestial mechanics
 Language
 eng
 Summary
 The guiding light of this monograph is a question easy to understand but difficult to answer: {What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a circle, as between Paris and New York, or take some other path, and if so, what would that path look like? If you accept that the model proposed here, which assumes a gravitational law extended to a universe of constant curvature, is a good approximation of the physical reality (and I will later outline a few arguments in this direction), then we can answer the above question for distances comparable to those of our solar system. More precisely, this monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. Physicists take the flatness of space for granted in regions of that size. But it is good to finally have a mathematical confirmation in this sense. Our main goals, however, are mathematical. We will shed some light on the dynamics of N point masses that move in spaces of nonzero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics: classical dynamics, nonEuclidean geometry, geometric topology, Lie groups, and the theory of polytopes
 Cataloging source
 GW5XE
 http://library.link/vocab/creatorDate
 19592018
 http://library.link/vocab/creatorName
 Diacu, Florin
 Dewey number
 530.14/4
 Illustrations
 portraits
 Index
 index present
 LC call number
 QA378
 LC item number
 .D53 2012
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Atlantis studies in dynamical systems
 Series volume
 v. 1
 http://library.link/vocab/subjectName

 Manybody problem
 Differential equations
 Celestial mechanics
 Mathematical Concepts
 Mathematics
 Mechanics
 Gravitation
 Physics
 SCIENCE
 Celestial mechanics
 Differential equations
 Manybody problem
 Label
 Relative equilibria of the curved Nbody problem, Florin Diacu
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Relative Equilibria (RE)
 Fixed Points (FP)
 pt. 3.
 Criteria and Qualitative Behavior
 Existence Criteria
 Qualitative Behavior
 pt. 4.
 Examples
 Positive Elliptic RE
 Positive EllipticElliptic RE
 Introduction
 Negative RE
 pt. 5.
 The 2dimensional case
 Polygonal RE
 Lagrangian and Eulerian RE
 Saari's Conjecture
 pt. 1.
 Background and Equations of Motion
 Preliminary Developments
 Equations of motion
 pt. 2.
 Isometries and Relative Equilibria
 Isometric Rotations
 Control code
 808632285
 Dimensions
 unknown
 Extent
 1 online resource
 File format
 unknown
 Form of item
 online
 Isbn
 9789491216688
 Lccn
 2014655006
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.2991/9789491216688
 Other physical details
 portraits.
 http://library.link/vocab/ext/overdrive/overdriveId
 394693
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)808632285
 Label
 Relative equilibria of the curved Nbody problem, Florin Diacu
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Relative Equilibria (RE)
 Fixed Points (FP)
 pt. 3.
 Criteria and Qualitative Behavior
 Existence Criteria
 Qualitative Behavior
 pt. 4.
 Examples
 Positive Elliptic RE
 Positive EllipticElliptic RE
 Introduction
 Negative RE
 pt. 5.
 The 2dimensional case
 Polygonal RE
 Lagrangian and Eulerian RE
 Saari's Conjecture
 pt. 1.
 Background and Equations of Motion
 Preliminary Developments
 Equations of motion
 pt. 2.
 Isometries and Relative Equilibria
 Isometric Rotations
 Control code
 808632285
 Dimensions
 unknown
 Extent
 1 online resource
 File format
 unknown
 Form of item
 online
 Isbn
 9789491216688
 Lccn
 2014655006
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.2991/9789491216688
 Other physical details
 portraits.
 http://library.link/vocab/ext/overdrive/overdriveId
 394693
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)808632285
Subject
 Celestial mechanics
 Celestial mechanics
 Differentiable dynamical systems.
 Differential Equations.
 Differential equations
 Differential equations
 Differential equations
 Dynamical Systems and Ergodic Theory.
 Gravitation
 Manybody problem  Numerical solutions
 Manybody problem  Numerical solutions
 Manybody problem  Numerical solutions
 Mathematical Concepts
 Mathematics
 Mathematics, general.
 Mathematics.
 Mechanics
 Ordinary Differential Equations.
 Physics  methods
 SCIENCE  Waves & Wave Mechanics
 Celestial mechanics
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