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caltech2003web - Collisions, Chaos and Periodic Orbits in...

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Unformatted text preview: Collisions, Chaos and Periodic Orbits in the Anisotropic Manev Problem Manuele Santoprete Department of Mathematics University of California Irvine California Institute of Technology October 15, 2003 Contents 2 Contents . . . . . . . . . . . 2 1 Preliminaries 3 The Manev Model . . . . 4 The Anisotropic Problems 5 Anisotropic Manev Problem 6 Motivation . . . . . . . . 7 The Hamiltonian . . . . . 9 2 Collision orbits and Chaos 11 Singularities . . . . . . . 13 McGehee Coordinates . . 14 The Collision Manifold . 15 Heteroclinic Orbits . . . 18 Homoclinic Manifold . . . 20 Perturbations . . . . . . 23 Melnikov Method . . . . 24 Homoclinics and Chaos . 26 Melnikov Integrals . . . . 27 3 Symmetric Periodic Or- bits 27 Symmetries . . . . . . . 29 Preliminaries . . . . . . . 32 The spaces of i . . . . . 33 Homotopy Classes . . . . 34 Hamilton Principle . . . . 36 The Variational Method . 37 Symmetrical Paths . . . . 38 Noncompactness . . . . . 39 Lower Semicontinuity . . 41 Periodic Orbits . . . . . . 42 Conclusions . . . . . . . 45 Bibliography . . . . . . . 46 Collisions, Chaos and Periodic Orbits in the Anisotropic Manev Problem 1 Preliminaries Manuele Santoprete The Manev Model 4 The Manev Model is a two body problem given by the potential energy U =- A r- B r 2 , where r is the distance between the particles and A, B are positive constants. This model describs the precession of the perihelion of Mercury with the same accuracy as general relativity. The Manev model also describes the relativistic dynam- ics of the Hydrogen atom , from a classical point of view. The Anisotropic Problems 5 The anisotropic problems , that replace the flat space with an anisotropic one, have been defined by Gutzwiller in the 1970s. Gutzwiller wanted to find an approximation of the quan- tum mechanical energy levels for a chaotic systems. He chose to study the Anisotropic Kepler Problem be- cause It is chaotic It is suitable to model physical phenomena en- countered in semiconductors Anisotropic Manev Problem 6 The Anisotropic Manev Problem (for example Craig et al. [1999] ) describes the motion of two point masses in an anisotropic space under the influence of the Manev law. It was first proposed in the mid 1990s by F. Diacu to have a deeper understanding of the connections between classical, relativistic and quantum mechanics. Motivation 7 fcbfh Physical Motivations Finding connections between Classical, Quantum and Relativistic Mechanics Describing a relativistic version of the Anisotropic Ke- pler Problem Describing gravitational models with anisotropic grav- itational constants. Can be used as a toy model for anisotropic problems in physics, astronomy, celestial mechanics,... Motivation 8 Mathematical Motivations Studying its peculiar collision manifolds and its colli- sion orbits Studying the mechanism responsible for the appear- ance of chaos Studying its discrete group of symmetry and the con- sequences of its existence. The Hamiltonian 9 The...
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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.

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caltech2003web - Collisions, Chaos and Periodic Orbits in...

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