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Advanced MechanicsFrom Euler's Determinism to Arnold's Chaos$
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S. G. Rajeev

Print publication date: 2013

Print ISBN-13: 9780199670857

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199670857.001.0001

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(p.55) 8 Geodesics
Advanced Mechanics

S. G. Rajeev

Oxford University Press

Riemannian geometry deals with distances of infinitesimally close points. A curve of minimal length is a geodesic. This variational principle happens to have the same structure as that of mechanics, therefore mechanical methods can be applied. Isometries of metrics lead to conservation laws. Conversely, the path of a Newtonian dynamical system can be reformulated as a geodesic of a metric in configuration space: the Maupertuis principle. Yet another application is to Fermat's principle of optics: the refractive index determines a metric, whose geodesics are light rays. But the deeper application of Riemannian geometry is to General Relativity: the gravitational field is the metric tensor. This chapter determines the orbits of a particle in the field of a black hole.

Keywords:   Riemannian geometry, geodesic, isometries, Maupertuis principle, Fermat principle, black hole

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