Mathematics-Geometry and Topology

The principle of least action in geometry and dynamics

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Title (user) : The principle of least action in geometry and dynamicsISBN : 3540219447,9783540219446ISSN : 0075-8434DDC : 510 s,516.3/5LCC : QA3 .L28 no. 1844GoogleBook ID : OVPOAENondwCOpenLibrary ID : OL3314197MEdition : 1Series : Lecture notes in mathematics 1844Aut...
Description

Title (user) : The principle of least action in geometry and dynamics

ISBN : 3540219447,9783540219446

ISSN : 0075-8434

DDC : 510 s,516.3/5

LCC : QA3 .L28 no. 1844

GoogleBook ID : OVPOAENondwC

OpenLibrary ID : OL3314197M

Edition : 1

Series : Lecture notes in mathematics 1844

Authors (user) : Karl Friedrich Siburg

Authors (google) : Richard Evan Schwartz

Publisher : Springer

Language : English

Publication Date : 2004

File Format : pdf

Categories : Dynamics


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Description (user) :
New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather?s minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.


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Description (google) :
"Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B.H. Neumann introduced this system in the 1950s, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system."--Publisher website.

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