The Principle of Least Action in Geometry and Dynamics
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, leng
- PDF / 2,395,935 Bytes
- 135 Pages / 431.04 x 666.24 pts Page_size
- 0 Downloads / 263 Views
1844
3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo
Karl Friedrich Siburg
The Principle of Least Action in Geometry and Dynamics
13
Author Karl Friedrich Siburg Fakult¨at f¨ur Mathematik Ruhr-Universit¨at Bochum 44780 Bochum, Germany e-mail: [email protected]
Library of Congress Control Number: 2004104313
Mathematics Subject Classification (2000): 37J , 53D, 58E ISSN 0075-8434 ISBN 3-540-21944-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 11002192
41/3142/du-543210 - Printed on acid-free paper
Preface
The motion of classical mechanical systems is determined by Hamilton’s differential equations: x(t) ˙ = ∂y H(x(t), y(t)) y(t) ˙ = −∂x H(x(t), y(t)) For instance, if we consider the motion of n particles in a potential field, the Hamiltonian function 1 2 y − V (x1 , . . . , xn ) 2 i=1 i n
H=
is the sum of kinetic and potential energy; this is just another formulation of Newton’s Second Law. A distinguished class of Hamiltonians on a cotangent bundle T ∗ X consists of those satisfying the Legendre condition. These Hamiltonians are obtained from Lagrangian systems on the configuration space X, with coordinates (x, x) ˙ = (space, velocity), by introducing the new coordinates (x, y) = (space, momentum) on its phase space T ∗ X. Analytically, the Legendre condition corresponds to the convexity of H with respect to the fiber variable y. The Hamiltonian gives the energy value along a solution (which is preserved for time–independent systems) whereas the Lagrangian describes the action. Hamilton’s equations are equivalent to the Euler–Lagrange equations for the Lagrangian: d ∂x˙ L(x(t), x(t)) ˙ = ∂x L(x(t), x(t)). ˙ dt These equations express the variational character of solutions of the Lagrangian system. A curve x : [t0 , t1 ] → Rn is a Euler–Lagrange trajectory if, and only if, the first variation of the action integral, with end points held fixed, vanishes: t1 x(t1 ) δ L(x(t), x(t)) ˙ dt
Data Loading...
