Cartan differential calculus9/22/2023 I have never regretted the effort that I expended in the pursuit of this hopeless dream-only that the book was published as a textbook and marketed as a textbook, with the result that the case for differential forms that it tried to make was hardly heard. My original plan had been to write a small supplementary textbook on differential forms, but overly optimistic publishers talked me out of this modest intention and into the wholly unrealistic objective (especially unrealistic for an unknown 30-year-old author) of writing a full-scale advanced calculus course that would revolutionize the way advanced calculus was taught and sell lots of books in the process. The book was born in 1969 as an "innovative textbook"-a breed everyone claims to want but which usually goes straight to the orphanage. With this new edition, I hope it has reached a secure middle age. Preface to the 1994 Edition My first book had a perilous childhood. Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the USA Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Edwards Reprinted 1994 with corrections from the original Houghton Mifflin edition.Ĭopyright is not claimed for works of U.S. Advanced calculus : a differential fonns approach / Harold M. Library of Congress Cataloging In-Publication Data Edwards, Harold M. Edwards Courant Institute New York University New York, NY 10012 ( This is a link.Advanced Calculus A Differential Forms Approach You could look at the EoM entry on integral invariants and the references therein. $\omega_L$ with Hamilton function $E_L:=\mathcal$ throgh fiber-wise homotopies.)Ībout the edited question: Truly my historical knowledge is very limited, but probably the origin of these concepts could be in the works by Poincarè on the celestial mechanics, (when the concept itself of differential forms was germinating,) and therefore at the origins of the differential topology as we know it nowadays. When the Lagrangian is not degenerate then $\omega_L$ is non degenerate, and the Euler-Lagrange vector field $\xi_L$ is the hamiltonian vector field w.r.t. There you would find that to any (Lagrangian) function $L$ on $TQ,$ there are associated the Poincaré-Cartan forms $\theta_L:=J^\ast dL$ and $\omega_L:=d\theta_L,$ where $J:T(TQ)\to T(TQ)$ is the vertical endomorphism associated to the canonical almost tangent structure on $TQ.$ I think that you could appreciate "Methods of Differential Geometry in Analytical Mechanics" by P.Rodriguez and M.deLeon ( this is a link)Īpart from its intrinsec interest as reference for both the constructions of differential geometry and the geometrization of Lagrangian/Hamiltonian mechanics, in particular, if you are looking for the role played by the Poincaré-Cartan forms in mechanics, you could find it in their Chapters 2 (for the canonical almost-tangent structure on $TQ$) and 9 (for the Lagrangian Mechanics).
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