Classical Mechanics

Classical Mechanics Joel A. Shapiro April 21, 2003 i Copyright C 1994, 1997 by Joel A. Shapiro All rights reserved. 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, or otherwise, without the prior written permission of the author. This is a preliminary version of the book, not to be considered a fully published edition. While some of the material, particularly the first four chapters, is close to readiness for a first edition, chapters 6 and 7 need more work, and chapter 8 is incomplete. The appendices are random selections not yet reorganized. There are also as yet few exercises for the later chapters. The first edition will have an adequate set of exercises for each chapter. The author welcomes corrections, comments, and criticism. ii Contents 1 Particle Kinematics 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Single Particle Kinematics . . . . . . . . . . . . . . . . 1.2.1 Motion in configuration space . . . . . . . . . . 1.2.2 Conserved Quantities . . . . . . . . . . . . . . . 1.3 Systems of Particles . . . . . . . . . . . . . . . . . . . . 1.3.1 External and internal forces . . . . . . . . . . . 1.3.2 Constraints . . . . . . . . . . . . . . . . . . . . 1.3.3 Generalized Coordinates for Unconstrained Systems . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Kinetic energy in generalized coordinates . . . . 1.4 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Dynamical Systems . . . . . . . . . . . . . . . . 1.4.2 Phase Space Flows . . . . . . . . . . . . . . . . 2 Lagrange’s and Hamilton’s Equations 2.1 Lagrangian Mechanics . . . . . . . . . . . . 2.1.1 Derivation for unconstrained systems 2.1.2 Lagrangian for Constrained Systems 2.1.3 Hamilton’s Principle . . . . . . . . . 2.1.4 Examples of functional variation . .