Analytical Mechanics, Lecture notes of Physics

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Analytical Mechanics

Analytical Mechanics

An Introduction

Antonio Fasano

University of Florence

Stefano Marmi

SNS, Pisa

Translated by Beatrice Pelloni University of Reading

3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York ©c 2002, Bollati Boringhieri editore, Torino English translation ©c Oxford University Press 2006 Translation of Meccanica Analytica by Antonio Fasano and Stefano Marmi originally published in Italian by Bollati-Boringhieri editore, Torino 2002 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published in English 2006 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Fasano, A. (Antonio) Analytical mechanics : an introduction / Antonio Fasano, Stefano Marmi; translated by Beatrice Pelloni. p. cm. Includes bibliographical references and index. ISBN-13: 978–0–19–850802– ISBN-10: 0–19–850802–

1. Mechanics, Analytic. I. Marmi, S. (Stefano), 1963- II. Title. QA805.2.F29 2002 531 ′.01—dc22 2005028822 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn ISBN 0–19–850802–6 978–0–19–850802– 1 3 5 7 9 10 8 6 4 2

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Contents

x Contents

10.8 Infinitesimal and near-to-identity canonical transformations. Lie series....................... 384 10.9 Symmetries and first integrals..................... 393 10.10 Integral invariants............................ 395 10.11 Symplectic manifolds and Hamiltonian dynamical systems............................ 397 10.12 Problems.................................. 399 10.13 Additional remarks and bibliographical notes........... 404 10.14 Additional solved problems...................... 405

11 Analytic mechanics: Hamilton–Jacobi theory and integrability.................................. 413 11.1 The Hamilton–Jacobi equation.................... 413 11.2 Separation of variables for the Hamilton–Jacobi equation....................... 421 11.3 Integrable systems with one degree of freedom: action-angle variables.......................... 431 11.4 Integrability by quadratures. Liouville’s theorem........ 439 11.5 Invariant l-dimensional tori. The theorem of Arnol’d...... 446 11.6 Integrable systems with several degrees of freedom: action-angle variables.......................... 453 11.7 Quasi-periodic motions and functions................ 458 11.8 Action-angle variables for the Kepler problem. Canonical elements, Delaunay and Poincar´e variables..... 466 11.9 Wave interpretation of mechanics.................. 471 11.10 Problems.................................. 477 11.11 Additional remarks and bibliographical notes........... 480 11.12 Additional solved problems...................... 481

12 Analytical mechanics: canonical perturbation theory............................... 487 12.1 Introduction to canonical perturbation theory.......... 487 12.2 Time periodic perturbations of one-dimensional uniform motions................................... 499 12.3 The equation Dω u = v. Conclusion of the previous analysis............................. 502 12.4 Discussion of the fundamental equation of canonical perturbation theory. Theorem of Poincar´e on the non-existence of first integrals of the motion........... 507 12.5 Birkhoff series: perturbations of harmonic oscillators..... 516 12.6 The Kolmogorov–Arnol’d–Moser theorem............. 522 12.7 Adiabatic invariants........................... 529 12.8 Problems.................................. 532

xii Contents

15.6 Maxwell–Boltzmann distribution and fluctuations in the microcanonical set........................ 627 15.7 Gibbs’ paradox.............................. 631 15.8 Equipartition of the energy (prescribed total energy)...... 634 15.9 Closed systems with prescribed temperature. Canonical set............................... 636 15.10 Equipartition of the energy (prescribed temperature)..... 640 15.11 Helmholtz free energy and orthodicity of the canonical set............................ 645 15.12 Canonical set and energy fluctuations................ 646 15.13 Open systems with fixed temperature. Grand canonical set........................... 647 15.14 Thermodynamical limit. Fluctuations in the grand canonical set....................... 651 15.15 Phase transitions............................. 654 15.16 Problems.................................. 656 15.17 Additional remarks and bibliographical notes........... 659 15.18 Additional solved problems...................... 662

16 Lagrangian formalism in continuum mechanics............. 671 16.1 Brief summary of the fundamental laws of continuum mechanics.......................... 671 16.2 The passage from the discrete to the continuous model. The Lagrangian function........................... 676 16.3 Lagrangian formulation of continuum mechanics......... 678 16.4 Applications of the Lagrangian formalism to continuum mechanics.................................. 680 16.5 Hamiltonian formalism......................... 684 16.6 The equilibrium of continua as a variational problem. Suspended cables............................. 685 16.7 Problems.................................. 690 16.8 Additional solved problems...................... 691

Appendices Appendix 1: Some basic results on ordinary differential equations............................... 695 A1.1 General results............................. 695 A1.2 Systems of equations with constant coefficients........ 697 A1.3 Dynamical systems on manifolds.................. 701 Appendix 2: Elliptic integrals and elliptic functions........... 705 Appendix 3: Second fundamental form of a surface........... 709 Appendix 4: Algebraic forms, differential forms, tensors........ 715 A4.1 Algebraic forms............................. 715 A4.2 Differential forms............................ 719 A4.3 Stokes’ theorem............................. 724 A4.4 Tensors.................................. 726

Contents xiii

Appendix 5: Physical realisation of constraints.............. 729 Appendix 6: Kepler’s problem, linear oscillators and geodesic flows................................. 733 Appendix 7: Fourier series expansions.................... 741 Appendix 8: Moments of the Gaussian distribution and the Euler Γ function............................ 745

Bibliography....................................... 749

Index............................................ 759

1 GEOMETRIC AND KINEMATIC FOUNDATIONS

OF LAGRANGIAN MECHANICS

Geometry is the art of deriving good reasoning from badly drawn pictures^1

The first step in the construction of a mathematical model for studying the motion of a system consisting of a certain number of points is necessarily the investigation of its geometrical properties. Such properties depend on the possible presence of limitations (constraints) imposed on the position of each single point with respect to a given reference frame. For a one-point system, it is intuitively clear what it means for the system to be constrained to lie on a curve or on a surface, and how this constraint limits the possible motions of the point. The geometric and hence the kinematic description of the system becomes much more complicated when the system contains two or more points, mutually constrained; an example is the case when the distance between each pair of points in the system is fixed. The correct set-up of the framework for studying this problem requires that one first considers some fundamental geometrical properties; the study of these properties is the subject of this chapter.

1.1 Curves in the plane

Curves in the plane can be thought of as level sets of functions F : U → R (for our purposes, it is sufficient for F to be of class C^2 ), where U is an open connected subset of R^2. The curve C is defined as the set

C = {(x 1 , x 2 ) ∈ U |F (x 1 , x 2 ) = 0}. (1.1)

We assume that this set is non-empty. Definition 1.1 A point P on the curve (hence such that F (x 1 , x 2 ) = 0) is called non-singular if the gradient of F computed at P is non-zero:

∇F (x 1 , x 2 ) =/ 0. (1.2)

A curve C whose points are all non-singular is called a regular curve.
By the implicit function theorem, if P is non-singular, in a neighbourhood of P the curve is representable as the graph of a function x 2 = f (x 1 ), if (∂F/∂x 2 )P =/ 0,

(^1) Anonymous quotation, in Felix Klein, Vorlesungen ¨uber die Entwicklung der Mathematik im 19. Jahrhundert, Springer-Verlag, Berlin 1926.

2 Geometric and kinematic foundations of Lagrangian mechanics 1.

or of a function x 1 = f (x 2 ), if (∂F/∂x 1 )P =/ 0. The function f is differentiable in the same neighbourhood. If x 2 is the dependent variable, for x 1 in a suitable open interval I,

C = graph (f ) = {(x 1 , x 2 ) ∈ R^2 |x 1 ∈ I, x 2 = f (x 1 )}, (1.3)

and

f ′(x 1 ) = −

∂F/∂x 1 ∂F/∂x 2

Equation (1.3) implies that, at least locally, the points of the curve are in one-to-one correspondence with the values of one of the Cartesian coordinates. The tangent line at a non-singular point x 0 = x(t 0 ) can be defined as the first-order term in the series expansion of the difference x(t) − x 0 ∼ (t − t 0 ) ˙x(t 0 ), i.e. as the best linear approximation to the curve in the neighbourhood of x 0. Since ˙x · ∇F (x(t)) = 0, the vector ˙x(t 0 ), which characterises the tangent line and can be called the velocity on the curve, is orthogonal to ∇F (x 0 ) (Fig. 1.1). More generally, it is possible to use a parametric representation (of class C^2 ) x : (a, b) → R^2 , where (a, b) is an open interval in R:

C = x((a, b)) = {(x 1 , x 2 ) ∈ R^2 | there exists t ∈ (a, b), (x 1 , x 2 ) = x(t)}. (1.4)

Note that the graph (1.3) can be interpreted as the parametrisation x(t) = (t, f (t)), and that it is possible to go from (1.3) to (1.4) introducing a function x 1 = x 1 (t) of class C^2 and such that x˙ 1 (t) =/ 0. It follows that Definition 1.1 is equivalent to the following.

x 2 F ( x 1 , x 2 ) = 0

∇ F

x ( t )

x 1

x· ( t )

Fig. 1.

4 Geometric and kinematic foundations of Lagrangian mechanics 1.

In the particular case of a graph x 2 = f (x 1 ), equation (1.5) becomes

l =

∫ (^) b

a

1 + (f ′(t))^2 dt. (1.6)

Example 1. Consider a circle of radius r. Since | x˙(t)| = |(−r sin t, r cos t)| = r, we have

l =

∫ (^2) π 0 r^ dt^ = 2πr.^

Example 1. The length of an ellipse with semi-axes a ≥ b is given by

l =

∫ (^2) π

0

a^2 cos^2 t + b^2 sin^2 t dt = 4a

∫ (^) π/ 2

0

a^2 − b^2 a^2

sin^2 t dt

= 4aE

a^2 − b^2 a^2

= 4aE(e),

where E is the complete elliptic integral of the second kind (cf. Appendix 2) and e is the ellipse eccentricity.

Remark 1. The length of a curve does not depend on the particular choice of paramet- risation. Indeed, let τ be a new parameter; t = t(τ ) is a C^2 function such that dt/dτ =/ 0, and hence invertible. The curve x(t) can thus be represented by

x(t(τ )) = y(τ ),

with t ∈ (a, b), τ ∈ (a′, b′), and t(a′) = a, t(b′) = b (if t′(τ ) > 0; the opposite case is completely analogous). It follows that

l =

∫ (^) b

a

| x˙(t)| dt =

∫ (^) b′

a′

dx dt

(t(τ ))

dt dτ

∣ dτ^ =

∫ (^) b′

a′

dy dτ

(τ )

∣ dτ.^

Any differentiable, non-singular curve admits a natural parametrisation with respect to a parameter s (called the arc length, or natural parameter ). Indeed, it is sufficient to endow the curve with a positive orientation, to fix an origin O on it, and to use for every point P on the curve the length s of the arc OP (measured with the appropriate sign and with respect to a fixed unit measure) as a coordinate of the point on the curve:

s(t) = ±

∫ (^) t

0

| x˙(τ )| dτ (1.7)

1.2 Geometric and kinematic foundations of Lagrangian mechanics 5

x 2

x 1

O S

P ( s )

Fig. 1.

(the choice of sign depends on the orientation given to the curve, see Fig. 1.2). Note that | s˙(t)| = | x˙(t)| =/ 0. Considering the natural parametrisation, we deduce from the previous remark the identity

s =

∫ (^) s

0

dx dσ

∣ dσ,

which yields ∣ ∣ ∣∣^ dx ds

(s)

∣ = 1^ for all^ s.^ (1.8)

Example 1. For an ellipse of semi-axes a ≥ b, the natural parameter is given by

s(t) =

∫ (^) t

0

a^2 cos^2 τ + b^2 sin^2 τ dτ = 4aE

t,

a^2 − b^2 a^2

(cf. Appendix 2 for the definition of E(t, e)).

Remark 1. If the curve is of class C^1 , but the velocity x˙ is zero somewhere, it is pos- sible that there exist singular points, i.e. points in whose neighbourhoods the curve cannot be expressed as the graph of a function x 2 = f (x 1 ) (or x 1 = g(x 2 )) of class C^1 , or else for which the tangent direction is not uniquely defined.

Example 1. Let x(t) = (x 1 (t), x 2 (t)) be the curve

x 1 (t) =

−t^4 , if t ≤ 0 , t^4 , if t > 0 ,

x 2 (t) = t^2 ,