Lectures on Mechanics

233 Pages·1992·1.355 MB·English

by  Marsden J.E.

Lectures on Mechanics
Second Edition
Jerrold E. Marsden
March 24, 1997
Contents
Preface iv
1 Introduction 1
1.1 The Classical Water Molecule and the Ozone Molecule . . . . . . . . 1
1.2 Lagrangian and Hamiltonian Formulation . . . . . . . . . . . . . . . 3
1.3 The Rigid Body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Geometry, Symmetry and Reduction . . . . . . . . . . . . . . . . . . 11
1.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Geometric Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 The Rotation Group and the Poincar¶e Sphere . . . . . . . . . . . . . 23
2 A Crash Course in Geometric Mechanics 26
2.1 Symplectic and Poisson Manifolds . . . . . . . . . . . . . . . . . . . 26
2.2 The Flow of a Hamiltonian Vector Field . . . . . . . . . . . . . . . . 28
2.3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Lie-Poisson Structures and the Rigid Body . . . . . . . . . . . . . . 30
2.6 The Euler-Poincar¶e Equations . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Symplectic and Poisson Reduction . . . . . . . . . . . . . . . . . . . 37
2.9 Singularities and Symmetry . . . . . . . . . . . . . . . . . . . . . . . 40
2.10 A Particle in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . 41
3 Tangent and Cotangent Bundle Reduction 44
3.1 Mechanical G-systems . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 The Classical Water Molecule . . . . . . . . . . . . . . . . . . . . . . 47
3.3 The Mechanical Connection . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 The Geometry and Dynamics of Cotangent Bundle Reduction . . . . 55
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 Lagrangian Reduction and the Routhian . . . . . . . . . . . . . . . . 65
3.7 The Reduced Euler-Lagrange Equations . . . . . . . . . . . . . . . . 70
3.8 Coupling to a Lie group . . . . . . . . . . . . . . . . . . . . . . . . . 72
i
ii
4 Relative Equilibria 76
4.1 Relative Equilibria on Symplectic Manifolds . . . . . . . . . . . . . . 76
4.2 Cotangent Relative Equilibria . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 The Rigid Body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 The Energy-Momentum Method 90
5.1 The General Technique . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Example: The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Block Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 The Normal Form for the Symplectic Structure . . . . . . . . . . . . 102
5.5 Stability of Relative Equilibria for the Double Spherical Pendulum . 105
6 Geometric Phases 108
6.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Cotangent Bundle Phases | a Special Case . . . . . . . . . . . . . . 111
6.4 Cotangent Bundles | General Case . . . . . . . . . . . . . . . . . . 113
6.5 Rigid Body Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6 Moving Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.7 The Bead on the Rotating Hoop . . . . . . . . . . . . . . . . . . . . 118
7 Stabilization and Control 121
7.1 The Rigid Body with Internal Rotors . . . . . . . . . . . . . . . . . . 121
7.2 The Hamiltonian Structure with Feedback Controls . . . . . . . . . . 122
7.3 Feedback Stabilization of a Rigid Body with a Single Rotor . . . . . 123
7.4 Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 The Kaluza-Klein Description of Charged Particles . . . . . . . . . . 130
7.6 Optimal Control and Yang-Mills Particles . . . . . . . . . . . . . . . 132
8 Discrete reduction 135
8.1 Fixed Point Sets and Discrete Reduction . . . . . . . . . . . . . . . . 137
8.2 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4 Sub-Block Diagonalization with Discrete Symmetry. . . . . . . . . . 148
8.5 Discrete Reduction of Dual Pairs . . . . . . . . . . . . . . . . . . . . 151
9 Mechanical Integrators 155
9.1 Deflnitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2 Limitations on Mechanical Integrators . . . . . . . . . . . . . . . . . 158
9.3 Symplectic Integrators and Generating Functions . . . . . . . . . . . 160
9.4 Symmetric Symplectic Algorithms Conserve J . . . . . . . . . . . . . 161
9.5 Energy-Momentum Algorithms . . . . . . . . . . . . . . . . . . . . . 163
9.6 The Lie-Poisson Hamilton-Jacobi Equation . . . . . . . . . . . . . . 164
9.7 Example: The Free Rigid Body . . . . . . . . . . . . . . . . . . . . . 168
9.8 Variational Considerations . . . . . . . . . . . . . . . . . . . . . . . . 169
iii
10 Hamiltonian Bifurcation 170
10.1 Some Introductory Examples . . . . . . . . . . . . . . . . . . . . . . 170
10.2 The Role of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.3 The One-to-One Resonance and Dual Pairs . . . . . . . . . . . . . . 182
10.4 Bifurcations in the Double Spherical Pendulum . . . . . . . . . . . . 183
10.5 Continuous Symmetry Groups and Solution Space Singularities . . . 185
10.6 The Poincar¶e-Melnikov Method . . . . . . . . . . . . . . . . . . . . . 186
10.7 The Role of Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.8 Double Bracket Dissipation . . . . . . . . . . . . . . . . . . . . . . . 200
References 204
Index 223
Preface
Many of the greatest mathematicians | Euler, Gauss, Lagrange, Riemann,
Poincar¶e, Hilbert, Birkhofi, Atiyah, Arnold, Smale | were well versed in
mechanics and many of the greatest advances in mathematics use ideas from
mechanicsinafundamentalway. Whyisitnolongertaughtasabasicsubject
to mathematicians? Anonymous
Iventuretohopethatmylecturesmayinterestengineers,physicists,andas-
tronomersaswellasmathematicians. Ifonemayaccusemathematiciansasa
classofignoringthemathematicalproblemsofthemodernphysicsandastron-
omy,onemay,withnolessjusticeperhaps,accusephysicistsandastronomers
of ignoring departments of the pure mathematics which have reached a high
degreeofdevelopmentandareflttedtorendervaluableservicetophysicsand
astronomy. It is the great need of the present in mathematical science that
the pure science and those departments of physical science in which it flnds
its most important applications should again be brought into the intimate
associationwhichprovedsofruitfulintheworkofLagrangeandGauss. Felix
Klein, 1896
These lectures cover a selection of topics from recent developments in the ge-
ometric approach to mechanics and its applications. In particular, we emphasize
methods based on symmetry, especially the action of Lie groups, both continuous
and discrete, and their associated Noether conserved quantities veiwed in the geo-
metriccontextofmomentummaps. Inthissetting,relative equilibria,theanalogue
of flxed points for systems without symmetry are especially interesting. In general,
relative equilibria are dynamic orbits that are also group orbits. For the rotation
group SO(3), these are uniformly rotating states or, in other words, dynamical
motions in steady rotation.
Some of the main points to be treated are as follows:
† The stability of relative equilibria analyzed using the method of separation of
internal and rotationalmodes, also referred to as the block diagonalization or
normal form technique.
† Geometric phases, including the phases of Berry and Hannay, are studied
using the technique of reduction and reconstruction.
† Mechanical integrators, such as numerical schemes that exactly preserve the
symplectic structure, energy, or the momentum map.
iv
Preface v
† Stabilizationandcontrolusingmethodsespeciallyadaptedtomechanicalsys-
tems.
† Bifurcation of relative equilibria in mechanical systems, dealing with the ap-
pearanceofnewrelativeequilibriaandtheirsymmetrybreakingasparameters
arevaried,andwiththedevelopmentofcomplex(chaotic)dynamicalmotions.
Aunifyingthemeformanyoftheseaspectsisprovidedbyreductiontheoryand
theassociatedmechanicalconnectionformechanicalsystemswithsymmetry. When
one does reduction, one sets the corresponding conserved quantity (the momentum
map) equal to a constant, and quotients by the subgroup of the symmetry group
that leaves this set invariant. One arrives at the reduced symplectic manifold that
itself is often a bundle that carries a connection. This connection is induced by a
basic ingredient in the theory, the mechanical connection on conflguration space.
This point of view is sometimes called the gauge theory of mechanics.
The geometry of reduction and the mechanical connection is an important in-
gredient in the decomposition into internal and rotational modes in the block diag-
onalization method, a powerful method for analyzing the stability and bifurcation
of relative equilibria. The holonomy of the connection on the reduction bundle
gives geometric phases. When stability of a relative equilibrium is lost, one can get
bifurcation, solution symmetry breaking, instability and chaos. The notion of
system symmetry breaking in which not only the solutions, but the equations
themselves lose symmetry, is also important but here is treated only by means of
some simple examples.
Two related topics that are discussed are control and mechanical integrators.
One would like to be able to control the geometric phases with the aim of, for ex-
ample,controllingtheattitudeofarigidbodywithinternalrotors. Withmechanical
integratorsoneisinterestedindesigningnumericalintegratorsthatexactlypreserve
the conserved momentum (say angular momentum) and either the energy or sym-
plectic structure, for the purpose of accurate long time integration of mechanical
systems. Suchintegratorsarebecomingpopularmethodsastheirperformancegets
tested in speciflc applications. We include a chapter on this topic that is meant to
be a basic introduction to the theory, but not the practice of these algorithms.
This work proceeds at a reasonably advanced level but has the corresponding
advantage of a shorter length. For a more detailed exposition of many of these
topicssuitableforbeginningstudentsinthesubject,seeMarsdenandRatiu[1994].
The work of many of my colleagues from around the world is drawn upon in
these lectures and is hereby gratefully acknowledged. In this regard, I especially
thank Mark Alber, Vladimir Arnold, Judy Arms, John Ball, Tony Bloch, David
Chillingworth, Richard Cushman, Michael Dellnitz, Arthur Fischer, Mark Gotay,
Marty Golubitsky, John Harnad, Aaron Hershman, Darryl Holm, Phil Holmes,
John Guckenheimer, Jacques Hurtubise, Sameer Jalnapurkar, Vivien Kirk, Wang-
Sang Koon, P.S. Krishnaprasad, Debbie Lewis, Robert Littlejohn, Ian Melbourne,
Vincent Moncrief, Richard Montgomery, George Patrick, Tom Posbergh, Tudor
Ratiu, Alexi Reyman, Gloria Sanchez de Alvarez, Shankar Sastry, Ju˜rgen Scheurle,
Mary Silber, Juan Simo, Ian Stewart, Greg Walsh, Steve Wan, Alan Weinstein,
Preface vi
Shmuel Weissman, Steve Wiggins, and Brett Zombro. The work of others is cited
at appropriate points in the text.
I would like to especially thank David Chillingworth for organizing the LMS
lectureseriesinSouthampton,April15{19,1991thatactedasamajorstimulusfor
preparing the written version of these notes. I would like to also thank the Mathe-
maticalSciencesResearchInstituteandespeciallyAlanWeinsteinandTudorRatiu
at Berkeley for arranging a preliminary set of lectures along these lines in April,
1989, and Francis Clarke at the Centre de Recherches Math¶ematique in Montr¶eal
for his hospitality during the Aisenstadt lectures in the fall of 1989. Thanks are
also due to Phil Holmes and John Guckenheimer at Cornell, the Mathematical
Sciences Institute, and to David Sattinger and Peter Olver at the University of
Minnesota, and the Institute for Mathematics and its Applications, where several
of these talks were given in various forms. I also thank the Humboldt Stiftung of
Germany, Ju˜rgen Scheurle and Klaus Kirchga˜ssner who provided the opportunity
and resources needed to put the lectures to paper during a pleasant and fruitful
stay in Hamburg and Blankenese during the flrst half of 1991. I also acknowledge a
varietyofresearchsupportfromNSFandDOEthathelpedmaketheworkpossible.
Ithankseveralparticipantsofthelectureseriesandothercolleaguesfortheiruseful
comments and corrections. I especially thank Hans Peter Kruse, Oliver O’Reilly,
Rick Wicklin, Brett Zombro and Florence Lin in this respect.
Very special thanks go to Barbara for typesetting the lectures and for her sup-
port in so many ways. Thomas the Cat also deserves thanks for his help with our
understanding of 180– cat manouvers. This work was not responsible for his unfor-
tunate fall from the roof (resulting in a broken paw), but his feat did prove that
cats can execute 90– attitude control as well.
Chapter 1
Introduction
Thischaptergivesanoverviewofsomeofthetopicsthatwillbecoveredsothereader
can get a coherent picture of the types of problems and associated mathematical
structures that will be developed.1
1.1 The Classical Water Molecule and the Ozone
Molecule
Anexamplethatwillbeusedtoillustratevariousconceptsthroughouttheselectures
is the classical (non-quantum) rotating \water molecule". This system, shown in
Figure 1.1.1, consists of three particles interacting by interparticle conservative
forces (one can think of springs connecting the particles, for example). The total
energy of the system, which will be taken as our Hamiltonian, is the sum of the
kinetic and potenial energies, while the Lagrangian is the difierence of the kinetic
and potential energies. The interesting special case of three equal masses gives the
\ozone" molecule.
We use the term \water molecule" mainly for terminological convenience. The
full problem is of course the classical three body problem in space. However,
thinking of it as a rotating system evokes certain constructions that we wish to
illustrate.
Imaginethismechanicalsystemrotatinginspaceand, simultaneously, undergo-
ing vibratory, or internal motions. We can ask a number of questions:
† How does one set up the equations of motion for this system?
† Is there a convenient way to describe steady rotations? Which of these are
stable? When do bifurcations occur?
† Is there a way to separate the rotational from the internal motions?
1We are grateful to Oliver O’Reilly, Rick Wicklin, and Brett Zombro for providing a helpful
draftofthenotesforanearlyversionofthislecture.
1
1. Introduction 2
m
m
z
r
r
1 2 M
R
y
x
Figure 1.1.1: The rotating and vibrating water molecule.
† Howdovibrationsafiectoverallrotations? Canoneusethemtocontrol overall
rotations? To stabilize otherwise unstable motions?
† Can one separate symmetric (the two hydrogen atoms moving as mirror im-
ages) and non-symmetric vibrations using a discrete symmetry?
† Does a deeper understanding of the classical mechanics of the water molecule
help with the corresponding quantum problem?
It is interesting that despite the old age of classical mechanics, new and deep
insightsarecomingtolightbycombiningtherichheritageofknowledgealreadywell
founded by masters like Newton, Euler, Lagrange, Jacobi, Laplace, Riemann and
Poincar¶e, with the newer techniques of geometry and qualitative analysis of people
likeArnoldandSmale. Ihopethatalreadytheclassicalwatermoleculeandrelated
systems will convey some of the spirit of modern research in geometric mechanics.
Thewatermoleculeisinfacttoohardanexampletocarryoutinasmuchdetail
as one would like, although it illustrates some of the general theory quite nicely. A
simpler example for which one can get more detailed information (about relative
equilibriaandtheirbifurcations,forexample)isthedouble spherical pendulum.
Here, instead of the symmetry group being the full (non-abelian) rotation group
SO(3), it is the (abelian) group S1 of rotations about the axis of gravity. The
doublependulumwillalsobeusedasathreadthroughthelectures. Theresultsfor
thisexamplearedrawnfromMarsdenandScheurle[1993]. Tomakesimilarprogress
with the water molecule, one would have to deal with the already complex issue of
flnding a reasonable model for the interatomic potential. There is a large literature
on this going back to Darling and Dennison [1940] and Sorbie and Murrell [1975].
Forsomeoftherecentworkthatmightbeimportantforthepresentapproach, and
for more references, see Xiao and Kellman [1989] and Li, Xiao and Kellman [1990].
1. Introduction 3
The special case of the ozone molecule with its three equal masses is also of
great interest, not only for environmental reasons, but because this molecule has
more symmetry than the water molecule. In fact, what we learn about the water
moleculecanbeusedtostudytheozonemoleculebyputtingm=M. Abigchange
thathasveryinterestingconsequencesisthefactthatthediscretesymmetrygroup
is enlarged from \re(cid:176)ections" Z to the \symmetry group of a triangle" D . This
2 3
situation is also of interest in chemistry for things like molecular control by using
laserbeamstocontrolthepotentialinwhichthemoleculeflndsitself. Somebelieve
that, together with ideas from semiclassical quantum mechanics, the study of this
systemasaclassicalsystemprovidesusefulinformation. WerefertoPierce,Dahleh
and Rabitz [1988], Tannor [1989] and Tannor and Jin [1991] for more information
and literature leads.
1.2 Lagrangian and Hamiltonian Formulation
Around 1790, Lagrange introduced generalized coordinates (q1;:::;qn) and their
velocities(q_q;:::;q_n)todescribethestateofamechanicalsystem. Motivatedbyco-
variance (coordinate independence) considerations, he introduced the Lagrangian
L(qi;q_i),whichisoftenthekineticenergyminusthepotentialenergy,andproposed
the equations of motion in the form
d @L @L
¡ =0; (1.2.1)
dt@q_i @qi
called the Euler-Lagrange equations. About 1830, Hamilton realized how to
obtain these equations from a variational principle
Z
b
– L(qi(t);q_i(t))dt=0; (1.2.2)
a
called the principle of critical action, in which the variation is over all curves
with two flxed endpoints and with a flxed time interval [a;b]. Curiously, Lagrange
knew the more sophisticated principle of least action, but not the proof of the
equivalence of (1.2.1) and (1.2.2), which is simple and is as follows. Let q(t;†) be a
family of curves with q(t)=q(t;0) and let the variation be deflned by
fl
fl
d fl
–q(t)= q(t;†)fl : (1.2.3)
d†
†=0
Note that, by equality of mixed partial derivatives,
–q_(t)=–_q(t):
R
Difierentiating bL(qi(t;†);q_i(t;†))dt in † at †=0 and using the chain rule gives
a
Z Z (cid:181) ¶
b b @L @L
– Ldt = –qi+ –q_i dt
@qi @q_i
a Za (cid:181) ¶
b @L d @L
= –qi¡ –qi dt
@qi dt@q_i
a