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FOLKMAR A . BORNEMANN

Homogenization in Time of Singularly Perturbed Conservative

Mechanical Systems

Preprint SC 97–48 (November 1997)

in memoriam Alfred Neumann (1902-1982)

Preface

Although the title might suggest it differently, this monograph is about a ertain method for establishing singular limits rather than about a clear

cut class of singularly perturbed problems. Using this part iular method I will address in a unified way such diverse topics as the micro-scale jus tification of the Lagrange-d'Alembert principle and the limit behavior of strong constraining potentials in classical mechanics on the one hand, and the adiabatic theorem of quantum mechanics on the other hand. I am confident that all these topics are paradigms of a larger class of singularl perturbed conservative mechanical systems which allow for the application of the method to be presented. Reflecting this, I have tried to apply the method to each case as directly as possible and refrained from studying an abstract superclass of problems which would leave the paradigms as mere examples. I believe that this "variationson-a-theme-style" of my presen-tation is more likely to make the method a working tool in the area than a "transformation-to-an-archetype-style" would be.

This monograph grew out of the attempt to understand the high fre-quency vibrations in classical molecular dynamics modeling. These non-linear vibrations are the major obstruction for an eficient and reliable nu-merical long term simulation. In the fall of 1994, I came up with the idea of studying the singular limit of these vibrations by means of the method of weak convergence which enjoys growing popularity in the study of sin-gularly perturbed nonlinear partial differential equations. Straightforward energy arguments led me to a qualitative understanding of the structural aspects of the limit system. However, additional ideas appeared to be nec essary for the explicit onstruction of the limit dynamics. I finally found these ideas in the physical conepts of virial theorems and adiabatic invari ants. For the single frequency case weak limit analogues and proofs were discovered soon, and the method was presented in the spring of 1996. In contrast, the multiple frequency case left two problems open: first, how to obtain a kind of component-wise virial theorem, and second, how to get rid of a certain perturbation term obstructing the multidimensional adiabatic ity argument. More than a year of struggle later, I discovered quite elegant solutions to both of these problems: the virial theorem generalizes to a matrix-ommutativity relation* that, after simultaneous diagonalization,

^To be found as Lemma I I 7 on p. 38 below.

I I PREFACE

implies the desired componentwise result. And the mentioned perturba-tion term vanishes as a onsequene of a resonan ondition by a strikingl short argument.*

In the meantime, I got involved in the study of mixed quantum-classical models in quantum chemistry. During the fall of 1996, my friend and col league Christof Schütte suggested to me to discuss the singular limit of a finite dimensional analogue of these models by transforming it to the kind of lassial mechanical systems that had already been studied by my method. However, the kinetic energy of the transformed system turned out to be of a more general type as considered before, making it necessary to generalize my results to mechanical systems on Riemannian manifolds. The Riemannian metric caused additional perturbation terms which, sur prisingly enough, vanished like magi beause of the already disovered generalization of the virial theorem.

Encouraged by this success I worked on a direct, untransformed version of the method of proof for these mixed quantum-classical models. The motivation was to deal also with the infinite dimensional ase involving partial differential equations. And, indeed, using appropriate concepts from physics and the right tools from functional analysis, I have not only been able to address this case but also to give a ne proof for the adiabati theorem of quantum mechanics.

All this endeavor shaped the paradigmatical point of view which I will pursue in this monograph. hope that the method presented here, i.e. the blend of weak onvergence techniques, virial theorems, and adiabati invariants, will find many interesting new applications and will help to establish, clarify, and unify results about singularly perturbed problems involving different time sales.

New York, Aprl 1991 Folkmar A. Bornemann

To be found as Lemma 1 1 1 on p. 42 below.

CKNOWLDGMENTS II

Acknowledgments^

It gives me pleasure to thank all those individuals who made this work possible by providing support, encouragement, advice, criticism, teaching, and friendship. Out of the many I would like to mention a fe to hom am especially thankful:

Olof Widlund, for inviting me to a one-year stay at the Courant In-stitute of Mathematical Sciences in New York. Relieving me from any obstructing obligations, he made possible an enjoyable and scientifically profitable visit at this lively pla ith its o n , uniquely stimulating re-searh atmosphere.

Peter Deuflhard, for his continuing support and promotion of many years. His foresighted interest in a contribution of applied mathematics to bio-technologies by the effiient numerial simulation of bio-molules initiated the beginnings of this work.

Christof Schütte, for fruitful and mutually stimulating discussions, for his encouragement and criticism. My work on the subject of this mono-graph started, in fact, as a joint enterprise with him, witnessed by our joint publications on the single frequen ase and on the mixed quantum-

lassial modeling issue. Luc Tartar and Frangois Murat, for their beautifully instructive and

stimulating series of lectures presented at the "DMV-Seminar on Compos ite Materials and Homogenization, Oberwolfach, Germany, Mar 12-18, 1995, which I had the unique opportunity to attend.1

And those, whose courses at the Freie Universität Berlin, some ten years ago, taught me several of the analytical skills which were essential for ac complishing this work: Volker Enß (real analysis), Michael Loss (differential geometry), and Dirk Werner (funtional analysis).

§My research on the subject of this monograph was s u p p o d in p a t by the U.S. epartment of Energy under contract DE-FG02-92ER25127. H This seminar was organized by the German mathematical society ( e u t c h e Mathe

matikerVereinigung)

IV

Contnts

Preface I cknoledgments Ill

I Introduction 1. The Basic Principles of Weak Convergence 2. An Illustration of the Method

2.1. The Model Problem 2.2. Step 1: qui-Boundedness (Energy Principle) . . . . 10 2.3. Step 2: The Weak Virial Theorem 12 2.4. Step 3: Adiabatic Invariance of the Normal Actions 12 2.5. Step 4: Identification of the Limit Mechanical System 13 2.6. Comments on the Notions ntrodued and the Result 14

II Homogenization of Natural Mechanical Systems 17 1. The Homogenization Result 18

1.1. Natural Mechanical Systems with a Strong Potential 18 1.2. The Critical Submanifold 19 1.3. Spectrally Smooth Constraining Potentials 20 1.4. Resonance Conditions 21 1.5. The Statement of the Homogenization Result . . . . 21 1.6. Homogenization of a Specific Class of Potentials . . 22 1.7. Remarks on Genericity 24 1.8. Counterexample for Flat Resonances 25 1.9. A Counterexample for Unbounded Energy 27 1.10. Bibliographical Remarks 28

2. The roof of the Homogenization Result 29 2.1. Step 1: qui-Boundedness (Energy Principle) . . . . 29 2.2. Step 2: The Weak Virial Theorem 36 2.3. Step 3: Adiabatic Invariance of the Normal Actions 40

§2.4. Step 4: Identification of the Limit Mechanical System 49 3. Realization of Holonomic Constraints 51

3.1. Conditions on the Initial Values 52 3.2. Conditions on the Constraining Potential 55

§3.3. Bibliographical Remarks 57 4. Spectrally Nonsmooth Constraining otentials: Takens Chaos 59

ONTENTS

III Applications 67 1. Magnetic Traps and Mirrors 68 2. Moleular Dynamics 70

2.1. An Example: The Butane Molecule 72 §2.2. Relation to the Fixman-Potential 77

3. Quantum-Classial Coupling: he Finite Dimensional Case 78

IV Adiabatic Results in Quantum Theory 83 1. The Adiabatic Theorem of Quantum Mechanics 85

1.1. he Result 85 1.2. The Proof 87 1.3. Extension to the Essential Spectrum 92 1.4. Remarks on the Limit Density Operator 94 1.5. The Example of a Two-Body Hamiltonian 96

§1.6. Bibliographical Remarks 97 2. Quantum-Classical Coupling: The Infinite Dimensional Case 98

2.1. he Singular Limit 99 2.2. The Proof 101 2.3. An Example 107 2.4. Remarks on the Born-Oppenheimer Approximation . 107

Appendix A: Eigenvalue Resonances of Codimension Tw 109

Appendix B: Advanced Tools from Functional Analysis 111 1. Weak* Convergence of OperatorValued Functions I l l

1.1. Trace Class Operators I l l 1.2. Lebesgue-Bochner Spaces 113 1.3. Spaes of Trace-ClassOperatorValued Functions . . 114 1.4. The Trace is Not Weakly* Sequentially Continuous . 117

§1.5. Trace Class Operators on Rigged Hilbert Spaces . . 118 2. SemiBounded Operators and Coercive Quadratic Forms . . 122

2.1. Rollnik-otentials 123

Appendix C: Asymptotic Studies of Two odel Problems 125 1. The Model Problem of the Introduction 125 2. Quantum-Classial Coupling (Finite Dimensional) 130

Appendix D: The Weak Vr ia l Theorem and Localization of emiclassical easures 137

Bibliography 139

ist of Symbols 145

ndex 14

oduti

Many problems of the applied sciences involve scales in time, or space, which are orders of magnitude different in size. The smallest scales, also called micro-scales, are caused, e.g., by dynamical effects or by the materi als involved, whereas the largest scales, also alled the macro-scales, involve the scales of observation or measurement. The micro-scales are frequently not measurable, or at least of no particular interest. Additionally, their presence poses severe problems for numerical simulations using todays or even future computing facilities. Thus, for first a deeper understanding of the underlying model, and second, for developing efficient and reliable nu-merical simulation methods, there is a strong need for macro-scale models

hich approximate the originally given model without involving the mi ro-ales.

Looking at many macro-scale models used in the natural sciences, one realizes that they were obtained phenomenologically, i.e., by analyzing the measurements, not by deriving them from a micro-scale model. For justi fying wellknown macro-scale models, or establishing even new ones, tech-niques for a mathematical modelderivation from the micro- to the macro-scale have beome i reas ing ly important and s ien t i fa l ly instrutive i r een t years.

A fundamental mathematical technique used in such a kind of model derivation is the identifiation of a scaleparameter e 0. Frequently, this limit changes, at least formally, the very nature of the mathematical model: Either some terms become formally ambiguous, or even non-sense, or, e.g., the order or type of a differential equation is changed. In these cases, one calls the model sin-gularly perturbed, and e the parameter of the singular perturbation. The analysis of the singular limit e —• 0 has to be either asymptotic or, in sense, oblivious to the micro-scale aspects of the solution.

In some models, singular perturbations cause rapid, micro-scale fluctua-tions in the solution. Therefore, an asymptotic description usually involves an explicit ansatz for the oscillatory part of the solution. This often requires much ingenuity and a lot of insight into the problem. Famous examples are provided by the perturbation theory of integrable Hamiltonian systems, cf. [6], the WKB method for semiclassical limits in quantum theory, cf. [68] the more recent technique of nonlinear geometric optics, cf. [67] [75], and the method of multiple-scale-asymptotics cf. [13] [53]

INTRODUCTION CHAP. I

On the other hand, macro-scale measurements can be viewed as a kind of averaging procedure, being oblivious of the rapid fluctuations on micro-scales. Mimicking this, one considers the limit of certain averages of the solutions or, equivalently, their weak limits. This method of weak conver gence has become increasingly popular in the study of nonlinear partial differential equations, since one has powerful tools from functional analysis at hand which allow to establish qualitative, or at least structural, infor mation about the limit system, cf. 28]. Even more, proving error bounds for a formal asymptoti (multiple-scale) analysis can be a hard problem

hich is sometimes a t taked by the method of weak onvergene.

Example 1. Both approaches to a singular limit problem can neatly be compared being applied to the so-called homogenization problem of elliptic partial differential equations. here, one studies the limit e 0 of the diffusion problems

divA(x/e)gr&dut(x) = f(x), xGflcW1, ue\on = 0,

where the 0, l ]per iodic diffusion matrix A() desribes a mi ro - sa le pe-riodi structure. For / g Ü_1(f2) we obtai

uf —>• Wo i n (f2)

and one might ask hether there is an effective diffusion matrix Aeg(x) such that

divA(x)gradu(x)=f(x), xGflcR, du=0-

It turns out that such an effective diffusion matrix exists and is constant indeed. However, this matrix is in general not simply given by some aver age value of the function A(). For this problem, a multiple-scale-analysis was set up by BENSOUSSAN, LIONS, and PAPANICOLAOU [13] he weak convergence method was pioneered by MURAT and TARTAR [70] [47]. In the latter reference one can also find a proof of an error bound for the multiple-

ale-expansion by means of the weak onvergene method, cf. [47, §1.]

The particularity of this example is provided by a nonlinear coupling, in fact a quadratic one, of the micro- and macro-scales, which leads to nontrivial problems and counterintuitive results in general. This effect of nonlinearities can be understood in a rather direct fashion by the method of weak convergence: nonlinear functionals are not weakly sequentially continuous in general.1 Describing the deviation from weak continuity, unexpected terms appear in the weak limit. In analogy to the elliptic example above, we will use the notion "homogenization," instead of the notion "averaging," if the derivation of a macro-sale model involves su unexpeted, ounter-intuitive, or nontrivial terms.

1For instance, in Example 1 above, the flux A(x/e) gradMe(x) is the product of two weakly converging sequences. Thus, although we have A(x/e) —>• A&r in L

rX}(Q,M.dxd) and grad?ie -*• grad^o in L

2(£l,l$L ) , there is not A(x/e) gradue(x) A a r g r a d M o i 2(fl,Md). This way, one understands why ^ r 7

INTRODUCTION

Purpose of this Monograph . It is our purpose in this monograph to present a particular method for the explicit homogenization of certain sin-gularly perturbed conservative mechanical systems. Caused by the singular perturbation and the conservation properties of the model, the solutions of these systems will show up rapid mi ro - sa le fluctuations. Our method will be based on energy priniples and weak convergence techniques. Sinc nonlinear functional are not weakly sequentially continuous, as mentioned above, we have to study simultaneously the weak limits of all those nonlin-ear quantities of the rapidly oscillating components which are of importance for the underlying problem. Using the physically motivated concepts of virial theorems, adiabatic invariants, and resonances, we will be able to establish sufficiently many relations between all these weak limits, a l lo ing to calculate them explicitly.

Our approach will be paradigmatical rather than aiming at the largest possible generality. This way, we can show most clearly how concepts and notions from the physical background of the underlying mathematical prob-lem enter and help to determine relations between weak limit quantities.

Example 2. This example presents a simplified version of the paradigm that will be discussed i Chapter II t is given by the singularly perturbed Ne ton ian equations

e + grad U(x) = 0,

describing a conservative mechanical system on the onfiguration s p a e Rm. If the potential U > 0 has a r i t ia l manifold

= {x G R : U(x) = 0} = {x G R : gradU(x) = 0},

e call it constraining to N. For fixed initial values x(0) = x* & and (0) = v* G K we obtai

in 1 ' 0 , r ] ]

where the limit average motion xo takes values in the manifold N. There-fore, naive intuition would expect that the limit X is dynamially desribed by the free, geodesi motion on

0 ± TX0

However, the rapidly oscillating velocities, being onl weakly convergent i general ause additional, unexpeted fore terms hich yield a limit

2In this restriction to just the smallest required class of nonlinearities our approach differs from utilizing and studying Young measures [28, § I.E.3] [97], or H-measures (mi-crolocal defect measures) [37] [95] [96] [97] and its scale-dependent variants like semiclas sical measures [36] or Wigner measures [65]. These advanced tools encode the weak limits of all possible quantities obtained from nonlinear substitutions or quadratic pseu-dodifferential operations, respectively. For the relation of our approach to semiclasical measures see Appendi

INTRODUCTION CHAP. I

dynamis of the for

x0 gradUhom(x0)

We call this limit system the homogenization of the given singularly per-turbed system. The nonlinearity responsible for the appearane of the homogenized potential Uhom is the quadratic approximation of U near to

. We ill ons t ru t t/hom explicitly for a large lass of potentials U.

Outline of Contents. For pedagogical reasons, in §2 of this introduc tory Chapter I we will introduce our method of proof for a very simple, illustrative special case of Example 2 above. This will help to clarify the basic structure of the argument. recedingly, we recall in §1 all those pre-requisites about weak convergen hi ill be needed in the first three chapters of this monograph.

The first major paradigm for the application of our method is subjec of Chapter II. There, we study natural mechanical systems on Riemannian manifolds, singularly perturbed by a strong onstraining potential We state and prove a homogenization result which considerably extends what is known from the only two references concerned ith this problem: i.e. work by TAKENS [94] from 1979, and by KELLER and RUBINSTEIN [52] from 1991. We show the necessity of resonance conditions on the one hand, but verify that genericity or transversality assumptions are sufficient on the other hand. As a special case of this paradigm we discuss the micro-scale justification of the Lagrange-d'Alembert principle by utilizing strong constraining potentials. Giving unified proofs for the results known about this justification, we additionally show the necessity of the conditions which prior to our work were only known to be sufficient. The chapter concludes with an explicit example that there is a strange sensitivity of the homogenization problem on the initial values, if the strong potential does not satisfy an important regularity assumption. For this effe was discovered by TAKENS [94] we call it Takens chaos.

Chapter III continues with a potpourri of appliations. We start by discussing the problem of guiding center motion in plasma physics and the elimination of fast vibrations in molecular dynamics. The closing ap-plication onsists of a simplified, finite-dimensional version of a model in quantum chemistry which describes the coupling of a quantum mechanical system with a lassical one. We sho that this model an be transformed to the paradigm of Chapter II.

The corresponding infinite-dimensional coupling model, in its original, untransformed guise leads to the seond major paradigm of this monograph,

3 The appearance of resonance-conditions for proving adiabaticity in multi-frequenc systems is a well-known fact in the perturbation theory of integrable Hamiltonian sys-tems, cf. [6, §5.4.2]. However, the resonance conditions employed there are far to res t i c tive for our purposes, see a l o Appendix C.

1] BAS PRINCIPLES OF EAK O N V E R G N C E

subject of Chapter IV. There, we relate the singular limit of the coupling model to the adiabatic theorem of quantum mechanics. By translating our method of proof to the appropriate concepts of quantum theory, we give a new proof for this theorem and, finally, discuss the singular limit of the coupling model. This way, we find ourselves in the curious situation having addressed adiabaticity in classical and quantum mechani with essential the same method of proof, thus adding new meaning to the orrespondence principle of Ehrenfest in the pre-Schrödinger, "old" quantum theory of the early twenties.

As the proofs in Chapter IV have to deal with operatorvalued function spaces and parameterdependent unbounded operators they require the ap-plication of more sophisticated tools from functional analysis than needed in the preceding chapters. These tools are not easily found and referenced in the literature, so that we have to provide some proofs for the reasons of logical completeness. However, for not obstructing the inherent simplic ity of the basic argument in Chapter IV, we p laed all the more general functional analyti material in Appendi B.

§1. The Basic Principles of Weak Convergence

Here, we present all those facts from real and functional analysis about weak* convergence that will be applied in the first three chapters. We group these facts into five basic principles, all well-known or trivial The experienced reader who is already familiar ith these onepts might ish to skip this section.

We consider sequenes {x€} of functions, indexed by a sequence {e} of real numbers which converge to zero, e > 0. Later on, e will be the singular perturbation parameter. All functions x will be defined on some bounded Lipschitz domain tt C M., the expression dx ill stand for any partial derivative djX, j = 1 , . . . , d.

We recall the fact [84, Theorem 6.16] that the funtion s p a e Lc(fl) is the dual of the funtion s p a e £1(fi),

L°°(fi) = ( i ^ ) )

Thus, the funtional analytic conept of weak*-onvergene speifies as f o l l s .

efinition 1. A sequene {xe} of L°(Q) converges weakly* to the limit x0 G i ° ( 0 ) , notated if and only if

(t)(j)(t)dt - (t) G i 1 ( f i ) . 4Additional material needed for Chapter IV can be found in Appendix B. 5We will write, for short {) fn (t)

INTRODUCTION CHAP. I

There is an alternative way of describing this kind of convergen hi larifies the onnetion to averaging or filtering onepts .

emma 1. A sequence {x} of Lc(fl) onverges weakly* t a limit (fl) and only i the follwing two properties hold:

(i) the sequence is bunded in (fl)

(ii) for every open rectangle I the corresponding integral mean value onverges

— xe(t)dt — x0(t)dt.

Proof. The linear span of the characteristic functions \i forms a dense subspace of L1^), cf. [98, §11, Theorem 4] and [98, §20, heorem 4]. This proves the "if"-part of the the assertion, since by property (i) the sequen {x} represents a sequence of equiontinuous linear forms on L1(f2).

The "only if" part follows from the uniform boundedness p r i i p l e of functional analysis [83, heorem 2.5] •

he weak* onvergence in L°(fi) helps to ignore rapid fluctuations in a convenient ay. As an example, we real l the w e l l k n o n Riemann-Lebesgue lemma 84, Se 5.14]: For xe(t) = os(t/e) one gets

xe i 0,l] 1)

The first principle, simple but extremely useful, relates the uniform on-vergene of funtions to the weak* onvergene of their derivatives.

Principle 1. Let {xe} be a sequence in C f i ) such that x€ — 0 in CQ). Then and only i the sequence dx is bunded in L0(fl) there holds

dx in (Q)

Proof. By the uniform boundedness principle [83, Theorem 2.5], a weakly* convergent sequence must be bounded, proving the necessity of the bound-edness condition. On the other hand, because C(Q) is a dense subspac of i1(f i ) , [63, Lemma 2.19], one can test for the weak* convergence of a bounded sequence in L(Q) with functions from C%(Q). Now, since by Holder's inequality uniform convergence in fi) implies weak* onvergen in L°(fl), e obtain by partial integration

dxe((j)) = xe(d4>)

for all testfunctions (j> G (fi), proving the s u f i e n y of the boundedness ondition. •

For instance, this principle provides quite a simple proof of the Rie-mann-Lebesgue lemma, q. ( 1 ) above: Si esin(t/e) — 0, uniforml

1] BAS PRINCIPLES OF EAK O N V E R G N C E

i C[0,1], the uniformly bounded sequene of its derivatives, os(t on-verges weakly* to zero, cos(i/e) 0.

A delicate point about weak* convergence is that nonlinear functionals are not weakly* sequentially continuous. Given nonlinear ontinuous function / : R R, there is, in general

A X f(x f(x).

A famous example is provided by the quadratic function f(x) = x2 and xe = cos(t/e). There, by a wel lknown generalization of the Riemann-Lebesgue lemma, 23, Lemma 1.2] we have tha t in L 0 , 1

xe o, but — os2 T - 0 . 2)

2T ;

This Zacfc of weak* sequential continuity is responsible for a lot of unex-pected results later on. It makes the study of singular perturbation prob-lems ith rapid flutuations a difficult but fascinating task.

However, if the weak* convergence enters an nonlinear expression onl linearly, passing to the limit is possible. A particular important instanc is provided by our second principle. This principle will be used most often in our work, and always without referring to it explicitly. Therefore, if w claim a passage to the weak* limit for a p rodut , the reader is cautioned to check carefull hether at most one factor is weakly* onverging.

Principle 2. Let there be the convergences xf XQ, weakly* in (Q,), and y uniforml in fl). Then we btain

in (Q).

Proof. Take (j> £ L1(n). Then, Lebesgue's theorem of dominated onver gene s h o s that strongly in L1(f2). herefore, we get

(xe • )() = xe{4>) ->• xo(yo0) = (xo • Vo)(4>),

hich proves the asserted weak*-onvergene. D

he next two principles establish a particularly convenient property of the weak*-topology: the Heine-Borel property, i.e., closed bounded sets are compact. Since the underlying topology is essentially induced by a metric, weak* convergene is almost as easy to handle as onvergene i 1R: bounded sequenes have onvergent subsequenes.

Principle 3. (Alaoglu Theorem). Let {xe} be a bounded sequence in the space L(Q). Then there is a subsequence {e'} and a functi XQ L°{ü), such that

in (Q).

INTRODUCTION CHAP. I

Proof. The Alaoglu theorem [83, heorem 3.15] of functional analysis states that a closed ball in L°(il) is compact with respect to the weak*-topology. Since the predual space L1(f2) is separable, cf. [63, Lemma 2.17], the weak*-topology is metrizable on closed balls [83, Theorem 3.16]. Hene , bounded sequenes have weak*-onvergent subsequenes. •

The next compactness principle is the classical Arzelä-Ascoli theorem, slightly extended to inlude some information on the derivative.

Principle 4. (An Extended Arzelä-Ascoli Theorem). Let {xe} be a bounded sequence in the space C 1 ( f2) of uniformly Lipschitz ontinuous functions Then there is a subsequence {e and functi 1(fi) such that

xe xo in fi), dxe dxo in (Q)

The partial derivatives dx and dx are classically defined almost every-where

Proof. The uniform bound on the Lipschitz constants of xe implies the equicontinuity of the sequence. Thus, the classical Arzelä-Asoli theorem [84, Theorem 11.28] shows that there is a subsequene e and a ontinuous function x0 € (Q), such that xe> —>• xo in C(fi).

By Rademacher's theorem [29, Theorem 3.1.2], a Lipschitz function x is differentiable almost everywhere. The thus obtained derivatives dx belong to L° and equal the weak derivatives of x in the sense of distributions. This way, one gets (e.g. 29, heorem 6.2.1] or 38, p. 154]) an isometri isomorphism

^ ) = ^ { Ü ) .

Hene, the sequences {dx} are bounded in L°(fl). By Priniple 3, w can choose the subsequence {e} in such a way that dx —*• % m L°(Q). Now, the very same argument as in the proof of Principle 1 s h o s that o = dxo- In particular, the above isomorphism implies that x 1(fi) hich finishes the proof •

By these compactness results, bounded sequences of functions turn out to be a mixture of sequences which converge in appropriate topologies. The fifth and final principle states a simple but very useful riterion for deiding about the onvergene of the given sequene itself

6 R U D I N [84, Theorem 11.29] gives a direct proof of this sequential v e r s in of the Alaoglu theorem—i.e., for spaces with a separable predual—which requires l i t l e mor than the ArzeläAscoli theorem [84, Theorem 11.28]. This sequential v e i o n is often called the Banach-Alaoglu theorem.

7It can be viewed as the Alaoglu theorem for the space C ^ ) = W ( n ) together with the compact Sobolev embedding (f2)

2] N ILLUSTRATION OF T E M E T O D

Principle 5. ("Uniqueness Implies Convergence") Let {xe} be a sequence in a sequentially compact Hausdorff space X. If every convergent subsequence of {xf} converges t one and the same element XQ € then the sequence converges itself

Proof. Suppose on the ontrary that xe does not converge to XQ. Then, there is a neighborhood (x) of x and a subsequen {e} such that

x e ( x 0 ) for all *)

Since is sequentially compact, there is a subsequence { e } of {e} su that { £ } onverges. By assumption » —>• x , a ontradition to *). D

As has been explicitly stated in the proof of rinciple 3, the weak*-topology of L° is metrizable on bounded sets. Thus, the convergen

r i i p l e 5 is appliable to the bounded sequenes of r i i p l e s 3 and 4.

§2 An Illustration of t e Me tod Here, we study a simple illustrative model problem which will serve as the skeleton of the arguments in Chapters II and IV. After having studied this model problem, the reader will more easily enjoy the inherent simplicity of our method which tends to be hidden by several technical difficulties later on. For instance, the technical difficulties in Chapter II are due to the dif ferential geometrical setting, in Chapter due to the infinite dimensional spaces and unbounded operators.

We have arranged this illustrative model problem in a way that the four basi steps of the argument ill be learly visible:

1. weak* ompactness based on an energy p r i i p l e ,

2. a weak virial theorem,

3. the adiabati invariane of the normal a t ions, and

4. the identifation of the limit mechanial system.

Surely, there would be shortcuts for this particularly simple problem which, however, are worth being sacrificed for a presentation of as many essential features and notions of the later proofs as possible.

2.1. he odel Problem

We consider the f o l l i n g singularly perturbed system of Newtonian equa-tions of motion,

\ + gradU(x) = 0, 3)

NTRODUCTION CHAP. I

describing a mechanical system with Euclidean onfiguration space M Rm. Splitting the coordinates according to x = (y, z) GW1 xW = Rm, speify the potential U by the quadrati expression

[x) = \ ( ) z , z) ) = d i a g K ) , . . c ) .

Here, {, •) denotes the Euclidean inner product on W; correspondingly, | • will denote Euclidean norms. We assume that the smooth funtions co\ are uniformly positive,8 ie. , there is a onstant w* > 0 s u h that

u\()>u*, e r , A l , . . . , r

The nonnegative potential U > 0 is alled to be constraining to the critical submanifold

= {x€M: U{x) = 0} = R {0} C M

Thus, for obvious reasons, we all y the tangential, and z the normal compo-nent of x. A omponen t i s e riting of the equations of motion, q. ( 3 ) , yields

W ^ ~ 2 ) z e , z ) , j = l,...,n,

(ii) e )z

We onsider initial values h i h are independent of e,

(0) = *, (0) = w*; z{0) = 0, ( 0 )=u* . 5)

Notice, that the particular choie ^e(0) = 0 is the only one that results in an eindependent bound for the conserved energy E€ of the system; in f a t , making the energy even independent of e,

Ee = i ^ l \%\ e - e , z ) = | h ^ = E 6)

2.2. Step 1: E q u i o u n d e d n e s s nergy Principle

In this step, energy and compactness arguments allow us to extract appro-priately converging quantities. Conservation of energy and H being positive definite immediately imply that the veloities are uniformly bounded,

e = 0 ( l ) e = 0 ( l )

as functions in C0 ,T] , given a certain final time 0 < T < o herefore, after integration, we also obtain the positional bounds

= 0 ( l ) , z = 0(l)

This asumpt ion is crucial cf. Footnot 13 on p. 18 as well as Eqs. (II6) and (1110)

2] ILUSTRATION OF T E M O D

he e s m a t

| e - 2 ± e - ) z z ) = ~2Uz E*

implies the uniform bound ze = e).

Inserting this into the first set (i) of the equations of motion ( 4 ) reveals the acceleration bound

(1).

N o , an application of the extended Arzelä-Ascoli theorem, Principle 4, and the Alaoglu theorem, Priniple 3, yields—after the extraction of a subsequen hi we denote by 0 again—the onvergenes

in ^ T ] ! ) i 0 , : ] E ) ,

in 0,T}R) " 1 in 0,T\W).

Multiplying the second set (ii) of the equations of motion ) by e and taking weak* limits gives, by realling riniple 1,

= H)n, i e . 0.

Because quadratic functional are not weakly* sequentially ontinuous i general annot expe t that the uniformly bounded matr ies

e ® e <

likewise converge weakly* to the zero matrix. Instead, after a further ap-plication of the Alaoglu theorem, Priniple 3, and an extration of subse-quenes, we obtain some limits and n

i 0,T]Wx), II£ A n 0 i 0,T]Wx).

These limit quantities S and n 0 will play a rucial role in the description of the limit dynamics of yo- As a first hint, we r e r i t e the first set (i) of the equations of motion 4) as

i t r ^ - f O - O .

By taking weak* limits on both sides, we obtain hat all the abstract limit equation for

^tv(dy0) - o ) . 7)

he next two steps establish sufficiently many relations between the limits yo, So, and n 0 , so that the force term of the abs t ra t limit equation an be expressed funtion of the limit alone.

NTRODUCTION CHAP. I

2.3. Step he Weak Virial heorem

First, we establish a relation between II0 and 0. he matri

e ® e)

converges uniformly to the zero matrix. herefore, taking weak* limits of its time derivative,

e ® e + e ®

yields, by r i i p l e 1, the equation

8)

n partiular, the diagonal entries of and I are related by

=c 9 )

This result is essentially about the energy distribution in the normal, os cillating component ze. he second set (ii) of the equations of motion (1.4 implies that each normal component z (A 1 , . . . , r) satisfies the equation of a fast harmonic oscillation hose frequeny is slowly perturbed,

) z 0 . 10)

We define the kinetic energy T ^ and the potential energy U^ of the normal A-omponent by

\ ^ | e - 2 =

the total energy is given by EjX T ^ + ^. The diagonal limit relation (1.9) implies that, in each omponent, the eak* limits of the kineti and potential energy are equal

abbreviating = a\. he limit of the total energy i

= c ) a x . 11)

The thus obtained equi-partitioning of energy into the kinetic and the po-tential part bears similarities with the virial theorem of classical mechanis. For this reason, all the result 8) the weak virial theorem.

2.4. Step 3 diabatic Invariance of the Normal Actions

Now, we establish a relation between 0 and ^ he normal actions are given by the energy-frequeny-ratios

— _ 1 r U\

2] ILUSTRATION OF T E M O D

We l sho heir adiabatc invarance, ie. , the unifor onvergen

onst.

This ill be accomplished by calulating the weak* limit of the time deriva-tive EQX in a twofold way. On the one hand, by using ( 1 0 ) , the time derivative of the energy E is

Thus, the time derivatives are bounded functions and a further application of the extended Arzelä-Ascoli theorem, rinciple 4, shows that the normal energies—and therefore the actions also—are in fact uniforml onverging. N o , the weak limit of the time derivative is given by

On the other hand, by a the direct differentiation of the limit ( 1 1 ) obtain

j t

A comparison of the two expressions obtained for implies the differen-tial equation

du)/dt du)\()/dt

Solving this expliitly s h o s that there are constants A = 1 , . . . , r) su that

— - — - , i e . — - — -u\ u\{y

he values of these onstants an be alulated at the initial time t = 0,

l i m ( 0 )

2.5. tep 4: dentifcation of the Lim Mechanical ystem

Finally, we reconsider the limit force field on the right hand side of the abs t ra t limit equation 7). By the results of the preeding set ion obtai

\ tr(d0) • 0) = | O) = o) = 9^hom(o)

NTRODUCTION CHAP. I

Here, we define the homogenized potental Uhom by

Notice that (7h0m—and therefore y—does not depend on the chosen sub-sequenes. By Principle 5, this a l l o s us to discard the extraction of sub-sequenes altogether. Summarizing, we have just proven the folloing the-orem.

Theorem 1. Let yhom be the soluti of the second order differential equ ti

/hom ^hom ( h o m ) , j = l , . 7 l ,

with initial values yhom(O) = j / * , yhom(O) = w*. Then, for every finite tim interval 0, T] btain the strong convergence

yhom in ^ T ] !

and the weak* convergences e~x and in L ° 0 , T ] R r ) .

2.6. omments on the Notions Introduced and the Resul

Let us comment on the notions "weak virial theorem" and "adiabati in-var i ane" as well as on one partiularity of the result, heorem 1.

Weak Virial Theorem. There is a much deeper relation of what we alled the weak virial theorem, Eq. (1.8), to the virial theorem of classi

cal mechanics than just equipartitioning of kinetic and potential energies. We follow the textbook of ABRAHAM and MARSDEN [1, p. 242ff.] for a recollection of the lassical virial theorem. Given a vector field X on the configuration spa , the assoiated momentum function P(X) is defined by

(X):TM M, {X){v) = X,v).

he virial function is defined by the oisson braket

{X) = {P{X),E}.

The virial theorem [1, Theorem 3.7.30] states that the time average of the virial funtion along a t rajetory is zero,

rT

— \ (X)(ydt - as

9For reasons of comparison, a further proof of this theorem, u t i l i ing asymp techniques, is subject of Appendi C.

2] ILUSTRATION OF T E M O D

Here, ve = ) denotes the velocity field of the trajectory under consid-eration. Now, we consider the following spe i f normal ve tor fields and momentum functions assoiated ith:

XX = z"-^ (XX)(v) = z? X,ß=l,...r

his yields the virial funtion

(Xx)(v) - ^ d _ E 1 E 1 L E 1 _ E 1

z» r2(H)z z?

hus, the assertion 8) an be r e r i t t e n in the form

(X)(ve) as O.

n f a t , this far reaching analogy with the virial theorem of lassial me-chanis motivated the name weak virial theorem.

Adiabatic nvariance. The time interval [0, T] under consideration is of the order 0(e~1rf), where re denotes a typical "period" of a small os cillation in the normal diretion. Thus, the usage of the notion "adiabati invariant" is in accordance with the usual definition given in textbooks on classical mechanics, as for instance in ARNOLD, KOZLOV, and NEISHTADT [6, Chap. 5.4]. In fact, the perturbation theory of integrable Hamiltonian systems is directly applicable for the single-frequency case r = 1. To this end, we resale time and positions and introdu orresponding momenta

~1t, q = ~1 ~1z, n = z.

Denoting derivatives with respect to the ne time by a prime, q. ( 4 ) is just the anonial system

dp€ dqe ^ rje

belonging to the energy E as defined i q. ( 6 ) hich, by suppresing the index ' 1 , ' transforms to

\ \v't \u{zq)rjt i u(eq)

Now, a result of the perturbation theory of integrable Hamiltonian systems, 6, Chap. 5.4, Theorem 24 and Example 20], s h o s that for times

e_1) there is the asymptoti

o + e) eq eq0 + e) = p0 + e),

NTRODUCTION CHAP. I

where 0Q is defined by the initial values and qo and po by the anonial equations of motion belonging to the limit energy function

£0 = i^ol 2 + to{eq0)-

These equations of motion are just the homogenized system of Theorem 1. Transforming ba we thus obtain, for times t = 0(1), the error estimates1

O + e), yhome), hom e).

The Result. Later on, the reader will notice that the analogues of The-orem 1 in the more complicated situations of Chapter II and Chapter IV require ertain resonance conditions to be imposed on the normal frequen-cies u\. This is due to the following fact: general nonlinear potentials U, position dependent eigenspaces of H, and general manifolds M and con-straints N introduce perturbations to the simple model of this introduction. Thus, instead of the harmoni osillator e q u a t i o n 1 0 ) ill only get something like

)z = 0(l)

While the equipartitioning of the kinetic and potential energy still holds true then, the adiabatic invariance of the action might suffer from reso-nanes . We illustrate this lai by the simple sa la r equation

\ + os(_1k;£),

with a constant frequeny U 0. For the initial values z(0) = z(0) = 0, we get the solution

(t) = ^ec~1tsm(e~1cot).

By q. ( 2 ) , the kineti and potential normal energy are equal in the limit,

(t) = \Xt)2 = \t2 os 2 (e -^ t ) e) (t) = ±t2,

Ht) = \

o m o g n i z i o f Na tu ra l Mechanical S y s t s w ong C o r a i n i n g n t a l

A natural mechanical system [6, p. 10] consists of a smooth Riemannian onfiguration manifold M with metric {, ) and a smooth potential funtion

K. he dynamis is desribed by the Lagrangian

(x,x) = ± x , ( x ) xeTx

We ill onsider a family of singularly perturbed potentials of the form

{x) =V{x)-2(x),

where the "strong" potential U is constraining to a smooth critical subman-ifold N C M. f we choose initial values with uniformly bounded energy, the solutions xe of the equations of motion are oscillating on a time-scale of order 0(e) within a distance of order 0(e) to the submanifold N. The se-quence of solutions onverges uniformly to some function of time taking values in N.

We will study the problem of a dynamical description for this limit x i.e., whether there is a mechanical system with configuration spac N such that xo is a solution of the corresponding equations of motion. We call this problem the homogenization problem for the given mechanical system.

For a large class of constraining potentials this homogenization prob-lem admits a surprisingly elegant and explicit solution which we will state in §1. In §2 w will present a proof of the homogenization result based on the method of weak convergene. Compared to the existing literature,11

this method of proof a l l o s to weaken the imposed resonan onditions onsiderably.

The so-called problem of realization of holonomic constraints provides a special case of the homogenization problem. Here, one studies the question, whether the limit Xo is just the solution of the equations of motion for the natural mechanical system with configuration spa N and potential V. By the Lagrange-d'Alembert principle, the limit e — would then "realize" the holonomic (positional) constraint x G N. In §3 we will establish necessary and s u f i e n t onditions for this to happen, first, on the initial values for

Short rv iews of the exi t ing literature can be found i §§1.10 and 3.3.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

general constraining potentials U, and seond, on the onstraining potential for general initial values. If the constraining potential does not belong to the class introduced

i §1, the limit dynamics can be of a ompletely different nature. In §4 we will present an explicit example for which the limit dynamics depends extremely sensitive on how the limit initial values are obtained. We will argue that in this case there is no solution of the homogenization problem

hich is omparably elegant to the result of §1.

§1 The Homogenization Result

Loosely speaking, the homogenization result states the following. Provided the potential U is "nice" and certain resonance conditions are fulfilled, the limit xo desribes the dynamics of a natural mechanical system on the submanifold N. The potential of this system an expliitly be onstruted from U and the given initial values.

The precise statement given in §1.5 requires the introduction of some notion first. The discussion of the model problem i an serve as a motivation for most of the definitions.

1.1. Natural Mechanical Systems w t h Strong otential

Let M be a smooth12 m-dimensional Riemannian manifold with metri ( ,•). For a sequence e — 0, we consider a family of mechanial systems on the onfiguration s p a e M given by the Lagrangians

{x,x) = \x, {x) fr2U{x), x G TX

with smooth potentials and U. We assume that V is bounded from below and U is non-negative. he corresponding singularly perturbed equation of motion is given by the ulerLagrange equation, [1, rop. 3.7.4]

V i 6 e + grad V(x ~2 grad U(x) = 0, I I l )

where the covariant derivative V denotes the LeviCivit onnetion of the Riemannian manifold M he energy, [1, Sect. 3.7]

\{ (x ~2U(x)

is a constant of motion. If we assume that the energy surfaes Ee = const are compact submanifolds of the tangent bundle TM, the flow of the equa-tion of motion (II.l) is complete, [1, Prop. 2.1.17]. This means that any orresponding initial value problem is solvable for all times

1 2The term "smooth" will mean "at least four times continuously differentiable" throughout this chapter.

13For potentials other than this there is no reasonable singular limit. For instance U(x) = —x2/2 yields (setting V = 0) an exponentially diveging family x€ esinh(t/e) of solutions with energy E 1/2, independent of e

1] E HOMOGENIZATION RESUL

Since we study the singular limit behavior of a family of mechanial problems it is physically reasonable to bound the energy uniformly in e, Ee < E*. In fact, this is a condition on the initial values, whi hoose to be fixed in the positions and onverging in the veloities,14

ar£(0) = x lim xe(0) = H2) - M

he equiboundedness of the energy directly implies that (x*) 0. herefore, energy itself onverges as a number in ffi,

±v*,v*(x*). II3)

1.2. he ritical Submanifold

The set where the potential U vanishes is of utmost importane for the limit behavior under study.

Definition 1. Let the potential U be non-negative, U > 0, and let iV {x e M : U(x) = 0} C M be a compact,15 smoothly embedded n-dimen-sional submanifold such that N = {x G M : DU(x) = } and the Hessian

of U, defined as a field of linear operators H : T TM by1

(x)u,v) = D(x)(u,v), u,v T^M, i e

fulfills the nondegenera ondition

kerF(a;) = 7 ; x e II4)

Then N will be called a nondegenerate critical manifold17 of [ and the potential U ill be alled constraining to

There are two equivalent formulations of the nondegeneracy condition (II.4) that we will frequently use. As the Hessian H is selfadjoint ith respet to the Riemannian met r i , a first equivalent is given by

rangeH(x) =T x € II5) 1 4This particular choice is for simplicity and elegance of the result only. At the expens

of a far more technical result, one could consider converging initial positions as well. 1 6There is no loss of geneali ty since we deal with compact energy surfaces and finit

time intervals only. 1 6The Hessian H is invariantly defined on the tangent bundle restricted to base-point

in N only. Note, that the second deivative D2 is no tensor field but coordinat dependent in general

dx dx

dxidxi dxkdxl dxi dxi dxk dxidxi

However, the second term of the right hand side vanishes on the critcal manifold N. 1 7This notion was introduced by B O T T 20] in his study of pa rametedependen t Mor

theory.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

Using the fact that U > 0 and a compatness argument, we obtain as a seond equivalent that there is a onstant CJ* 0 s u h that

(x)u,u)>Lu,u), u&Tx xe II6)

Parts of these onditions an be described conveniently using certain pro-jections. Let P : TM and Q : TM be the bundle maps defined by letting

{x) : TXM TX (x) : TXM Ts a; G

the orthogonal projections of TXM onto T^1- and onto T ^ , respectively.

N o , the nondegenera onditions impl

PH = H, QH 0. II7)

1.3. Spectrall mooth Constraining otentials

We now introduce the class of constraining potentials U for hich the homogenization problem is solvable. For each x £ N the Hessian H(x) is a selfadjoint linear operator on TXM. Therefore, it is diagonizable. We ill need that the spe t rum an be arranged in a ontiguous way.

Definition 2. Let U be a potential onstraining to the n-dimensional non-degenerate critical submanifold If the Hessian H of U has a smooth spetra l deomposition on

(x) = (x)(x), xG II8)

the potential U will be called to onstrain spectrally smooth to N. Here, the smooth bundle maps Px : TM\N J T define by Px(x) : TXM > TxN (x G N) orthogonal projetions of T onto mutually orthogonal subspaes of T

The nondegeneracy condition (II.5) implies that P = J2\ P\- F° r refer e n e , we state the orthogonality properties of these projetions expliitly,

PX, Px ß, PA, II9)

denoting by the adjoint linear operator ith respet to the Riemannian metric.

The smooth scalar fields LJ\ : M — represent all the nonzero eigen-values of the Hessian; the nondegenera ondition II6) is equivalent to the uniform lower bound

> 0, x G 1110)

1] E HOMOGENIZATION RESUL

Therefore, the square root u)\ of each eigenvalue consitutes a smooth func tion. For reasons which will beome clear later on, we call these functions the normal frequencies of H. Without loss of generality we may assume that they are mutually non-identical on N since otherwise we ould om-bine the orresponding eigenprojetions. Moreover, the integers

tr P\ = dim range PA \ = r n,

are constants on , summing up to the codimension r m n of the crit ical submanifold N. We call n\ the smooth multiplicity of the ydependent family LJ\ of eigenvalues. Notie , that the multiplicity of to\() at a partic-ular point y € N might be accidentally greater than \. Such points are alled resonance points.

1.4. Resonance Conditions

The homogenization result we are going to prove relies on imposing certain resonance conditions on the normal frequenies

22 HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

we set

) = wA(a;) xG II13)

The potential Uhom will be called the homogenization of the onstraining potential U ith respet to the initial values v , x

Physically speaking, the constants 9$ have the dimension of an action as a ratio of energy and frequency. Later, in §2, our proof will identify them as the adiabatic invariants of the motion normal to the onstraint manifold

Theo rem 1. F r a sequence e — 0, sider the famil of echanical systems given by the Lagrangian

{x,x) = \x, {x) fr2U{x), x G TX

The potential U is assumed strain sectrally smooth to a nondegen-erate critical submanifold N c M Let the initial positions be fixed on the critical submanifold, xe(0) = i , £ JV, and the initial velocities convergent in TXtM, xe(0) -> v* G TXtM. Then, for a finite time interval [0,T], there exists a unique sequence xe of oluti of the EulerLagrange equati orresponding to J?e.

Let Uhom be the homogenizatin of U with resect t the limit ini-tial values (x*,t>*). We denote by hom the unique solution of the Euler-Lagrange equatis corresponding t the hogenized Lagrangian

fhom(x,x) = \{x, V(x) Uhom(x) x G

with initial data £hom(0) = x* G N and Xhom(0) = (x*)v* G TXt If a;hom is non-Hatly resonant u order three the sequence on-

verges uniforml o Xhom 0,T]

1.6. Homogenization of a ecifc Class of Potentials

In appliations, one frequently enounters onstraining potentials of the form

with smooth scalar functions tpj : K. his way, the r i t ia l submani fold is given by the set

= {x G M (x) = 0} = {x G : ̂ ( x ) = = '(x) = 0}.

This manifold N has codimension r if and onl if the gradient vectors, grad^j(x), are linearly independent for all x hi ill assume

1] E HOMOGENIZATION RESUL 23

o be the ase. hen, they for a basis field for the normal bundle of

dimTx = r span{gradV>j(a;) : j = 1 , . . . , r} = Tx x G N. 1114)

oordinates, the seond derivative of is given by

{x) = 'tpjixfipjix), xG

Recalling that gradtpj = G~1DipJ, cf. [1, Def. 2.5.14], where G denotes the metric tensor of the oordinate system, we get the f o l l i n g expression for the Hessian H:

~l • . g r a d ^ g r a d V j .

Notice, that the last expression is independent of a chosen coordinate sys tem. Using the span relation 11.14), we obtain the nondegeneracy ondi tion II.5), range H = TN. Thus, U onstrains to the submanifold i the technical sense of Definition 1.

Now, let to(x) > 0 be a nonzero eigenvalue of (x) and the orresponding eigenvetor,

(x) (x)X. 1115)

Beause of (1114), there is a unique representation of of the form

- g r a d ( x ) .

Inserting this into the eigenvalue problem 1115) yields an equivalent, r-dimensional problem,

(x) = u(x) 1116)

for the Grammian matrix of the gradient vetors ,

{x)jk gv&d^^ic&diix)). H17)

The matrix-valued field Hr : N W will be called the reduced Hessian of the onstraining potential U. Notie , that the normalization of is given according to

X,) = (x)

Solving the reduced r x r eigenvalue problem (11.16) is all one needs to know for, first, deciding about whether U constrains spectrally smooth, and seond, establishing the homogenized potential Uhom

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

Example 1. (The Codimension One Case). The case of codimension r = 1 provides the simplest possible example. Suppressing the index ' 1 , ' we diretly read off that

co(x) = grad^(a;) x G

where || || denotes the norm on the tangent space TXM indued by the Riemannian metri he adiabati invariant 0Q is given by

!>„,, g rad^x*) ) 2

2^{x

Finally, (7hom takes the form Uh0ra{x) =Q-U{X).

1.7. Remarks on Generici

We now study the "genericity" of the assumption that U constrains spec-trally smooth. By definition, generic properties are structurally stable un-der perturbations, i.e., are typical for a whole neighborhood of problems. Otherwise, a property would be highly improbable in the presence of some unertainty in the problem data like modeling or measurement errors.

The reader should be cautioned that the term "generi" is always un-derstood relatively to a specific class of perturbations.

Now, the genericity of the spectral smoothness of U depends on two different influences: first, the presence of resonances on JV; second, the lass of perturbations of U allowed for by the underlying physial problem.

The Nonresonant Case. If there are no resonances of order two of the normal frequencies co\ on N, the potential U constrains spectrally smooth. This claim can be proved by representing the spectral projetions by a Cauchy integral, a standard argument of perturbation theory [51]. We omit the technial details here, the reader ill find them in the first part of the proof of Theorem II.3 later on.

Accordingly, because having no resonances of order two is certainly a generic property with respect to any class of perturbations, the potential

onstrains spetrally smooth generically.

The Resonant Case. Here, certain resonances of order two exist on the manifold N, a far more subtle ase which requires a careful study of the perturbations that are allowed. These perturbations determine the so-called codimension1 of a resonance, i.e., the codimension of the set of all matrices having that eigenvalue-resonance in the set of all those matr ies that are consistent with the given class of perturbations.

For real symmetric matrices, as the Hessian H, we typically have reso-nanes of odimension one or two, depending on hether the perturbations

1] E HOMOGENIZATION RESUL

preserve ertain structures besides the symmetry H = H or not. For a complete classification of generic resonances in a problem class related to quantum mechanics we refer to the remarks given in §IV.2.4.

Instead of developing a general theory we just provide some examples and remarks hich, however, seem to reflet the general situation quite well

xample 2. Let the onstraining potential U have the speial s t ru ture

{x) = \ { x xGM

Suppose that the underlying problem only admits perturbations which pre-serve this structure. The results of §1.6 show that the reduced Hessian Hr is always diagonal for this specifc class of potentials, implying that U is generically spectrally smooth. Now, the set of resonant diagonal matr ies obviously has codimension one in the set of all diagonal matr ies .

The discussion of the butane molecule in §1112.1 will provide a less trivial example here a potential is generically spetrally smooth beause just a restricted class of perturbations applies.

The most general class of perturbations of a potential leads to per turbations of the Hessian that preserve the symmetry of the matrix only. Here, a generic resonance has codimension two and, unavoidably, the po-tential does not constrain spectrally smooth. A proof of these claims can be found in Appendix A. If we have to admit general perturbations of the constraining potential, either a resonance or the spectral smoothness are non-generic.

An example of a generi resonane of codimension two is subjet of §4. We will show that the convergence assertion of Theorem 1, hich is neessarily not appliable there, suffers a dramati b reakdon .

1.8. ounterexample for Flat Resonances

The short discussion in §1.2.6 has motivated why resonance conditions have to be employed at a certain stage of the proof we designed for Theorem 1. Here, e will show that the result itself demands resonance conditions.

We onsider the lidean s p a e M = K3 and the potential

(x) = | T k ) \ 2 \ \ x = z\z2) G M3,

with smooth functions Wi,cj2 > which ill be specified below. This potential is obviously constrainingin the technial sense of Definition 1 to the one-dimensional submanifold N = {x € : z 0}. he Hessian H is given by the diagonal matri

O = d i a g ( 0 , w O w ) ) .

26 HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

he initial values shall be given by x* = 0 £ N and v* = (w*,0,2) . No we speify and u- Let to G (K) be a funtion such that

u()

1/2,

2,

5/2

and L ^ 0 for 1/2 1 and 2 5/2. We set

1, U.

Defining the projections Pi = diag(0,1,0) and Pi = diag(0,0,1), the Hes sian H trivially has the smooth spetral deomposition

)= 1118)

According to Definition 4, this and the initial values yield the homogenized potential

However, because of the resonance there are other smooth spectral deom-positions of H which follow the paths of the eigenvalues in a different ay. Let 6 G (K) be a function such that

) os 4> s n ( ± os 0 s n 0

and

W) p os2

sin 4>

os cf) sin <

n particular, we have Pi(0) = Pi and P2(0) = P2, but P(TT/2) = P2 and P2 (TT/2) = . A short alulation reveals the smooth spectral deomposi tion

) = (0( (4,()), II19)

here

) /2 ,

/2 )

3/ 3/2,

3/2

1] E HOMOGENIZATION RESUL

Corespondingly, Definion 4 yields the homogenized potental

Uhom

he fore fields U{i0m and Ü{i0m differ, s i e for ]2,5/2 we have

Kom) = 4 ) = w = u ) = Uiom).

For an initial veloity w* > 0 which is large enough, the corresponding solutions Xhom and Xhom will hit this interval at some time t = 1.

herefore, they must be different, Xhom ^ Xhom-Clearly now, Theorem 1 cannot hold true after dropping the assumption

of non-flat resonane. For then we would get the contradiction that both —> Xhom and y€ —> hom, i-e., Xhom ^hom on the time interval [0,1]. The setting of this counterexample fits into the framework of the model

problem of §1.2. Therefore, heorem 1.1 is applicable. It teaches that the homogenized potential t/h0m

w2 is, in fact, the "corret" one, desribing the limit dynamics and yielding the uniform convergence y> Xhom- Thus, the spectral decomposition (1118) of the Hessian is somewhat more "nat ural" than the other, constructed one, Eq. (11.19). However, it seems to be difficult defining a general notion of "natural" smooth spectral decom-positions in a way whih would allow to relax the resonance conditions of Theorem 1 in any significant fashion. Moreover, the author conjectures that the resonance conditions of Theorem 1 do not only reflect the possibility of a non-uniqueness of the homogenized potential, but also the possibility of a omplete b reakdon of the entire limit s t ruture .

1.9. ounterexample for nbounded Energy

If we drop the assumption of uniformly bounded energy, i.e., the assump-tion x £ AT, e cannot expect a homogenization result similar to Theo-rem 1. To show this, we reprodue a ounterexample of BORNEMANN and

HÜTTE [18].

We onsider on the onfiguration s p a e M = f the Lagrangian

ith the potential'

I dj • rij I e~2U(x)

( ^ / 2 ( X ) =

0

>0

which is onstraining to the manifold i = {0}. Given the initial position x* and the initial v e l i t y v* 0, the energy E e~

2 0 The limited differentiability of U is not essential for this counterxample. It coul be smoothed out at the cost of sacificing the simplicity of the result

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

cannot be bounded uniformly in e. The solution of the equation of moton is given by the rapidly osillating funtion x(t) = x(t/e), here

x{t) os(2i) TT/4,

2 sin(t TT/4) TT/4 5TT/4,

sin(2i TT/2) 5TT/4 3TT/2.

Here, we get merely weak convergence of xe in L°°, namely by a well k n o n generalization, 23, Lemma 1.2], of the Riemann-Lebesgue lemma

3-H/2

TT/2

l x0 2/TT = —- X(

which is not on the constraint manifold N. Trivially, this limit annot be desribed by a mechanial system ith onfiguration spa

1.10. ibliographical emarks

Strikingly, the homogenization problem of this chapter has found only little systematic attention in the mathematical and physical literature—at least much less than the special case provided by the realization-ofonstraints problem that will be discussed in §3.

All work we know of about this particular homogenization problem con-siders an Euclidean space M = R as the onfiguration manifold. The gen-eralization to Riemannian manifolds, as accomplished by us, is not straight forward since the metric introduces a further source of nonlinearity. As we

ill see, the d i f u l t y of the proof consists in ontrolling the nonlinearities ith respect to weak* convergence.

To our knowledge, the first mathematial work on the homogenization problem was done by RUBIN and UNGAR 82, p. 82f.] in 1957. These authors consider constraint manifolds N of codimension r = 1 only. The result, s o m e h a t hidden in the paper, is stated in a speial oordinate system.

Independently, for codimension r 1, the result can be found by means of an example in the work of the physicists K O P P E and JENSEN [58, Eq. (7)] from 1971. The argument of these authors is basically physical and in-volves averaging in an informal way. However, we owe the backbone of our proof to the physical ideas presented in their paper: a virial theorem and adiabaticity. The notion of weak* convergence puts the informal averaging process on a firm mathematical basis. A orresponding mathematical proof of the codimension ase has been worked out by BORNEMANN and

HÜTTE [18].

The codimension r = 1 case was also discussed in the work of the physicist VAN KAMPEN [49] in 1985. This author utilizes the WKB method. However, the proof is mathematically i o m p l e t e as has been pointed out by BORNEMANN and ÜTTE [18]

2] E P R O O F OF T HOMOGNIZATION ESUL

he f i r s t a n d until our present work onlycomplete study of the gen-eral case for M = R was given by TAKENS [94] in 1979. This author revealed the importance of resonance conditions and proved the result un-der the assumptions that there are no resonance hypersurfaces of second and third order, all eigenvalues of the Hessian having smooth multiplicity one. The method of proof set up by TAKENS starts with a rather explicit representation of the normal oscillations, as one would do for an asymptoti analysis, and then proceeds by using the Riemann-Lebesgue lemma, which is in fact a result about weak convergence. The presentation of the proof is in parts only sketchy, cf. Remark 2 on p. 43 below

Referring to the work of TAKENS, an informal discussion of the general homogenization result can be found in an article by KoiLLER 57] from 1990. This author uses action-angle variables and identifies the constants #o in the expression for the homogenized potential Uhom as adiabatic in-variants.

One should also mention the work of KELLER and RUBINSTEIN [52] from 1991. The oncern of these authors is the homogenization problem for the semilinear wave equation vtt = A e~

2gradU(v). Their argument—an ingenious multiple scale asymptot ics is directly applicable to our problem and correctly establishes the homogenization potential Uhom- However, the expansion presented in the paper is only formal and no estimate of the remainder term is given. These authors cannot predict difficulties at resonances like in §1.8. It should be stressed that no approximation further than the zero-order term , the singular limit, is provided.

§2 Th roof of t Homogenization esult

The proof of Theorem 1 proceeds along the lines we have discussed in § 2 . However, in contrast to the simple example given there, we now need a considerable amount of notions from differential geometry. For purposes of reference, we collect some of the intermediate results in a series of sixteen lemmas. he assumptions of heorem 1 shall be valid throughout.

2.1. tep 1: quioundedness nergy Principle

The proof will be given in local coordinates. Since we work on compa time intervals [0, T] an res t r i t ourselves to a single oordinate pat of M here we have

x = {x\...x e f t c

In these coordinates the metric is represented as usual by a covariant tensor of seond order, (gt) he equation of motion (II1) an be r e r i t t e n 2 1

as xe + Txe,xe) + Fv(xe -

2Fv(xe) = 0. 1120)

Cf. [ , Proof of P o p . 3.7.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

he fores grad and grad U are given by22

dx dx

where the contravariant tensor of second order (g) = G~l represents the inverse matrix of he term denotes the Christoffel symbol

*«.•> I K % *" ( I f ^ ^ ) We will frequently use a slightly nonsymmetri version of the Christoffel symbols,

i §9i 9jk \ jk dx dxl )

which obviously gives Y{u,u) = (u,u). oordinate representation the Hessian is given by the tensor

82T (H)

dxdx

and the energy as the expression

Ee = yi(xe) V(x€) f-(xe)

The nondegenerate r i t ia l submanifold an no be viewed as a sub-manifold of Rm.

emma 1. There is a subsequence of 0, dented by again such that

xc^x0 in 0,T]R), xe x0 in 0,T]).

The limit function is at least Lipschitz continuus and takes values in the submanifold N, x0 ^(0,T]).

Proof. Let a > 0 denote the smallest eigenvalue of the metric G, V* a lwer bound of the potential V, and E* a uniform bound for the energy

Conservation of energy gives

a\xe(t)/2

and integration

M*) X,\ TJ^-E,V,)

for all t € [0, T]. These equiboundedness results allow the application of the extended Arzelä-Asoli theorem, r i i p l e 4: here is a subsequen

2] E P R O O F OF T HOMOGNIZATION ESUL

of e, denoted by e again, and a limit funtion x 1 (0 ,T] lR m ) su that the asserted limit relations hold.

Multiplying the equation of motion 1120) by e2 and taking the weak* limit s h o s that

gradL(xo(i)) = 0, ie . xo(t) G

for alH G 0, T]. Here, we have used riniple 1 for establishing the weak* onvergence e2x 0. D

From now on consider the subsequence of this lemma. Further ex-tractions will follow and they will always be denoted by e again.

The uniform convergence xe —• XQ implies that for sufficiently small e all trajectories are within a tubular neighborhood of N. A point x G is an element of s u h a neighborhood, if there is a unique representation

exp z, , zG H21)

Here, "exp" denotes the geodesic exponential map, [1, p. 149], and a well known theorem of differential geometry states the existence of such a tubu-lar neighborhood of N, [1, Theorem 2.7.5]. We consider a smooth field ( e , . , e) of orthonormal bases of the normal bundle T-1, ie. , for

) G ),ej()) = < y .

hus, in the tubular neighborhood we get the unique representation23

x = exp n+l .. n+

W e l l k n o n properties of geodesis, [1, p. 150], imply that

distfo z, z) = ^^ n 2 2 )

here the left hand side is independent of the chosen bases field. For a set of given local coordinates (1,..., n) of the manifold AT w

define the tubular coordinates

( x . . x ) = ...,;zn+1,...zn+)

Putting y = (y...,y;0,...,0) and z = (0 , . . . ,0; zn+1,... ,zn+r), this oordinate system has an obvious linear structure and ill frequentl rite in short form24

z, , z

2 3This particular numbering of the coefficients z will simplify the notation following below.

24Certainly, this is "abus de langage", compared to the invariant relation (1121) However, when we use this short form, it should be clear that we are working i thi pecific coordinate system.

32 HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

n the folloing onsider e su f ien t ly small such that

xe = exp e, resp. xe = e,

defines a time-dependent functions ye with values in and ze with values in . We all the constrained motion of x and its normal motion.

emma 2. The limit relati eci as follws:

^ in 0,T]) in 0,T]),

for the cstrained mti and25

=0e) in 0,T]

for the normal mti

Proof Lemma 1 and the distane relation 1122) proves that

->• x0, x 0 , Ze-0, 0.

Conservation of energy and a Taylor expansion s h o s t h a t

) > (xe) = ± ) z e , z ) (\e\3).

The nondegeneracy condition (II.6) and the uniform onvergen imply, for su f ien t ly small e, the estimate

e\ e\ ±e\2,

here denotes some positive onstant. his proves = 0 e ) . •

In what follows, we will always abbreviate fe = f(e) for any smooth tensor field defined on the submanifold AT, including e = 0, i.e., /o = f(xo).

We now take a closer look on the constrained and normal motion. T this end, we apply the orthogonal projetions of §1.2. Using the notation just introdued, we get

= z 0.

Within the tubular coordinate system, we can view as a time dependent matrix. Thus, we can extend the a t ion of in a oordinate dependent fashion and get for the veloities

Pex€ = Pee = ze Pfze = e) 1123)

and for the accelerations

Jt(P e + g) = e + (1). 1124

25Here and in what follows, we denote an estimate for a t imedependent function by a Landau symbol, i the estimate holds uniformly in time.

2] E P R O O F OF T HOMOGNIZATION ESUL 33

Lemma 3. After a farther extraction of subsequences the ollwing limi relati are satisfied by the cstrained mti

->• x

In particular, one gets the regularity 1 '1 (0 , T] ) . The initial values of x are given by

(0) = x* e (0)=Q(x*)v* £TX

The normal mtion satisfies a second order equati of the form

e+ {l). 1125)

Proof. I we multiply the equation of motion 1120) by we get

e + (x) ( {x r 2 { x ) = 0 .

he equiboundedness of xe and xe shows that < 5 e r ( x e ) ( ) = 0(1) and QeF(xe) = 0(1). The projection relation (11.24) yields Qexe — y\ = 0(1). Below, Lemma ill state that ~ 2 { x ) = (1). Summarizing, obtai

e = ( 1 ) .

An application of the extended Arzelä-Asoli theorem, r i i p l e 4, proves the limit assertions.

Likeise, if we multiply the equation of motion 1120) by we get

Pexe \) Pe(xe)(xe,xe PeFv(xe) ~2PeFv(xe) = 0.

Below, Lemma 5 will state that e~PeFu(xe) = e~2Heze + 0(1). The same

arguments as for the onstrained motion yield that the middle terms are equibounded. Thus, we obtain the asserted seond order equation.

Since xe(0) = x* we obtain by the uniform onvergen that (0) = x*. From q. (1123) follo

(x*)v* = lim lim (0).

he uniform onvergen implies (0) = (x*)v*. D

As a result, we now kno that all quantities fe = f() are strongly onvergent in O,!]

Lemma 4. L ° [ 0 , T ] W ) the bunded quantity e~xz onverges weak ly* t zero,

rx 0.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

Proof Lemma 2 s h o s that

Ve = ze = (1)

After an application of the Alaoglu theorem, r i i p l e 3, and an extra tion of subsequenes, we obtain that

for some r) G L c ( [0 ,T] ,K r ) . Multiplying the relation (11.25) by e and taking weak* limits gives, by Principle 1, H(x 0. hus, the nonde-genera ondition I I ) yields that

r/o

On the other hand, sinc € Ty^N implies 0, we get by taking weak* limits that (x)r 0, ie .

m

Hence, we obtain that r/o 0. Since this limit is unique we may disard the extration of subsequenes, realling riniple 5. •

Despite the fact that zf J 0 and z/e 0, we cannot expect that

quadratic expressions of these quantities converge weakly* to zero. This lack of weak* sequential onvergence constitutes the core of the homoge-nization result and, as we will see in §3.1, the general obstruction for real ization of constraints. For this reason, we expliitly introdue the quadrati expressions2

~2e ® € < ) r2

k j ( ) ,

and

,® ) 0y ( ) .

By Lemma 2, both quadrati expressions are uniformly bounded,

= o ( i ) (i).

We may therefore assume—by an application of the Alaoglu theorem, rin-ipl 3, and after a further extraction of subsequenes—that

Later on we will see that in general S ^ 0 and n 0 0.27 There is a note-

worthy differene in the definitions of e and IIe: Whereas e invariantl 2 6 The metric tensor Ge is included to make the resulting matrices selfadjoi with

espect to the Riemannian metric. This simplifies the calculations later on. 2 7To be specific, we will prove in §3.1, Lemma 17, that So = 0, or Ilo = 0, if and onl

f the limit initial velocity v i angenial to the critical submanifold N, v* 6 Tx N.

2] E P R O O F OF T HOMOGNIZATION ESUL

defines a tensor field along y ovariant of order one and contravariant of order one, the definition of depends on the ho ie of the oordinate system.

We finish this step of the proof by stating a Taylor expansion of the strong fore term Fv. Parts of this statement have been used in the proof of Lemma 3. For conveniene, we introdue some further notation: he quantity gradi denotes

For A = (A) and we define the t r a e s

( g r a d ^ ^ f

: A %jk g r a d F gradH

Lemma 5. The Taylor eansions of the force term e~ second order is given by

~2{x) = ~ 2 e + (l)

~2Hz e : { H :1 ±gradF e : e)

The projectins int the normal bundle and its orthonal cplement can be estimated by

~ 2 ( x € ) = (1), resp. Pe{xe) = ~2 (1).

Proof We use the f a t that e) and a long omputation:

[ f dxdx dxdxdx

)

here we have split into the f o l l i n g three terms:

fasek = "^

36 HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

hus, a Taylor expansion of (x ) in the expresion of yields

dx dxdx %^SS dx

Likeise, by differentiating

d2U

^ ^ ^ ) .

dxdx

we obtain g dH[

dxdxdx dx dx nserting that into the expression for S yields

^ ^ ^ ) ^ | ^ ) .

Altogether, after regrouping and renaming some summation indies we get

(x) =

Jz < ) J K ^ ^ ^ d ),

which immediately gives the desired formula. The assertions on the pro-jet ions foll from the nondegenera onditions (II7) •

2.2. Step he Weak Virial heorem

This step of the proof is about the distribution of energy in the onstrained and the normal motion.

Definition 5. The kinetic energy < , the potential energy , and the total energy of the onstrained motion are defined by

\gM = v(),

The kinetic energy T^, the potential energy , and the total energy E of the normal motion are defined by

ha( l--2)zz)

Notice that only the potential parts are denned in a coordinate inde-pendent fashion. This ambiguity, however, will disappear in the limit e — as an impliation of the weak virial theorem.

file:///gM

2] E P R O O F OF T HOMOGNIZATION ESUL

The quadratic expressions IIe and S allow elegant expressions for the energies of the normal motion and thei weak* limits in L°0,T]

±tr i t r 1126)

E^ i t r ( J Uj ±tr(H00).

Real l that we use the notation iJe = H(yf). The following lemma shows that we essentially have split all the energy

into the onstrained and normal motion.

e m m a 6. The ttal energy decmposes int

Ee=E e).

th arts converge uniforml as functions in 0,T]

i V(x), = E

Proof Differentiation of the orthogonality relation

z) = j{ = 0

ith respet to time yields

M i(M ^ ^ e).

hus, the kineti energy deomposes into

~2 \*^e? Xe) — ^di ^ e ) CJ .

Taylor expansion of the potentials gives

(x) = e)

and

^(xf) = l ^ ^ l e) = e)

Hence, the energy decomposes as claimed. Lemma 3 implies the asserted uniform onvergene of the onstrained

part Ej! of the energy. By Eq. (II.3), the total energy converges as a number in K, Ee> EQ . This readily implies the uniform onvergene of the normal part E of the energy. D

Now, we state and prove a entral result of our argument. For reasons we have disussed at length i 2.6, all it the weak virial theorem.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

Lemma 7. (Weak Vr ia l Theorem). The limit of the quadratic epres si of ormal positi and velcity are related by the Hessian

n 0 = H00. 1127)

the limit the quadratic epressions ommute with the Hessian

F o , ] = 0 , flo,n0] = 0, IL28)

and there is an equiartitioning of the kinetic and potential energy of the ormal mti

Proof 28 By Lemma 2, we obtai

e = e = e) 0,

and that this quantity has a bounded time derivative,

c < e + e ® f + c < e + e) 1129)

In the last step, we have made use of the second order equation (11.25) of Lemma 3. By riniple 1, taking weak* limits on both sides of q. (1129) yields

This readily implies that, in the limit, the kineti and potential parts of the normal energy are equal

|tr(ILj) = i t r ( F 0 o ) = U

Since all the tensors n 0 , S0 , and H0 are linear operators TXoM > TXoM and they are obviously selfadjoint with respet to the Riemannian metri we get the ommutation relations (1128). •

From the relation (11.27) of the weak virial theorem, an important fact follows: In contrast to the coordinate-dependent definition of IIe, the limit n 0 r epresen tsas its counterparts H and S0 do—an invariantly defined field of linear maps TXoM> TXoM. In the same way, the limit normal part T^ of the kinetic energy is a scalar field invariantly defined along xo

The weak virial theorem allows us to give a partial, abstract, and general answer to the homogenization problem.

2 8This proof owes its formal structure to the traditional proof of the v i a l t h e o r m of classical mechanics which may be found for instance in the textbooks of A B R A A M and MARSDEN 1, Theore 3.7.30] GOLDSTEIN 39, Chap. 34 ] , and LANDA and L I F S I T Z 62, p. 23].

A different proof, using the localization principle of semiclassical measures, can be found in Appendix D. That considerably more technical, though still short proof sys tematizes the result and broadens the p e p e c t i v e by elating it, for instance, t the theory of compensated compactness

2] E P R O O F OF T HOMOGNIZATION ESUL

emma 8. ( h e Abstract Limit quation). The limit bey

o + ( x ) ( (x *om(t) TX0 1130)

which is the second order equati of tin for a mechanical system that is cstrained the manifold The ogenized force ^}om(£) is given by

b°m(t) = \ g r a d t (xo(t)) : 0(t). 1131)

Proof. As in the proof of Lemma 3, we multiply the equation of motion 11.20) by the projection f. However, this time we are interested in the

weak* limit behavior of each term and not only in their boundedness. We get

e + e + {x) ( {x) { (x) (

Fv(xe) e-Fu(xe) = 0. (1132)

Because of the weak* onvergen 0 and the uniform onvergenes —> x and we get

(x)( o.

Using q. (1123) and q. (1124), we obtai

,z\ = 2P€ Pfxe e)

sin = 0 by Lemma 9 b e l . We get further

( a ) ( ) = Q ( a ) : ( , " 1 f0 : ( Q -1 ) .

Lemma 5 and the nondegenera ondition II 7) give

- ' 2 ( x f £ : ( f r1 \ g™dH e + e)

Q 0 0 : (Hoö1 i g r a d F o : 0.

hus, taking weak* limits i q. (1132) yields the equation

Qo xo (xo)(xox0) (x0) ±gvadH(x0) :

f o ^ 0 ,

here the last term is zero2 beause of the weak virial theorem (1127). D

We call the limit equation (1130) abstract since we have no real access to the time-dependent force field F^fm It involves the weak* limit S , an expression that depends on the particular subsequence we have chosen. Therefore, we cannot even conlude that XQ is unique, ie. , independent of the chosen subsequene of e.

2 9This ter would be trivially zero for a flat manifold M, like the Euclidean space, since then T = 0. Here is the first of two places in our proof, where such a m e t i c dependent ter drops out like magic because of the weak v i i a l theorem.

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

Remark 1. For M = Mm endowed with the Eulidean met r i , Lemma 8 has been proven by BORNEMANN and S C Ü T T E [18, Theorem 2.1]. For cri t ial submanifolds N with codimension r = 1 this lemma appears in somewhat different form—using suitable averaging operatorsin the work of the physiists K O P P E and JENSEN 58, q. (5)] and VAN KAMPEN 49,

q. (8.33)]

For the purposes of Chapter IV, we state the following lemma more general than we have needed in the proof of Lemma 8 above.

emma 9. Let P be a projection operator in some Hubert space Jtf, and Q be an arbitrary bounded operator, th deending ontinuusly differ-entiable arameter Assum that

= Q 0.

Then, if we dente differentiation with resect the arameter by a prim btain

= 0, 0.

Proof Differentiation of PQ = 0 gives PQ + ' = 0. Because of QP = 0, multiplication by P from the right yields PQ = 0. Likeise, by differentiating the projection relation 2 = P, we get Multipliation by from the right gives 0. •

2.3. Step 3 diabatic Invariance of the Normal Actions

Until now we have not made any use of the smooth spectral decomposition of the Hessian H introduced in §1.3. Recall that the corresponding eigen-projetions P\ induce an orthogonal decomposition of the normal spaes

N n partiular, we deompose the normal motion into

e = ith Pf

As we have done with the projection P in §2.1, we may view the projec tions PA in the tubular coordinate system as parameterdependent matri ces. This allows to extend their action to all vectors of Rm in a coordinate-dependent fashion. Analogously to the relations 1123) and 1124), we no get

e + e + e) 1133)

and

XZe + e + e) = XZe + (1). 1134

As usual, we abbrviate Pe\ () and eX (, including 0.

2] E P R O O F OF T HOMOGENIZATION ESULT

Therefore, multiplying the differential equation 1125) by X immediatel yields the omponen t i s e relation

-2 = 0(l). 1135)

This equation explains why we have called cox the normal frequencies of the Hessian H. The components zfX are thus perturbed harmonic oscillations.

The next lemma shows that the limit quadratic expressions IIo and So have a block-diagonal form ith respect to the eigenspaces of H. This is a direct consequene of the weak virial theorem and a ertain resonan ondition.

Lemma 10. Let there be essentially n resonances of order tw along the limit

^

fo Then, one gets the blck-diagonalizati

A, =0 ß, II36)

and

A, ß, II37)

as functi in L ° 0 , T ] R m x m ) .

Proof. Multiplying the commutativity relation H0T,0 = T>0H0 of the weak virial theorem by the projetion X from the left and by from the right gives

- P O O P O = ^ ^ o P o

Thus, by the resonance assumption, for A ^ ß we get POAS0POM = 0 in L°([0,T]Rmm). The same argument applies to n 0 . he representation n 0 = H0o of the weak virial theorem yields

\H

By e = 0, we obtain QeT,e = T,€Qf = 0 and, taking weak* limits, QoT

HOMOGENIZATION OF NATURAL ECANICAL SYSTEMS CHAP. I I

These relations can easily be proven i oordinates by using q. 1133) and the selfadjointness P\, i e .

jP 1139)

Thus, Lemma 10 shows the weak* ontinuity of ertain, but ertainly not all, quadrati expressions,

Real l that e _ 1

0, 0, li. 40)

Z j O and ze 0. Surprisingly, only very little is needed to establish full cubic weak conti

nuity. A further resonance condition and the perturbed harmoni osillator equation 1135) turn out to be enough.

Lemma 11 . Let there be essentiall resonances of order three along the limit