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Preface
This textbook is intended for use by students of physics, physical chemistry, and theoretical
chemistry. The reader is presumed to have a basic knowledge of atomic and quantum physics
atthe levelprovided,forexample,bythe firstfewchaptersinourbookThe Physics of Atoms
and Quanta. The student of physics will find here material which should be included in the
basic educationofeveryphysicist. This book shouldfurthermore allowstudents to acquirean
9 appreciationof the breadth and variety within the field of molecular physics and its future as
0 a fascinating area of research.
0
Forthestudentofchemistry,theconceptsintroducedinthisbookwillprovideatheoretical
2
framework for that entire field of study. With the help of these concepts, it is at least in

n principle possible to reduce the enormous body of empirical chemical knowledge to a few
a
basic principles: those of quantum mechanics. In addition, modern physical methods whose
J
fundamentals are introduced here are becoming increasingly important in chemistry and now
6
represent indispensable tools for the chemist. As examples, we might mention the structural
2
analysis of complex organic compounds, spectroscopic investigation of very rapid reaction

] processes or, as a practical application, the remote detection of pollutants in the air.
L
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April 1995 Walter Olthoff
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ProgramChair

1 ECOOP’95
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Organization
ECOOP’95 is organized by the department of Computer Science, Univeristy of ˚Arhus and
AITO(associationInternationapourlesTechnologieObject)incooperationwithACM/SIGPLAN.
Executive Commitee
Conference Chair: Ole LehrmannMadsen(˚Arhus University,DK)
ProgramChair: Walter Olthoff (DFKI GmbH, Germany)
Organizing Chair: Jørgen Lindskov Knudsen (˚Arhus University,
DK)
Tutorials: Birger Møller-Pedersen
(Norwegian Computing Center, Norway)
Workshops: Eric Jul (University of Kopenhagen,Denmark)
Panels: Boris Magnusson (Lund University, Sweden)
Exhibition: Elmer Sandvad (˚Arhus University, DK)
Demonstrations: Kurt Nørdmark (˚Arhus University, DK)
Program Commitee
Conference Chair: Ole LehrmannMadsen(˚Arhus University,DK)
ProgramChair: Walter Olthoff (DFKI GmbH, Germany)
Organizing Chair: Jørgen Lindskov Knudsen (˚Arhus University,
DK)
Tutorials: Birger Møller-Pedersen
(Norwegian Computing Center, Norway)
Workshops: Eric Jul (University of Kopenhagen,Denmark)
Panels: Boris Magnusson (Lund University, Sweden)
Exhibition: Elmer Sandvad (˚Arhus University, DK)
Demonstrations: Kurt Nørdmark (˚Arhus University, DK)
Referees
V. Andreev Del Buono P. Goritchev
B¨arwolff P. Dingus H. Greif
E. Barrelet H. Duhm E.M. Hanlon
H.P. Beck J. Ebert R. Haydar
G. Bernardi S. Eichenberger R.C.W. Henderso
E. Binder R.J. Ellison P. Hill
P.C. Bosetti Feltesse H. Hufnagel
Braunschweig W. Flauger A. Jacholkowska
F.W. Bu¨sser A. Fomenko Johannsen
T. Carli G. Franke S. Kasarian
A.B. Clegg J. Garvey I.R. Kenyon
G. Cozzika M. Gennis C. Kleinwort
S. Dagoret L. Goerlich T. K¨ohler
S.D. Kolya J.V. Morris A. Semenov
III
P. Kostka B. Naroska G. Siegmon
U. Kru¨ger Nguyen J.R. Smith
J. Kurzho¨fer U. Obrock M. Steenbock
M.P.J. Landon G.D. Patel U. Straumann
A. Lebedev Ch. Pichler C. Thiebaux
Ch. Ley S. Prell P. Van Esch
F. Linsel F. Raupach from Yerevan Ph
H. Lohmand V. Riech L.R. West
Martin P. Robmann G.-G. Winter
S. Masson N. Sahlmann T.P. Yiou
K. Meier P. Schleper M. Zimmer
C.A. Meyer Scho¨ning
S. Mikocki B. Schwab
Sponsoring Institutions
Bernauer-Budiman Inc., Reading, Mass.
The Hofmann-International Company, San Louis Obispo, Cal.
Kramer Industries, Heidelberg, Germany
Table of Contents
Hamiltonian Mechanics unter besonderer
Beru¨cksichtigung der h¨ohreren Lehranstalten
Ivar Ekeland1, Roger Temam2 Jeffrey Dean, David Grove, Craig Chambers, Kim B. Bruce,
and Elsa Bertino
1
Princeton University,Princeton NJ 08544, USA,
[email protected],
WWW homepage: http://users/~iekeland/web/welcome.html
2
Universit´ede Paris-Sud,Laboratoire d’AnalyseNum´erique,Bˆatiment 425,
F-91405 Orsay Cedex, France
Abstract. The abstract should summarize thecontentsof thepaperusingat least 70
and at most 150 words. It will be set in 9-point font size and be inset 1.0 cm from the
right andleft margins. Therewill betwoblank linesbeforeand aftertheAbstract....
1 Fixed-Period Problems: The Sublinear Case
With this chapter, the preliminaries are over, and we begin the search for periodic solutions
to Hamiltonian systems. All this will be done in the convex case; that is, we shall study the
boundary-value problem
x˙ =JH′(t,x)
x(0)=x(T)
with H(t,·) a convex function of x, going to +∞ when kxk→∞.
1.1 Autonomous Systems
In this section, we will consider the case when the Hamiltonian H(x) is autonomous. For the
sake of simplicity, we shall also assume that it is C1.
We shall first consider the question of nontriviality, within the general framework of
(A ,B )-subquadraticHamiltonians.Inthesecondsubsection,weshalllookintothespecial
∞ ∞
case when H is (0,b )-subquadratic, and we shall try to derive additional information.
∞
TheGeneralCase:Nontriviality. WeassumethatH is(A ,B )-subquadraticatinfinity,
∞ ∞
for some constant symmetric matrices A and B , with B −A positive definite. Set:
∞ ∞ ∞ ∞
γ :=smallest eigenvalue of B −A (1)
∞ ∞
d
λ:=largest negative eigenvalue of J +A . (2)
∞
dt
Theorem ?? tells us that if λ+γ <0, the boundary-value problem:
x˙ =JH′(x)
(3)
x(0)=x(T)
has at least one solution x, which is found by minimizing the dual action functional:
T 1
ψ(u)= Λ−1u,u +N∗(−u) dt (4)
Z (cid:20)2 o (cid:21)
o (cid:0) (cid:1)
on the range of Λ, which is a subspace R(Λ)2 with finite codimension. Here
2 L
1
N(x):=H(x)− (A x,x) (5)
∞
2
is a convex function, and
1
N(x)≤ ((B −A )x,x)+c ∀x . (6)
∞ ∞
2
Proposition 1. Assume H′(0)=0 and H(0)=0. Set:
−2
δ :=liminf2N(x)kxk . (7)
x→0
If γ <−λ<δ, the solution u is non-zero:
x(t)6=0 ∀t . (8)
Proof. Condition (??) means that, for every δ′ >δ, there is some ε>0 such that
δ′
2
kxk≤ε⇒N(x)≤ kxk . (9)
2
It is an exercise in convex analysis, into which we shall not go, to show that this implies
that there is an η >0 such that
1
fkxk≤η ⇒N∗(y)≤ kyk2 . (10)
2δ′
Fig.1. This is thecaption of thefigure displaying a white eagle and a white horse on a snow field
Sinceu1 isasmoothfunction,wewillhavekhu1k∞ ≤η forhsmallenough,andinequality
(??) will hold, yielding thereby:
h2 1 h2 1
2 2
ψ(hu1)≤ 2 λku1k2+ 2 δ′ ku1k . (11)
If we choose δ′ close enough to δ, the quantity 1 + 1 will be negative, and we end up
λ δ′
with (cid:0) (cid:1)
ψ(hu1)<0 for h6=0 small . (12)
On the other hand, we check directly that ψ(0) = 0. This shows that 0 cannot be a
minimizer of ψ, not even a local one. So u6=0 and u6=Λ−1(0)=0. ⊓⊔
o
Corollary 1. Assume H is C2 and (a∞,b∞)-subquadratic at infinity. Let ξ1,...,ξN be the3
equilibria, that is, the solutions of H′(ξ)=0. Denoteby ω the smallest eigenvalue of H′′(ξ ),
k k
and set:
ω :=Min {ω1,...,ωk} . (13)
If:
T T T
b <−E − a < ω (14)
∞ ∞
2π (cid:20) 2π (cid:21) 2π
then minimization of ψ yields a non-constant T-periodic solution x.
We recall once more that by the integer part E[α] of α ∈ IR, we mean the a ∈ ZZ such
that a <α ≤a+1. For instance, if we take a =0, Corollary 2 tells us that x exists and is
∞
non-constant provided that:
T T
b <1< (15)
∞
2π 2π
or
2π 2π
T ∈ , . (16)
(cid:18) ω b (cid:19)
∞
Proof. ThespectrumofΛis 2πZZ+a .Thelargestnegativeeigenvalueλisgivenby 2πk +a ,
T ∞ T o ∞
where
2π 2π
k +a <0≤ (k +1)+a . (17)
o ∞ o ∞
T T
Hence:
T
k =E − a . (18)
o ∞
(cid:20) 2π (cid:21)
The condition γ <−λ<δ now becomes:
2π
b −a <− k −a <ω−a (19)
∞ ∞ o ∞ ∞
T
which is precisely condition (??). ⊓⊔
Lemma 1. Assume that H is C2 on IR2n \{0} and that H′′(x) is non-degenerate for any
x6=0. Then any local minimizer x of ψ has minimal period T.
Proof. We know that x, or x+ξefor some constant ξ ∈ IR2n, is a T-periodic solution of the
Hamiltonian system:
e e x˙ =JH′(x) . (20)
There is no loss of generality in taking ξ =0. So ψ(x)≥ψ(x) for all x in some neighbour-
hood of x in W1,2 IR/TZZ;IR2n .
e e
But this index(cid:0)is precisely th(cid:1)e index iT(x) of the T-periodic solution x over the interval
(0,T), as defined in Sect. 2.6. So
e e
i (x)=0 . (21)
T
Now if x has a lower period, T/k say, weewould have, by Corollary 31:
e i (x)=i (x)≥ki (x)+k−1≥k−1≥1 . (22)
T kT/k T/k
This would contradicte(??), andethus cannotehappen. ⊓⊔
Table 1. This is theexample table taken out of The TEXbook, p.246
4
Year World population
8000 B.C. 5,000,000
50 A.D. 200,000,000
1650 A.D. 500,000,000
1945 A.D. 2,300,000,000
1980 A.D. 4,400,000,000
Notes and Comments. The results in this section are a refined version of [?]; the minimality
result of Proposition 14 was the first of its kind.
Tounderstandthenontrivialityconditions,suchastheoneinformula(??),onemaythink
ofaone-parameterfamily x ,T ∈ 2πω−1,2πb−1 ofperiodic solutions,x (0)=x (T),with
T ∞ T T
xT going awayto infinity when T →(cid:0) 2πω−1,which(cid:1) is the period ofthe linearizedsystemat0.
Theorem 1 (Ghoussoub-Preiss). Assume H(t,x) is (0,ε)-subquadratic at infinity for all
ε>0, and T-periodic in t
H(t,·) is convex ∀t (23)
H(·,x) is T−periodic ∀x (24)
H(t,x)≥n(kxk) with n(s)s−1 →∞ as s→∞ (25)
ε
2
∀ε>0 , ∃c : H(t,x)≤ kxk +c . (26)
2
Assume also that H is C2, and H′′(t,x) is positive definite everywhere. Then there is a
sequence x , k ∈IN, of kT-periodic solutions of the system
k
x˙ =JH′(t,x) (27)
such that, for every k ∈IN, there is some p ∈IN with:
o
p≥p ⇒x 6=x . (28)
o pk k
⊓⊔
Example 1 (External forcing). Consider the system:
x˙ =JH′(x)+f(t) (29)
where the Hamiltonian H is (0,b )-subquadratic, and the forcing term is a distribution on
∞
the circle:
d
f = F +f with F ∈L2 IR/TZZ;IR2n , (30)
o
dt
(cid:0) (cid:1)
where f :=T−1 T f(t)dt. For instance,
o o
R
f(t)= δ ξ , (31)
k
kX∈IN
where δ is the Dirac mass at t = k and ξ ∈ IR2n is a constant, fits the prescription. This
k
means that the system x˙ =JH′(x) is being excited by a series of identical shocks at interval
T.
Definition 1. Let A (t) and B (t) be symmetric operators in IR2n, depending continuously
∞ ∞ 5
on t∈[0,T], such that A (t)≤B (t) for all t.
∞ ∞
A Borelian function H : [0,T]×IR2n → IR is called (A ,B )-subquadratic at infinity if
∞ ∞
there exists a function N(t,x) such that:
1
H(t,x)= (A (t)x,x)+N(t,x) (32)
∞
2
∀t , N(t,x) is convex with respect to x (33)
N(t,x)≥n(kxk) with n(s)s−1 →+∞ as s→+∞ (34)
1
∃c∈IR : H(t,x)≤ (B (t)x,x)+c ∀x . (35)
∞
2
If A (t) = a I and B (t) = b I, with a ≤ b ∈ IR, we shall say that H is
∞ ∞ ∞ ∞ ∞ ∞
(a ,b )-subquadratic at infinity. As an example, the function kxkα, with 1 ≤ α < 2, is
∞ ∞
(0,ε)-subquadratic at infinity for every ε>0. Similarly, the Hamiltonian
1
H(t,x)= kkkk2+kxkα (36)
2
is (k,k+ε)-subquadratic for every ε>0. Note that, if k <0, it is not convex.
Notes and Comments. The first results on subharmonics were obtained by Rabinowitz in [?],
who showed the existence of infinitely many subharmonics both in the subquadratic and su-
perquadraticcase,withsuitablegrowthconditionsonH′.Againthedualityapproachenabled
Clarke and Ekeland in [?] to treat the same problem in the convex-subquadratic case, with
growth conditions on H only.
Recently, Michalek and Tarantello (see [?] and [?]) have obtained lower bound on the
number of subharmonics of period kT, based on symmetry considerations and on pinching
estimates, as in Sect. 5.2 of this article.
References
1. Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian sys-
tems. Arch. Rat.Mech. Anal.78 (1982) 315–333
2. Clarke, F., Ekeland, I.: Solutions p´eriodiques, du p´eriode donn´ee, des ´equations hamiltoniennes.
Note CRAS Paris 287 (1978) 1013–1015
3. Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonau-
tonomous Hamiltonian systems. J. Diff. Eq. 72 (1988) 28–55
4. Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a ZZp pseudoindex theory.
Annali diMatematica Pura (to appear)
5. Rabinowitz, P.:Onsubharmonicsolutions ofaHamiltonian system.Comm. PureAppl.Math. 33
(1980) 609–633
Hamiltonian Mechanics2
Ivar Ekeland1 and Roger Temam2
1 Princeton University,Princeton NJ 08544, USA
2
Universit´ede Paris-Sud,Laboratoire d’AnalyseNum´erique,Bˆatiment 425,
F-91405 Orsay Cedex, France
Abstract. The abstract should summarize thecontentsof thepaperusingat least 70
and at most 150 words. It will be set in 9-point font size and be inset 1.0 cm from the
right andleft margins. Therewill betwoblank linesbeforeand aftertheAbstract....
1 Fixed-Period Problems: The Sublinear Case
With this chapter, the preliminaries are over, and we begin the search for periodic solutions
to Hamiltonian systems. All this will be done in the convex case; that is, we shall study the
boundary-value problem
x˙ =JH′(t,x)
x(0)=x(T)
with H(t,·) a convex function of x, going to +∞ when kxk→∞.
1.1 Autonomous Systems
In this section, we will consider the case when the Hamiltonian H(x) is autonomous. For the
sake of simplicity, we shall also assume that it is C1.
We shall first consider the question of nontriviality, within the general framework of
(A ,B )-subquadraticHamiltonians.Inthesecondsubsection,weshalllookintothespecial
∞ ∞
case when H is (0,b )-subquadratic, and we shall try to derive additional information.
∞
TheGeneralCase:Nontriviality. WeassumethatH is(A ,B )-subquadraticatinfinity,
∞ ∞
for some constant symmetric matrices A and B , with B −A positive definite. Set:
∞ ∞ ∞ ∞
γ :=smallest eigenvalue of B −A (1)
∞ ∞
d
λ:=largest negative eigenvalue of J +A . (2)
∞
dt
Theorem 21 tells us that if λ+γ <0, the boundary-value problem:
x˙ =JH′(x)
(3)
x(0)=x(T)
has at least one solution x, which is found by minimizing the dual action functional:
T 1
ψ(u)= Λ−1u,u +N∗(−u) dt (4)
Z (cid:20)2 o (cid:21)
o (cid:0) (cid:1)