Selected Chapters in the Calculus of Variations

149 Pages·2002·1.333 MB·English

by  Jürgen Moser

Selected chapters in the calculus of variations
J.Moser
2
Contents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2 On these lecture notes . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 One-dimensional variational problems 7
1.1 Regularity of the minimals. . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 The acessoric Variational problem . . . . . . . . . . . . . . . . . . 22
1.4 Extremal (cid:12)elds for n=1 . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 The Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . 32
1.6 Exercices to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Extremal (cid:12)elds and global minimals 41
2.1 Global extremal (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 An existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Properties of global minimals . . . . . . . . . . . . . . . . . . . . . 51
2.4 A priori estimates and a compactness property for minimals . . . . 59
2.5 for irrational (cid:11), Mather sets . . . . . . . . . . . . . . . . . . . 67
(cid:11)
M
2.6 for rational (cid:11) . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
(cid:11)
M
2.7 Exercices to chapter II . . . . . . . . . . . . . . . . . . . . . . . . . 92
3 Discrete Systems, Applications 95
3.1 Monotone twist maps . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 A discrete variational problem. . . . . . . . . . . . . . . . . . . . . 109
3.3 Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.3.1 The Standard map . . . . . . . . . . . . . . . . . . . . . . . 114
3.3.2 Birkho(cid:11) billiard. . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3.3 Dual Billard. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.4 A second variational problem . . . . . . . . . . . . . . . . . . . . . 122
3.5 Minimal geodesics on T2 . . . . . . . . . . . . . . . . . . . . . . . . 123
3.6 Hedlund’s metric on T3 . . . . . . . . . . . . . . . . . . . . . . . . 127
3.7 Exercices to chapter III . . . . . . . . . . . . . . . . . . . . . . . . 134
3.8 Remarks on the literature . . . . . . . . . . . . . . . . . . . . . . . 137
3
4 CONTENTS
0.1 Introduction
Theselecturenotesdescribeanewdevelopmentinthecalculusofvariationscalled
Aubry-Mather-Theory. The starting point for the theoretical physicist Aubry
wasthedescriptionofthemotionofelectronsinatwo-dimensionalcrystalinterms
ofasimplemodel.Todoso,Aubryinvestigatedadiscretevariationalproblemand
the corresponding minimals.
Ontheotherhand,Matherstartedfromaspeci(cid:12)cclassofarea-preservingannulus
mappings,thesocalledmonotone twist maps.Thesemapsappearinmechanics
asPoincar(cid:19)emaps.SuchmapswerestudiedbyBirkho(cid:11)duringthe1920’sinseveral
basicpapers.Mathersucceededin1982tomakeessentialprogressinthis(cid:12)eldand
to prove the existence of a class of closed invariant subsets, which are now called
Mather sets. His existence theorem is based again on a variational principle.
Evensothesetwoinvestigationshavedi(cid:11)erentmotivations,theyarecloselyrelated
andhavethesamemathematicalfoundation.Inthefollowing,wewillnownotfol-
low those approaches but will make a connection to classical results of Jacobi,
Legendre, Weierstrass and others from the 19’th century. Therefore in Chapter I,
we will put together the results of the classical theory which are the most impor-
tant for us. The notion of extremal (cid:12)elds will be most relevant in the following.
In chapter II we investigate variational problems on the 2-dimensional torus. We
look at the corresponding global minimals as well as at the relation between min-
imals and extremal (cid:12)elds. In this way, we will be led to Mather sets. Finally, in
Chapter III, we will learn the connection with monotone twist maps, which was
the starting point for Mather’s theory. We will so arrive at a discrete variational
problem which was the basis for Aubry’s investigations.
Thistheoryadditionallyhasinterestingapplicationsindi(cid:11)erentialgeometry,namely
for the geodesic (cid:13)ow on two-dimensional surfaces, especially on the torus. In this
context the minimal geodesics as investigated by Morse and Hedlund (1932)
play a distinguished role.
As Bangert has shown, the theories of Aubry and Mather lead to new results for
the geodesic (cid:13)ow on the two-dimensional torus. The restriction to two dimensions
is essential as the example in the last section of these lecture notes shows. These
di(cid:11)erential geometric questions are treated at the end of the third chapter.
ThebeautifulsurveyarticleofBangertshouldbeathandwiththeselecturenotes.
Ourdescriptionaimslesstogeneralityasrathertoshowtherelationsofnewerde-
velopments with classical notions with the extremal (cid:12)elds. Especially, the Mather
sets appear like this as ’generalized extremal (cid:12)elds’.
0.2. ON THESE LECTURE NOTES 5
For the production of these lecture notes I was assisted by O. Knill to whom I
want to express my thanks.
Zu(cid:127)rich, September 1988, J. Moser
0.2 On these lecture notes
TheselecturesweregivenbyJ.Moserinthespringof1988attheETHZu(cid:127)rich.The
students were in the 6.-8’th semester (which corresponds to the 3’th-4’th year of
a 4 year curriculum). There were however also PhD students (graduate students)
and visitors of the FIM (research institute at the ETH) in the auditorium.
In the last 12 years since the event the research on this special topic in the
calculus of variations has made some progress. A few hints to the literature are
attachedinanappendix.Becauseimportantquestionsarestillopen,theselecture
notes might maybe be of more than historical value.
In March 2000, I stumbled over old (cid:13)oppy diskettes which contained the lec-
ture notes which I had written in the summer of 1998 using the text processor
’Signum’on an AtaryST. J.Moserhad looked carefully through the lecturenotes
in September 1988. Because the text editor is now obsolete, the typesetting had
to be done new in LATEX. The original has not been changed except for small,
mostly stylistic or typographical corrections. The translation took more time as
anticipated, partly because we tried to do it automatically using a perl script. It
probablywouldhavebeenfasterwithoutthis"help"butithastheadvantagethat
the program can now be blamed for any remaining germanisms.
Austin, TX, June 2000, O. Knill
Cambridge, MA, September 2000-April 2002, (English translation), The (cid:12)gures
were added in May-June 2002, O. Knill
6 CONTENTS
Chapter 1
One-dimensional variational
problems
1.1 Regularity of the minimals
Let(cid:10)beanopenregioninRn+1fromwhichweassumethatitissimplyconnected.
A point in (cid:10) has the coordinates (t;x ;:::;x ) = (t;x). Let F = F(t;x;p)
1 n
2
Cr((cid:10) Rn) with r 2 and let (t ;a) and (t ;b) be two points in (cid:10). The space
1 2
(cid:2) (cid:21)
(cid:0):= (cid:13) :t x(t) (cid:10) x C1[t ;t ]; x(t )=a;x(t )=b
1 2 1 2
f ! 2 j 2 g
consists of all continuous di(cid:11)erentiable curves which start at (t ;a) and end at
1
(t ;b). On (cid:0) is de(cid:12)ned the functional
2
t2
I((cid:13))= F(t;x(t);x_(t))dt:
Zt1
De(cid:12)nition: We say (cid:13) (cid:0) is minimal in (cid:0), if
(cid:3)
2
I((cid:13)) I((cid:13) ); (cid:13) (cid:0):
(cid:3)
(cid:21) 8 2
We (cid:12)rst search for necessary conditions for a minimum of I, while assuming
the existence of a minimal.
Remark.
A minimum does not need to exist in general:
It is possible that (cid:0)= .
(cid:15) ;
It is also possible, that a minimal (cid:13) is contained only in (cid:10).
(cid:3)
(cid:15)
7
8 CHAPTER 1. ONE-DIMENSIONAL VARIATIONAL PROBLEMS
Finally, the in(cid:12)mum could exist without that the minimum is achieved.
(cid:15)
Example: Let n = 1 and F(t;x;x_) = t2 x_2;(t ;a) = (0;0);(t ;b) = (1;1).
1 2
(cid:1)
We have
1
(cid:13) (t)=tm; I((cid:13) )= ; inf I((cid:13) )=0;
m m m
m+3 m N
2
but for all (cid:13) (cid:0) one has I((cid:13))>0.
2
If (cid:13) is minimal in (cid:0), then
(cid:3)
t
F (t;x ;x_ )= F (s;x ;x_ )ds=const
Theorem 1.1.1 pj (cid:3) (cid:3) xj (cid:3) (cid:3)
Zt1
for all t t t and j = 1;:::;n. These equations are
1 2
(cid:20) (cid:20)
called integrated Euler equations.
De(cid:12)nition: One calls (cid:13) regular, if det(F ) = 0 for
(cid:3) pipj 6
x=x ;p=x_ .
(cid:3) (cid:3)
If (cid:13) is a regular minimal, then x C2[t ;t ] and one has
(cid:3) (cid:3) 1 2
2
for j =1;:::;n
Theorem 1.1.2 d
F (t;x ;x_ )=F (t;x ;x_ ) (1.1)
dt pj (cid:3) (cid:3) xj (cid:3) (cid:3)
This equations called Euler equations.
De(cid:12)nition: An element (cid:13) (cid:0), satisfying the Euler equa-
(cid:3)
2
tions 1.1 are called a extremal in (cid:0).
Attention: not every extremal solution is a minimal!
Proof of Theorem 1.1.1:
Proof. Weassume,that(cid:13) isminimalin(cid:0).Let(cid:24) C1(t ;t )= x C1[t ;t ] x(t )=
(cid:3) 2 0 1 2 f 2 1 2 j 1
x(t )=0 and (cid:13) :t x(t)+(cid:15)(cid:24)(t). Since (cid:10) is open and (cid:13) (cid:10), then also (cid:13) (cid:10)
2 (cid:15) (cid:15)
g 7! 2 2
for enough little (cid:15). Therefore,
d
0 = I((cid:13) )
(cid:15) (cid:15)=0
d(cid:15) j
t2 n
= F (s)(cid:24)_ +F (s) (cid:24) ds
pj j xj j
Zt1 Xj=1(cid:16) (cid:17)
t2
= ((cid:21)(t);(cid:24)(t))dt
Zt1
1.1. REGULARITY OF THE MINIMALS 9
with (cid:21) (t) = F (t) t2F (s) ds. Theorem 1.1.1 is now a consequence of the
j pj (cid:0) t1 xj
following Lemma. 2
R
If (cid:21) C[t ;t ] and
1 2
2
t2
Lemma 1.1.3 ((cid:21);(cid:24)_)dt=0; (cid:24) C1[t ;t ]
8 2 0 1 2
Zt1
then (cid:21)=const.
Proof. De(cid:12)ne c=(t t ) 1 t2(cid:21)(t)dt and put (cid:24)(t)= t2((cid:21)(s) c)ds. We have
2(cid:0) 1 (cid:0) t1 t1 (cid:0)
(cid:24) C1[t ;t ] and by assumption we have:
2 0 1 2 R R
t2 t2 t2
0= ((cid:21);(cid:24)_)dt ((cid:21);((cid:21) c))dt= ((cid:21) c)2 dt;
(cid:0) (cid:0)
Zt1 Zt1 Zt1
wherethelastequationfollowedfrom t2((cid:21) c)dt=0.Since(cid:21)wasassumedcon-
t1 (cid:0)
tinuous this implies with t2((cid:21) c)2 dt = 0 the claim (cid:21) = const. This concludes
t1 (cid:0) R
the proof of Theorem 1.1.1. 2
R
Proof of Theorem 1.1.2:
Proof. Put y = F (t;x ;p ). Since by assumption det(F ) = 0 at every
j(cid:3) pj (cid:3) (cid:3) pipj 6
point (t;x (t);x_ (t)), the implicit function theorem assures that functions p =
(cid:3) (cid:3) (cid:3)k
(cid:30) (t;x ;y ) exist, which are locally C1. From Theorem 1.1.1 we know
k (cid:3) (cid:3)
t
y =const F (s;x ;x_ )ds C1 (1.2)
j(cid:3) (cid:0) xj (cid:3) (cid:3) 2
Zt1
and so
x_(cid:3)k =(cid:30)k(t;x(cid:3);y(cid:3))2C1 :
Therefore x C2. The Euler equations are obtained from the integrated Euler
(cid:3)k 2
equations in Theorem 1.1.1. 2
If (cid:13) is minimal then
(cid:3)
n
Theorem 1.1.4 (Fpp(t;x(cid:3);y(cid:3))(cid:16);(cid:16))= Fpipj(t;x(cid:3);y(cid:3))(cid:16)i(cid:16)j (cid:21)0
i;j=1
X
holds for all t <t<t and all (cid:16) Rn.
1 2
2
10 CHAPTER 1. ONE-DIMENSIONAL VARIATIONAL PROBLEMS
Proof. Let (cid:13) be de(cid:12)ned as in the proof of Theorem 1.1.1. Then (cid:13) :t x (t)+
(cid:15) (cid:15) (cid:3)
7!
(cid:15)(cid:24)(t);(cid:24) C1.
2 0
d2
0 II := I((cid:13) ) (1.3)
(cid:20) (d(cid:15))2 (cid:15) j(cid:15)=0
t2
= (F (cid:24)_;(cid:24)_)+2(F (cid:24)_;(cid:24)_)+(F (cid:24);(cid:24))dt: (1.4)
pp px xx
Zt1
II iscalledthesecond variationofthefunctionalI.Lett (t ;t )bearbitrary.
1 2
2
We construct now special functions (cid:24) C1(t ;t ):
j 2 0 1 2
t (cid:28)
(cid:24) (t)=(cid:16) ( (cid:0) );
j j
(cid:15)
where(cid:16) Rand C1(R)byassumption, ((cid:21))=0for (cid:21) >1and ( )2 d(cid:21)=
j 2 2 j j R 0
1. Here denotes the derivative with respect to the new time variable (cid:28), which
0
R
is related to t as follows:
t=(cid:28) +(cid:15)(cid:21); (cid:15) 1dt=d(cid:21):
(cid:0)
The equations
t (cid:28)
(cid:24)_j(t)=(cid:15)(cid:0)1(cid:16)j 0( (cid:0) )
(cid:15)
and (1.3) gives
0 (cid:15)3II = (F (cid:16);(cid:16))( )2((cid:21))d(cid:21)+O((cid:15))
pp 0
(cid:20)
ZR
For (cid:15)>0 and (cid:15) 0 this means that
!
(F (t;x(t);x_(t))(cid:16);(cid:16)) 0:
pp
(cid:21)
2
Remark:
Theorem1.1.4tells,thatforaminimal(cid:13) theHessianofF ispositivesemide(cid:12)nite.
(cid:3)
De(cid:12)nition: We call the function F autonomous, if F is
independent of t, i.e. if F =0 holds.
t
IfF isautonomous,everyregularextremalsolutionsatis(cid:12)es
n
H = F + p F =const::
Theorem 1.1.5 (cid:0) j pj
j=1
X
The function H is also called the energy. In the au-
tonomous case we have therefore energy conservation.