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Renormalization theory for Hamiltonian systems [Elektronische Ressource] / von Mikhail Pronine

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Published 01 January 2002
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Renormalization Theory
for
Hamiltonian Systems
Mikhail Pronine
Universitat Bremen
2002Renormalization Theory
for
Hamiltonian Systems
Vom Fachbereich fur Physik und Elektrotechnik
der Universit at Bremen
zur Erlangung des akademischen Grades
DOKTOR DER NATURWISSENSCHAFTEN (Dr. rer. nat.)
genehmigte Dissertation
von
Dipl.-Math. Mikhail Pronine
aus Simferopol
Referent: Professor Dr. P. Richter
Korreferent: Dr. H. Schwegler
Tag des Promotionskolloquiums: 16.12.2002Preface
This thesis is devoted to the study of the breakup of invariant tori of irrational winding
numbers in Hamiltonian systems with two degrees of freedom. Due to topological reasons,
the decay of invariant tori in such systems is closely related to the onset of widespread
chaos. To give an estimate of where in parameter space the breakup of an invariant torus
occurs, an approximate renormalization scheme is derived. The scheme is applied to a
number of systems (the paradigm Hamiltonian of Escande and Doveil, the Walker and
Ford model, a model of the ethane molecule, the double pendulum, the Baggott system,
lima con billiards).
The work is organized as follows.
Chapter 1 describes the behavior of a generic Hamiltonian system with more than
one degree of freedom. The emphasis is put on systems with two degrees of freedom.
We introduce the main problem of the work, i.e., the problem of nding the threshold
to global chaos in terms of the breakup of the "last" invariant KAM torus. There exist
a number of analytical and numerical methods to deal with the problem. We review
these methods in Chapter 2. Our version of the renormalization group approach to the
problem is discussed in Chapter 3. The method is applied to various systems in Chapter
4. Chapter 5 summarizes the results of the work.
Appendix A is devoted to the normal form for Hamiltonian systems with two degrees
of freedom. In Appendix B we discuss classical perturbation theory and its application
to the normal form. Appendix C contains some useful formulae for the calculation of
derivatives of implicit functions. The realization of the RG approach to the study of the
breakup of invariant tori in the Maple computer algebra system is presented in Appendix
D.
I would like to thank my scienti c advisor Prof. Peter H. Richter for supervising the
work. I have bene ted from useful discussions with former and current members of the
group Nichtlineare Dynamik. I particularly thank Dr. Holger Dullin, Jan Nagler, Dr.
Hermann Pleteit, Dr. Holger Waalkens.
vContents
1 Introduction 1
2 Criteria for the Breakup of KAM Tori 7
2.1 Sup map analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The method of overlapping resonances . . . . . . . . . . . . . . . . . . . . 11
2.3 Greene’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 The renormalization group approach . . . . . . . . . . . . . . . . . . . . . 16
2.5 Comparison between the di erent methods . . . . . . . . . . . . . . . . . . 18
3 Renormalization Theory 19
3.1 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The renormalization operator . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 The map for the normal form . . . . . . . . . . . . . . . . 23
3.4 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Applications 31
4.1 Application to the paradigm Hamiltonian . . . . . . . . . . . . . . . . . . . 31
4.2 to the Walker and Ford model . . . . . . . . . . . . . . . . . . 36
4.3 Application to a model of the ethane molecule . . . . . . . . . . . . . . . . 41
4.4 to the double pendulum problem . . . . . . . . . . . . . . . . . 53
4.4.1 The Lagrange function . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.2 Integrable cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.3 The Hamilton function . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.4 The integrable limit of high energies . . . . . . . . . . . . . . . . . 57
4.5 The Baggott H O Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 652
4.6 Lima con billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Conclusions 89
A The Normal Form 91
B Classical Perturbation Theory 95
B.1 Application to the normal form . . . . . . . . . . . . . . . . . . . . . . . . 101
C Derivatives of Implicit Functions 105
viiviii
D Maple Program 107
D.1 Renormalization operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.1.1 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.1.3 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . 109
D.1.4 Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
D.1.5 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
D.2 The paradigm Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
D.3 Application to a model of the ethane molecule . . . . . . . . . . . . . . . . 113
D.4 to the Baggott Hamiltonian . . . . . . . . . . . . . . . . . . . 114Chapter 1
Introduction
This thesis deals with the dynamic behavior of classical Hamiltonian systems with two
degrees of freedom. Such systems and their stability properties are of interest in diverse
elds (celestial mechanics [28, 45], plasma physics [12, 21], chemical physics [28, 24, 25]
to name just a few). Hamiltonian systems can be divided into two classes, integrable and
non-integrable. Let us recall the notion of integrability which is of utmost importance in
the study of Hamiltonian systems. Consider a Hamiltonian system de ned by a function
H(p;q;t) in the phase spaceT Q(p;q). The functionH is called the Hamiltonian of the
system. The equations of motion are Hamilton’s equations
dp @H dq @H
= ; = : (1.1)
dt @q dt @p
A functionF in the phase space is said to be a constant of motion if along any trajectory its
value is constant. Constants of motion are also referred to as rst integrals . In autonomous
systems (i.e., H is explicitly time independent, H = H(p;q)) the Hamiltonian H is a
constant of motion. In what follows we restrict ourselves to the case of
systems.
The Poisson bracket of two functions F = F(p;q;t) and G = G(p;q;t) is de ned to
be nX @F @G @F @G
[F;G] = : (1.2)
@p @q @q @pi i i i
i=1
The time dependence of a function F =F(p;q;t) is given by
dF @F
= [H;F] + : (1.3)
dt @t
Assume that the functionF does not depend on time explicitly. It follows from (1.3) that
F is a constant of motion if and only if [H;F] = 0. Two functions F and G are said to
be in involution if their Poisson bracket [F;G] vanishes.
Two functionsF(p;q) andG(p;q) are called (functionally) independent if their gradi-
ents (@F=@p ;:::;@F=@q ) and (@G=@p ;:::;@G=@q ) are linearly independent for almost1 n 1 n
every point (p;q).
We are ready now to introduce the notion of integrability for Hamiltonian systems. A
Hamiltonian system ofn degrees of freedom is called integrable if there existn independent
12 CHAPTER 1. INTRODUCTION
constants of motion F ;:::;F which are in involution [3]. Integrable systems are also1 n
referred to as completely integrable ones.
The geometric description of integrable systems is given by the Liouville-Arnold the-
orem [3]. According to this result the phase space of an integrable Hamiltonian system
H of n degrees of freedom is foliated by the invariant sets0
f(p;q)2T Q: F (p;q) =c ;:::;F (p;q) =cg; (1.4)1 1 n n
with F ;:::;F being constants of motion. Moreover, in generic situations the motion1 n
on these invariant sets is periodic or quasiperiodic. If the energy surface H (p;q) = h0 0
is compact, then connected components of the invariant sets are just n-dimensional tori.
Locally there exists a canonical transformation (p;q) ! (I;) such that in the new
coordinates (I;) the Hamiltonian H does not explicitly depend on the angle variables0
:
H =H (I): (1.5)0 0
The coordinates (I;) are referred to as action-angle coordinates. The equations of motion
are readily solvable in the action-angle coordinates. Indeed, Hamilton’s are
dI @H d @H0 0
= = 0; = : (1.6)
dt @ dt @I
Thus, the actionsI are constants of motion, and the time evolution of the angle variables
is periodic or quasiperiodic with constant frequencies @H =@I.0
A generic Hamiltonian system with two or more degrees of freedom is non-integrable.
In this case there is no simple geometric description of motion. Moreover, the dynamic
behavior of a generic Hamiltonian system is at least partially chaotic.
Recall some relevant de nitions from the theory of dynamical systems [13]. Let M be
tan arbitrary set. Consider a one-parametric family of maps f : M ! M from M into
titself. The pair (M;f ) is called a dynamical system. The set M is referred to as the
t tphase space of the dynamical system (M;f ). The family f is said to be the dynamics.
tIf the parameter t is continuous, then the dynamical system (M;f ) is called the ow .
In the case of Hamiltonian systems we de ne the corresponding ow in the following
way. Choose the phase space T Q =f(p;q)g as the phase space of the ow. Given a
tpoint (p ;q ) from T Q, the map f assigns to it the solution of Hamilton’s equations0 0
with initial conditions (p ;q ) at time t = 0.0 0
0 t 0 0A subset M of the phase space M is called an invariant set if f (M ) lies in M for
every t. The notion of invariant set plays an important role in the study of dynamical
systems. Indeed, having identi ed all invariant sets of a given dynamical system, we can
tinvestigate the behavior of the dynamicsf on the invariant sets independently from each
other.
tAssume now that the phase space M of some dynamical system (M;f ) is a metric
space. For example, the phase space T Q of a Hamiltonian with the Euclidian
metric is a metric space.
tThe dynamical system (M;f ) is said to be topologically transitive if for any two open
tsetsU andV fromM there existst> 0 such that the intersectionf (U)\V is not empty.
Note that a topologically transitive dynamical system has no non-trivial open invariant