Vous aimerez aussi
hP
ep
can
be
the
case
be
vior.
are
resp
ect
the
suited
for
in
dev
ey
ym
tp
ys
the
one
10
)n
ha
two
ws
haotic
cien
can
be
in
ativ
es
the
yL
necessitates,
=(
;q
;:::;q
)a
ta
=(
;p
;:::
;p
y
losed
i
curv
n
t
f
r
has
o
c
vides
o
a
textb
v
ordinates
ery
e
simple
conserv
e
strictly
xample
analyzed
of
then
a
an
c
S
onserv
freedom
is
of
e
F
c
degrees
lassical
f
system
p
w
he
ith
o
nontrivial,
h
c
ysics
haotic
systems,
dynamics.
where
The
ely
limiting
discussed
cases
ten
o
as
f
description
s
N
trictly
o
regular
a
(\in
een
tegrable")
a
and
systems
strictly
a
irregular
r.
(\ergo
a
dic")
eld,
systems
canonical
billiard
ha
plane
f
illustrated,
ystems,
as
tic
w
the
system
haos"
as
eried.
a
Korsc
t
so
ypical
is
on
y
whic
c
h
e
sho
latter
ws
uction
an
n
he
Hans
tricate
Lieb
mixture
a
of
r
regular
3]
and
with
irregular
e
particle
N
h
1
a
d
a
e
Irregular
as.
orbits
w
of
that
c
e
haracterized
can
b
b
y
merely
an
freedom,
extremely
momen
sensitivit
b
y
motion
t
particle
motion
w
frictionless
e
to
in
The
tial,
initial
sho
conditions.
b
Suc
Since
h
indication
billiard
haos
systems
eterministic
are
torren
exemplarily
s
Abstract.
h
for
whic
educational
of
purp
\deterministic
oses
b
as
further
mo
in
dels
h
y
t
simple
There
systems
dissipativ
with
here
complicated
t,
dynamics
urgen
as

w
is
ell
t
as
otion.
for
hall
farreac
with
hing
here.
f
tro
tal
o
German
of
v
systems,
estigations.
oks
1
c
In
erg
tro
[1]
duction
uster
In
w
the
excellen
past
N
decades,
erry
classical
recommended.
ph
a
ysics
degrees
has
as
w
able
itnessed
ordinates,
an
osition
unexp
Chaotic
ected,
r
imp
f
etuous
egrees
Kaiserslautern,
freedom,
elopmen
.g.,
t,
g
o
It
h
no
has
b
led
established
olution
suc
an
b
en
h
timeev
vior
The
b
understanding
exhibited
).
y
N
with
classical
Zimmer
dynamics
of
of
hence
simple
v
systems,
small
an
um
area
e
in
The
ph
o
ysics
a
t
in
hat
t
h
odimensional
as
ativ
usually
force
b
i.e.
een
a
presumed
oten
to
typic
b
N
e
rank
generally
F
understo
e
o
vior.
d
t
a
rst
nd
o
concluded.
c
I
in
t
d
b
s
ecame
a
eviden
t
t,
f
ho
nd
w
studies
c
as
set
a
n,
in
t
h
2
existence
{
d
con
a
trary
c
t
has
o
een
the
and
concepts
v
y
Chaotic
systems
v
ph
D67653
a
ed
devided
b
to
Kaiserslautern,
w
o
groups:
s
are
Univ.
called
h
e
ysics
w
t
friction
extb
presen
o
and
oks
conserv
{
e
s
v
tems,
en
energy
simplest,
a
completely
onstan
deterministic
o
systems
m
ma
W
ysik,
s
h
exclusiv
o
deal
w
the
irregular,
case
c
As
1
in
b
d
e
t
ha
J
vior,
dynamics
t
conserv
hat
e
i
the
s
o
as
b
unpredictable
i
p
h
h
b
tossing
and
of
ermann
a
and
b
c
h
Commonly
[2]
ereic
s
accepted
ell
a
the
random,
t
b
b
2
b
hastic
B
b
[
eha
are
vior
The
only
of
for
system
a
N
system
of
with
Billiards
a
w
l
kno
arge
2
(
co
p
namely
h
p
23
co
ac
q
ounded
q
b
e
sto
c
w,
article
eview
ativ
suc
ly
al
ery
um
coin.
as
haotic
con
hat
er,
ev
the
of
new
tirely
to
whic
fundamen
with
in
ell
ativHans
y(
;p
)),
curv
ei
na
@H
@H
=1
;:::
;N
@q
@p
=1
;:::
;N
q;
anish:
@F
@F
@F
@F
;F
=0
@p
@q
@p
@q
the
q;
the
is
an
ust
rue
In
ativ
the
is
the
ativ
ys
in
tit
.e.
with
H;
=0
.I
al
dic
this
phase
Suc
or
only
as
een
en.
the
pe
the
yi
tire
rbit,
;p
=0
;:::
;p
;p
hi
po
ar
all
d
The
of
textb
y
o
b
oks
the
in
ev
classical
(
mec
fully
hanics,
n
w
w
ith
more
s
n
ome
an
rare
chaotic
exceptions,
with
deal
F
with
and
so
tudying
called
with
inte
q
gr
f
able
to
systems,
try:
i.e.
e
t
f
here
almost
e
an
xist
for
N
i
indep
is
enden
b
t
u
c
In
onstan
one
ts
section
of
ne
motion
n
F
surface
j
q
,
whic
j
n
;
hosen
i
of
=
y
,
e
whic
a
h
a
are
t
functions
The
tersection
n
phase
system,
s
a
pace
densely
whose
system
v
q
in
dicit
(1)
ro
ne
not
billiard,
c
The
hange
h
along
haotic
the
irregular
tra
double
jectory
successor.
.
of
I
the
n
implify
a
of
ddition,
a
one
instead
m
and
ust
o
demand
n
t
;
hat
oin
t
ection.
hese
a
functions
)
F
n
j
n
(
the
i
y
p
randomly
)
ystem
are
one
\in
is
in
In
v
ho
olution",
often
i.e.
to
their
motion
P
article
set
c
The
eld
k
t
ets
few
v
tegrable
q
osite
f
n
F
system
j
go
_
whic
k
ery
:
g
ailable
=
)
X
t
i
suc
gular
;
There
j
Hamiltonian
i
h
i
h
=
rigorously
k
giv
i
these
i
the
p
a
k
ends
_
ypical
i
system
motion
s
j
nor
of
con
i
regular
W
equations
are
:
n
(2)
u
Because
threeb
of
p
tial
mec
N
to
conditions
omplicated
F
o
j
ts
(
a
dieren
b
p
d
)
urface
=
phase
f
f
j
e
=
h
const:
eeps
,
o
Hamiltonian
(
motion
n
he
;
restricted
to
tersection
t
with
N
f
dimensional
this
m
o
anifold
mapping
in
J
2
!
N
+1
dimensional
)
phase
asso
s
probabilit
pace
for
whose
i
t
tegrabilit
op
o
ology
a
is
c
that
s
o
with
f
than
a
degree
n
freedom
N
equal
torus
zero.
[3{5].
tegrabilit
Moreo
map
v
w
er,
er,
this
b
m
related
y
symme
b
the
e
of
t
p
b
i
f
a
or
e
any
force
tra
b
jectory
longs
and
o
therefore
he
the
examples
phase
in
space
systems.
is
opp
densely
o
lled
a
w
i
ith
tegrable
s
is
uc
er
h
t
n
for
ested
h
tori.
S
orbit
uc
Poinc
h
v
a
a
dynamics
energy!)
is
space
called
.
regular.
h
etermined
h
a
called
socalled
e
conserv
(
d
.
e
exist
Hamiltonian
few
system,
tems
s
whic
energy
ergo
i
y
one
b
of
en
e
p
constan
v
ts
O
of
of
motion
systems
and
s
consequen
stadium
tly
i.e.
a
rectangle
onedimensional
semicircular
conserv
[6].
curv
t
e
case
s
a
ystem
whic
i
i
s
neither
alw
regular
a
c
This
and
space.
tains
tegrable.
oth
In
and
the
orbits.
f
a
ollo
examples
preserving.
therefore
w
Zimmer
e
d
s
l
hall
m
discuss
rank
the
o
simplest
y
n
roblem
ontrivial
celestial
c
hanics.
ase,
order
namely
visualize
t
c
w
dynamics
odimensional
t
conserv
s
area
i
e
handling,
systems,
uses
whic
reduction
h
information
are
F
th
tro
us
ucing
in
s
tegrable
of
if
in
there
space:
exists
o
y
s
et
the
another
n
i
o
phase
o
n
k
i
trac
k
conserv
nly
ed
f
quan
sequence
socalled
q
y
Korsc
F
),
,
urgen
i
1
the
2
space,
its
f
t
dimensional
p
F
ts
is
this
N
o
2
s
n
In
T
manner
grable
e
systems
btain
are,
discrete
opp
(
o
n
sed
n
t
T
o
(
c
n
ommon
2
b
+1
elief,
(3)
extremely
h
rare.
ciates
T
he
ndep
tersection
enden
i
t
g
of
or
eac
in
its
of
and
wn
ellkno
sys
irr
is
orbit
(for
the
lls
ev
entr
can,
te
ativ
wing
brac
oisson
do
alues
ones
tt
as
(as
ta
an
[6,7]
po
Bet
particle
the
angle
oi
es
olv
yt
the
po
the
enien
er,
Pα
r
Sϕ
)o0
)a
=c
ound
incidence
the
,
dieren
or
called
o
yp
w
es
ev
of
y
orbits.
ngle
Billiard
v
systems
eads
are
are
exemplarily
ts
s
t
uited
Fig.
is
een
mo
r
dels
reection.
for
t
e
the
ducational
=
purp
undary
oses
system,
ts
he
w
data
ell
i
a
hic
s
t
f
on
or
and
farreac
can
hing
undary
fundamen
w:
tal
the
in
hall
v
i
estigations!)
o
since
p
t
and
he
lar
billiard
)
motion
smo
is
c
easy
6,8].
to
comprehend
naturally
and
n
the
b
n
the
umerical
'
treatmen
of
t
after
d
b
o
e
tangen
s
s
n
w
ot,
o
or
pp
easily
osed
(
to
w
man
to
y
t
other
h
s
equal
ystems,
o
r
e
equire
with
n
billiards
umerical
a
i
f
n
tangen
tegration
n
of
ts
dieren
oundary
tial
h
equations.
p
This
ordinates
is
(
an
(4)
imp
sucien
ortan
b
oin
s
d
b
v
dic
p
the
curv
P
ary
s
s
es
distinguish
rom
oundary
means
e.
h
the
a
p
computation
represen
i
h
s,
impact
ev
o
en
orbit.
b
t
y
t
u
o
sing
tra
mo
impact,
dern
c
com
measured
puters,
he
comparativ
resp
ely
t
timeconsuming,
(
esp
t
ecially
ore
since
ction
c
e
haotic
se
phenomena
Poinc
a
he
re
the
exhibited
(
in
d
the
s
l
=
ongtime
e
b
o
e
according
ha
the
vior
eection
of
a
an
a
orbit.
of
F
is
urthermore,
to
n
angle
umerical
f
metho
W
ds
s
for
deal
solving
con
dieren
ex
tial
here,
equations
.e.
a
straigh
re
l
not
h
exact,
t
and
t
s
uc
ho
tersection
w
oin
i
with
nstabilities,
b
whic
curv
h
jectory
can
r
not
in
alw
o
a
co
ys
r
b
r
e
'
clearly
:
distinguished
F
from
a
the
tly
t
oth
rue
o
c
curv
haotic
u
b
h
eha
illiards
vior.
nonergo
2
[
Billiard
In
Systems
billiard
The
the
t
oincar
w
e
odimensional
ection
billiard
n
problem
quite
3
f
describ
t
es
b
a
curv
illiards
This
i
that
n
i
t
tersection
b
oin
mo
are
he
ted
without
I
friction
e
on
at
a
with
plane
b
billiard
undary
t
namely
able,
ra
b
at
ounded
the
b
n
y
and
a
direction
closed
f
curv
he
e.
jectory
B
the
w
w
een
h
the
an
impacts
e
at
b
the
t
b
angle
o
with
undary
ect
,
o
t
he
he
t
Chaotic
see
ne
1).
I
1.
i
c
m
e
c
section
v
o
t,
ho
a
ev
system,
t
n
u
nitial
the
of
ar
billiard
ra
(
Fig.
mo
Boundary
v
urv
es
r
on
'
straigh
f
t
billiard
l
a
ines
i
with
part
constan
a
t
t
v
jectory
elo
t
c
co
it
dinates:
y
arc
.
S
I
'
t
n
i
p
s
o
r
eected
a
a
t
b
t
t
he
b
l
e).
length
this
angle
whic
e,
most
at
as
ine
ving
particle
uc
since
tage,