90
3 Area-Preserving Maps
To finally write the Hamiltonian in the form of a paradigm Hamiltonian, we make the
following transformation from variables
( ¯
p α+1 , ¯
x α+1 , t α ) to variables (p α+1 , x α+1 , t α+1 ):
t α+1 = v α+1 t α , p α+1 =
¯
p α+1
m α+1 v α+1
, x α+1 = ¯
x α+1 ,
(3.90)
H (α + 1) = −
¯
H (α + 1) − E o (I c
nα +1 )
m α+1 ((v α+1 ) 2
.
(3.91)
We can now use the above results to write the Hamiltonian in Eq. (3.78) in the
form of a paradigm Hamiltonian
H (α + 1)=
1
2
p
2
α+1 +U
(0)
α+1 cos(x α+1 )+U
(1)
α+1 cos[ν α+1 (x α+1 −t α+1 )], (3.92)
with amplitudes given by
U
(0)
α+1 =
U
(1)
α V n α +1 (I c
n α +1 )
m α+1 ((v α+1 ) 2
and U
(1)
α+1 =
U
(1)
α V n α (I c
n α
)
m α+1 ((v α+1 ) 2
. (3.93)
where we assume that V n α < V n α +1 . The Eqs. (3.85) and (3.93) define a special case
of the renormalization map between two successive scales in the phase space. Note
that Eq. (3.85) which maps the relative wave number ν α from one scale to another is
independent of Eq. (3.93), which maps the cosine wave amplitudes from one scale
to another, although the converse is not true.
Although we have rescaled the time and momentum (and hence energy) so that
our two resonances have velocities ˙
x α = 0 and 1 at each level, we have not rescaled
the space coordinates. Thus, on each scale the Hamiltonian will appear to oscillate
with a different period, 2πM α , and different relative number of oscillations, ν α =
N α
M α
, over that period.
The renormalization map determines when a given KAM torus is “broken” by
its rational approximates. Iteration of the renormalization map moves us between
different spatial scales in the phase space by mapping between rational approximates
of the KAM torus that exist at different spatial scales. Iteration of the map for the
relative wave number, Eq. (3.85), determines the sequence of rational approximates
and therefore the KAM torus of interest.
3.8.2 Fixed Points of the Renormalization Map
The renormalization map is a map in the parameter space, (ν, U, V ), and not in
phase space. Our first task is to determine the flow of orbits in parameter space as
the map is iterated. The nature of this flow can be largely determined by the nature of
3 Area-Preserving Maps
To finally write the Hamiltonian in the form of a paradigm Hamiltonian, we make the
following transformation from variables
( ¯
p α+1 , ¯
x α+1 , t α ) to variables (p α+1 , x α+1 , t α+1 ):
t α+1 = v α+1 t α , p α+1 =
¯
p α+1
m α+1 v α+1
, x α+1 = ¯
x α+1 ,
(3.90)
H (α + 1) = −
¯
H (α + 1) − E o (I c
nα +1 )
m α+1 ((v α+1 ) 2
.
(3.91)
We can now use the above results to write the Hamiltonian in Eq. (3.78) in the
form of a paradigm Hamiltonian
H (α + 1)=
1
2
p
2
α+1 +U
(0)
α+1 cos(x α+1 )+U
(1)
α+1 cos[ν α+1 (x α+1 −t α+1 )], (3.92)
with amplitudes given by
U
(0)
α+1 =
U
(1)
α V n α +1 (I c
n α +1 )
m α+1 ((v α+1 ) 2
and U
(1)
α+1 =
U
(1)
α V n α (I c
n α
)
m α+1 ((v α+1 ) 2
. (3.93)
where we assume that V n α < V n α +1 . The Eqs. (3.85) and (3.93) define a special case
of the renormalization map between two successive scales in the phase space. Note
that Eq. (3.85) which maps the relative wave number ν α from one scale to another is
independent of Eq. (3.93), which maps the cosine wave amplitudes from one scale
to another, although the converse is not true.
Although we have rescaled the time and momentum (and hence energy) so that
our two resonances have velocities ˙
x α = 0 and 1 at each level, we have not rescaled
the space coordinates. Thus, on each scale the Hamiltonian will appear to oscillate
with a different period, 2πM α , and different relative number of oscillations, ν α =
N α
M α
, over that period.
The renormalization map determines when a given KAM torus is “broken” by
its rational approximates. Iteration of the renormalization map moves us between
different spatial scales in the phase space by mapping between rational approximates
of the KAM torus that exist at different spatial scales. Iteration of the map for the
relative wave number, Eq. (3.85), determines the sequence of rational approximates
and therefore the KAM torus of interest.
3.8.2 Fixed Points of the Renormalization Map
The renormalization map is a map in the parameter space, (ν, U, V ), and not in
phase space. Our first task is to determine the flow of orbits in parameter space as
the map is iterated. The nature of this flow can be largely determined by the nature of