Vilasi G. Hamiltonian Dynamics
Подождите немного. Документ загружается.
354
with Hamiltonian functionals given by
1
+OD
Np
=
2
f_,
(Dpv)2dz.
(13.24)
In order that
Eqs.
(13.23)
and
(13.24)
make sense, some assumptions on the
functional space
M
must be made, for example to assume that
M
consists of
fast
decreasing infinitely differentiable functions. Then clearly
TIvJ
=
I)
is
a
Nijenhuis A-invariant tensor operator for the heat equation hierarchy.
In the present geometrical approach,
Eq,
(13.21)
plays the role of a coordi-
nate transformation, and thus a N~jenhuis A-invariant tensor operator for the
Burgers' hierarchy is readily obtained from
f"u]
byg4
which easily yields
T[u]
=
I)
+
DUD-'
.
(13.25)
The Burgers' hierarchy
is
then obtained by repeated applications,
on
the
translation group generator A0
=
u,,
of the tensor operator expressed by
Eq.
(13.25),
Ah
=
TkAO.
(13.26)
The first fields of the hierarchy are
A0
=
Us,
A1
=
2~~3:
+
UX~,
A,
=;
(3u3
+
~UU,
-+
u,,),
.
This hierarchy
is
just the transcription in the new coordinate frame
of
the
linear one and, apart from some technical points on the phase manifold
N,
one can translate what has been said for
Eq.
(13.22)
to
the Burgers' hierarchy.
More precisely,
Eq.
(13.26)
splits
into
the following two sub hierarchies:
*
~~ss~~ut~~e
~~e~u~~~~
TAo,
T3A0,.
.
.
,
Y'2n+1A
oi...
9
Recursion Operators
in
Dissipative
Dynamics
355
which are, respectively,
a
sequence
of
dissipative and Hamiltonian vector fields.
The foregoing statement
can
be understood by examining the spectral proper-
ties
of
If,
whose
bl~ck
~~a~a~a~
~~~~’
is
where the vector fields
such
that
6
e’
-
-
6
e(&)
=;
-
6qW
’
-
&A&)
‘
In
the bidi~ensio~a~ integral manifold of
{e(k),eik)),
the operator
T
can be
projected
to
356
General
Stmctwes
where
are action-angle type variables. Then,
f'
transforms a dissipative integrable
field
of
the type
into
a
Hamiltonian one
and
vice versa.
(13.26)
into two subhierarchies. F'urthermore, we observe that
This alternating character
of
T
is
responsible for the splitting
of
hierarchy
T
has a bidimensional invariant spaces, but
is
not diagonalizable with-
Pz,
which characterizes the Hamiltonian subhierarchy,
is
diagonaliz-
out complexification.
able with doubly degenerate constant eigenvalues,
Thus, for none
of
the subhierarchies one can use the integrabi~ity criterion
to establish their integrability.
However, we observe that the projections of dissipative dynamics on the
bidimensional invariant spaces simply are
1
degree
of
freedom dynamics, while
for the Hamiltonian ones, the existence of a functional
J('))(U],
which
is
not
trivially conserved on each bidimensional space, ensures its integrability.
It
is
worthwhile remarking that this same functional
J(kj
[u]
obviously plays the role
of
a Ljapunov** functional for the projection
of
the dissipative dynamics on the
bidimensional invariant ~ubmanifoId, thus ensuring the asymptotic stability of
the solution
J(k)[u]
=
0.
The
~a~~ltu~~~~
su~~~era~c~~
We
discuss
in
more details the ~ami~tonian character
of
subhierarchy
(13.22).
In
order to do that, some care is needed for the appropriate choice
of
the
**Alexander Ljapunov
was
born in Jarosiav (central Russia) in
1857
and
died
in
S.
Petersburg
in
1918,
He
has
been
professor
of
mathematics at Kharkov University and after,
member
of
the
S.
Petersburg Academy
of
Science.
Recursion
Operators
in
Dissipative Dynamics
357
functional space
M
on which dynamics is defined. The most natural one would
be to take
M
as
the functional space whose elements
u
go to
a
constant
as
t
-+
&too,
as
it
is the space on which there lies the typical solitary wave of
Burgers’ hierarchy. However, with such a choice it would not be possible to
introduce
a
Hamiltonian structure on
M.
This can be understood easily by going back to the linear hierarchy for
which
M
becomes, via the transforma~ion
(13.21),
the space of functions which
as
I):
+
foo behave like exp[lct), and the Hamiltonian becomes meaningless.
One
is
then tempted to restrict
M
in such a way, that both symplectic struc-
tures and the Hamiltonian one be well-defined. This can be accomplished by
considering only function
.(I):)
tending to some nonvanishing fixed constants
as
I):
4
fco
or, equ~valently, functions
.(I):)
vanishing
as
2
+
fcm,
whose in-
tegral has fixed value. More precisely,
as
for what refers to tangent spaces, the
derivative of the Hopf-Cole map
is
a bijection
6v
-+
6u
between
S(R);
i.e. the
space
of
all
fast
decreasing
test functions, and the space
of
functions which
are derivatives of elements
of
S(%),
this ensuring the existence of
a
symplectic
structure with respect to which the subhierarchy
is
Hamiltonian.
The previous analysis shows the role played by the spectral hypothes~s on
the invariant mixed tensor field
T
in characterizing dynamical systems. The
violation
of
the diagonalizability hypothesis allowed the inclusion
of
dissipative
dynamics into the geometrical scheme. Moreover, the example shows that even
if
the eigenvaiues
of
T2
are trivially constant, sequences of constants
of
motion
can be constructed by it.
Chapter
14
Meaning and
Existence
of
Recursion Operators
Some confusion exists in the literature about recursion operators. This chapter
will be addressed
to
clarify
the
meaning and the existence of recursion operators
for completely integrable Hamiltonian systems.
In previous chapters it
has
been shown that completely integrable Hamil-
tonian dynamical systems may admit more than one Hamiltonian description.
It has been also shown that, usually, with these alternative descriptions, one
associates a (1,l)-tensor field which can be used under suitable conditions,
as
a
recursion operator, namely
as
an operator which generates enough constants
of the motion
in
involution. It seems to be an open question whether it
is
possible to find
a
recursion operator for any completely integrable system.
In
the hypothesis of nonresonance, it has been shown that
a
recursion oper-
ator can always be constructed, even for some infinite dimensional systems.80
Some
authors claimed however that this is not always the case.
So
it seems to us that it is of some interest to further comment on possible
meanings of recursion operators and to show that, in condition
of
nonresonance,
any integrable system can be reduced to a linear normal form via
a
nonlinear
noncanonical transformation. For these normal forms, it is straightforward to
construct recursion operators.130
359
360
Meaning and Existence
of
Recursion
Operators
14.1
Integrable Systems
Let
M
be a smooth Z~-dimensional manifoId. Let
us
suppose we
can
find
n
vector fields
XI,.
. .
,X,
E
X(M)
and
n
functions
PI,.
.
.
,
F,
E
T(M)
with
the following properties:
[xi,xjl=
0,
(14.1)
CxiFj=O,
i,jE
{l,...,?z].
(14.2)
Let us suppose also that, on an open dense submanifold
of
M,
we
have
XI
A*.*AXn
$0,
(14.3)
dFIA*-.AdFn
#O.
(14.4)
We shall show that any dynamical system
A
on
M,
which
is
of the form
n
A
=
EviXi,
d
=
vi(F1,.
.
.
,Fn),
(14.5)
is
explicitly integrable on the submanifo~d on which
Eqs.
(14.3)
and
(14.4)
are
satisfied.
2=
f
We assume finally, that the level sets of the submersion
F
:
M
--+
32"
,
F
=
(P,
..
,
,
F")
(14.6)
are
compact. Then the vector fields
Xi
are complete
on
each leaf
F-l(a),
a
E
W,
and they integrate to a locally free action of the Abelian group
Rn.
Moreover, each leaf is parallelizable and we can find closed differential
1-forms
d,.
. .
,a",
dai
=
0,
such that
d(Xj)
=
a;,
i,j
E
(1,.
.
*
,n)
*
(14.7)
With all previous construction, the vector field
A
in
Eq.
(14.5)
can be
explicitly integrated in
a
neighborhood of each leaf
F-'(a),
where we take
as
coordinates the functions
{Fi,
(8)
with
d@
=
a$,
The equations
of
motion
of
A
are given by
@
=
vi(Ff,.
. .
,
F")
,
pz
=o.
(14.8)
Integrable
Systems
361
Therefore, the corresponding solutions are
cPi(t)
=
t.i(F(P*))
+
d(P0)
1
Pi@)
=
&(Po)
7
(14.9)
with
po
E
M
the initial point. We see that the functions
Y'
play the role
of
frequencies.
We stress the fact that up to now we have not used any Hamiltonian struc-
ture.
For
an
algebraic characterization of complete integrability, see Refs.
77
and
126.
14.1.1
Alternative Hamiltonian descriptions
for
integmble systems
We shall now investigate under which conditions a dynamical system, which
is
integrable in the sense stated before, admits infinitely many alternative
Hamiltonian descriptions.
With the n-functions
F1,.
. .
,
F"
obeying the condition expressed
by
Eq.
(14.4),
we can define a closed differential
2-form
by
Wf
=
Cdfi(F')
A
d,
(14.10)
which is nondegenerate
as
long
as
dfi
A-
+
.Adfn
#
0.
Any one
of
these symplectic
forms makes the action of
9"
a Hamiltonian one. Indeed, by construction of
Wf
i
ix,wf=-dfj,
j€{l,
...,
n}.
(14.11)
As for the vector field
A
in Eq.
(14.5),
we shall have that
iawf
=
-
uidfi.
i
(14.12)
A necessary condition for
~AWF
to be exact is that it is closed, namely that
Cdv'
Adfi
=O.
(14.13)
i
All sets of solutions of this equation for
f
l,
. . .
,
f
"
satisfying
df1
A.
A
df,,
#
0
will give alternative Hamiltonian descriptions for the dynamical systems
A
362
Meaning and Existence
of
Recursion Operators
in Eq.
(14.4).
Moreover, any such
A
will be completely integrable in the
Liouville-Arnold sense, the functions
fl,
.
. .
,
fn
being constants of the motion
(by assumption
of
Eq.
(14.2))
in involution,
{fi,
fj}A
=
uf
(xi,
xj)
=
cX,fj
=
0.
(14.14)
There are two
limiting case where it is easy to exhibit solutions of
Eq.
(14.13).
The constant case
All the frequencies
vi
are constant numbers
so
that
dui
=
0
and Eq.
(14.13)
is
automatically satisfied.
Any differential 2-form in Eq.
(14.10)
is an admissible symplectic structure,
and the corresponding Hamiltonian function is given by
Wf
=
pfi.
(14.15)
i
An example of system
for
which this happens is given by the n-dimensional
harmonic oscillator written
as
i
1
a a
pi
-
-
aqi
,
no
sum
over
i
A.
-
___
-
&7&
34.2
(
14.16)
Here
mi
and
ki
are the mass and the elastic constant
of
the ith oscillator. Now
the functions
Fi
are just given by the partial Hamiltonians
Fi=I(g+kiq:),
2
mi
i~{l,
...ln}.
(14.17)
The nonresonant case
None of the frequencies
ui
is constant and we have that
dv'
A
Adu"
#
0.
In this case we may think of the
ui
as "coordinates" and of the
fj
as
€unctions
of
the
v'.
Integrable
Systems
363
In this second case, very simple solutions of Eq.
(14.13)
are given by linear
functions
fi
=
Cj
Aijuj,
i
E
(1,.
. .
n},
Aij
E
9.
The corresponding Hamilto-
nian description for
A
can given with quadratic Hamiltonian functions by
(14.18)
(14.19)
Moreover, any other symplectic structure of the form
Wf
=
xdfi(ui)
A
d
,
(14.20)
in which any
fi
depends only on the corresponding frequency
ui,
will be admis-
sible
ELS
long
as
wf
is nondegenerate; that is,
as
long
as
dfi
A.
.
A
dfn
#
0.
The
associated Hamiltonian functions depend on the explicit form of the functions
fi.
For instance, if
fi
=
(8Gi/8iji)(iji),
the corresponding Hamiltonian can be
written
it5
a
(14.21)
A
simple example for these case is given again by the n-dimensional har-
monic oscillator written
as
where
Fi
and
Ai
are given
A
=
C
F~A~,
i
(14.22)
by Eqs.
(14.17)
and
(14.16),
respectively. Now the
partial Hamiltonians
Fi
play the role of frequencies.
we refer to Ref.
80.
for
A
that cannot be derived by using the previous construction.
The intermediate cases are more involved. For further comments on them
It
is
worth stressing that there may be admissible Hamiltonian structures
14.1.2
We shall now
show
how to construct recursion operators for the integrable
systems that we have considered in the previous sections.
As
we have seen,
Recursion operators for integrable
systems