Involutive orbits of non-Noether symmetry groups
George Chavchanidze
Department of Theoretical Physics,
A. Razmadze Institute of Mathematics,
1 Aleksidze Street, Tbilisi 0193, Georgia
We consider set of functions on Poisson manifold related by continues one-parameter group
of transformations. Class of vector fields that produce involutive families of functions
is investigated and relationship between these vector fields and non-Noether
symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated
with sample models: modified Boussinesq system and Broer-Kaup system.
Non-Noether symmetry; Conservation laws; Modified Boussinesq system; Broer-Kaup system;
70H33; 70H06; 58J70; 53Z05; 35A30
J. Phys. A: Math. Gen. 38 (2005) 6517-6524
In Hamiltonian integrable models, conservation laws often form involutive orbit of
one-parameter symmetry group. Such a symmetry carries important information about
integrable model and its bi-Hamiltonian structure. The present paper is an attempt to
describe class of one-parameter group of transformations of Poisson manifold
that possess involutive orbits and may be related to Hamiltonian integrable systems.
Let be algebra of smooth functions on manifold equipped with Poisson bracket
where is Poisson bivector satisfying property .
Each vector field on manifold gives rise to one-parameter group of transformations of
algebra
where denotes Lie derivative along the vector field .
To any smooth function this group assigns orbit that goes through
the orbit is called involutive if
Involutive orbits are often related to integrable models where
plays the role of involutive family of conservation laws.
Involutivity of orbit depends on nature of vector field and function
and in general it is hard to describe all pairs that produce involutive orbits
however one interesting class of involutive orbits can be outlined by the following theorem:
For any non-Poisson vector field
satisfying property
and any function such that
one-parameter family of functions is involutive.
By taking Lie derivative of property
(6) along the vector field
we get
where
is real constant which is neither zero nor positive integer.
Taking into account
(5) one can rewrite result as follows
that after
iterations produces
Now using this property let us prove that functions
are in involution.
Indeed
Suppose that
and let us rewrite Poisson bracket as follows
Thus we have
Using this property
times produces
and since Poisson bracket is skew-symmetric we finally get
So we showed that functions
are in involution.
In the same time orbit
is linear combination of functions
and thus it is involutive as well.
In many infinite dimensional integrable Hamiltonian systems Poisson bivector has nontrivial kernel,
and set of conservation laws belongs to orbit of non-Noether symmetry group that goes through
centre of Poisson algebra. This fact is reflected in the following theorem:
If non-Poisson vector field satisfies property
then every orbit derived from centre
of Poisson algebra is involutive.
If function belongs to centre of Poisson algebra
then by definition . By taking Lie derivative of this condition along vector field
one gets
that according to Theorem 1 ensures involutivity of orbit.
The theorems proved above may have interesting applications in theory of infinite dimensional
Hamiltonian models where they provide simple way to construct involutive family of conservation laws.
One non-trivial example of such a model is modified Boussinesq system
[2],
[5],
[6] described by the following
set of partial differential equations
where
are smooth functions on
subjected to zero boundary conditions
This system can be rewritten in Hamiltonian form
with the following Hamiltonian
and Poisson bracket defined by Poisson bivector field
where
are vector fields that for every smooth functional
are defined
via variational derivatives
and
.
For Poisson bivector
(24) there exist vector field
such that
this vector field has the following form
Applying one-parameter group of transformations generated by this vector field to centre of Poisson algebra
which in our case is formed by functional
where
are arbitrary constants, produces involutive orbit that recovers
infinite sequence of conservation laws of modified Boussinesq hierarchy
Another interesting model that has infinite sequence of conservation laws lying on single
orbit of non-Noether symmetry group is Broer-Kaup system
[3],
[5],
[6], or more precisely special case
of Broer-Kaup system formed by the following partial differential equations
where
are again smooth functions on
subjected to zero boundary conditions
Equations
(29) can be rewritten in Hamiltonian form
with the Hamiltonian equal to
and Poisson bracket defined by
One can show that the following vector field
has property
and thus group of transformations generated by this vector field transforms centre of Poisson algebra
formed by functional
into involutive orbit that reproduces well known infinite set of conservation laws
of modified Broer-Kaup hierarchy
Two samples discussed above are representatives of one interesting family of infinite dimensional
Hamiltonian systems formed by
partial differential equations of the following type
where
is vector with components
that are smooth functions on
subjected to zero boundary conditions
is constant symmetric nondegenerate matrix,
is constant skew-symmetric matrix,
is constants vector that satisfies condition
and
denotes scalar product
System of equations
(37) is Hamiltonian with respect to Poisson bivector equal to
where
is vector with components
that are vector fields defined
for every smooth functional
via variational derivatives
.
Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivector
that is compatible with
or in other words
Corresponding Hamiltonians that produce Hamiltonian realization
of the evolution equations
(37) are
and
The most remarkable property of system
(37) is that it possesses
set of conservation laws that belong to single orbit obtained from
centre of Poisson algebra via one-parameter
group of transformations generated by the following vector field
Note that centre of Poisson algebra (with respect to bracket defined by
) is formed by
functionals of the following type
where
is arbitrary constant vector and applying group of transformations generated by
to this functional
yields the infinite sequence of functionals
One can check that the vector field
satisfies condition
and according to Theorem 2 the sequence
is involutive.
So
are conservation laws of bi-Hamiltonian dynamical system
(37)
and vector field
is related to non-Noether symmetries of evolutionary equations
(see Remark 1).
Note that in special case when
have the following form
model
(37) reduces to modified Boussinesq system discussed above.
Another choice of constants
gives rise to Broer-Kaup system described in previous sample.
Groups of transformations of Poisson manifold that possess involutive orbits play important
role in some integrable models where conservation laws form orbit of non-Noether symmetry group.
Therefore classification of vector fields that generate such a groups would create good background
for description of remarkable class of integrable system that have interesting geometric origin.
The present paper is an attempt to outline one particular class of vector fields that are
related to non-Noether symmetries of Hamiltonian dynamical systems
and produce involutive families of conservation laws.
The research described in this publication was made possible in part by
Award No. GEP1-3327-TB-03 of the Georgian Research and Development Foundation (GRDF)
and the U.S. Civilian Research & Development Foundation for the
Independent States of the Former Soviet Union (CRDF).
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