Because of their exceptional properties the non-Noether symmetries could be
effectively used in analysis of Hamiltonian dynamical systems.
From the geometric point of view these symmetries are important
because of their tight relationship with geometric structures on phase space
such as bi-Hamiltonian structures, Frölicher-Nijenhuis operators,
Lax pairs and bicomplexes [1]. The correspondence
between non-Noether symmetries and conservation laws is also interesting and
in regular Hamiltonian systems on *2n* dimensional Poisson manifold
up to *n* integrals of motion could be associated with each generator
of non-Noether symmetry [1] [3].
As a result non-Noether symmetries could be especially useful in analysis of
Hamiltonian systems with many degrees of freedom, as well as infinite dimensional
Hamiltonian systems, where large (and even infinite) number of conservation laws
could be constructed from the
single generator of such a symmetry. Under certain conditions satisfied by the
symmetry generator these conservation laws appear to be involutive and ensure
integrability of the dynamical system.

The n-particle non periodic Toda model is one of integrable models
that possesses such a nontrivial symmetry. In this model non-Noether symmetry
(which is one-parameter group of noncannonical transformations)
yields conservation laws that appear to be functionally independent,
involutive and ensure the integrability of this dynamical system.
Well known bi-Hamiltonian realization
of the Toda model is also related to this symmetry.

Nonlinear Schrödinger equation is another important example
where symmetry (again one-parameter group) leads to the infinite sequence of
conservation laws in involution. The KdV and mKdV equations also possess
non-Noether symmetries which are quite nontrivial (but symmetry group is
still one-parameter) and in each model the infinite set of conservation laws is
associated with the single generator of the symmetry.

Before we consider these models in detail we briefly remind some basic facts
concerning symmetries of Hamiltonian systems. Since throughout the article
continuous one-parameter groups of symmetries play central role let us remind that
each vector field *E* on the phase space *M* of the
Hamiltonian dynamical system defines continuous one-parameter group of
transformations (flow)
*L*_{E} denotes Lie derivative along the
vector field *E*. Action of this group on observables (smooth
functions on *M*) is given by expansion
*M* is *2n* dimensional
symplectic manifold and the group of transformations *g*_{z}
will be called symmetry of Hamiltonian system if it preserves manifold of solutions
of Hamilton's equation
*{ , }* denotes Poisson bracket defined in a standard manner
by Poisson bivector field *{f , g} = W(df ∧ dg)* and *h*
is smooth function on *M* called Hamiltonian) or in other words if for
each *f* satisfying Hamilton's equation *g*_{z}(f)
also satisfies it. This happens when *g*_{z} commutes
with time evolution operator
*E* of the group *g*_{z}
does not preserve Poisson bracket structure
*[E , W] ≠ 0* then the *g*_{z} is called
non-Noether symmetry. Let us briefly recall some basic features of non-Noether
symmetries. First of all if *E* generates non-Noether symmetry
then the *n* functions
*ω* is symplectic form obtained by inverting Poisson
bivector *W* and *s* denotes contraction) are integrals
of motion (see [1] [3])
and if additionally the symmetry generator
*E* satisfies Yang-Baxter equation
*Y*_{k} appear to be in involution
*{Y*_{k}, Y_{m}} = 0
while the bivector fields *W* and *[E , W]*
(or in terms of 2-forms *ω* and *L*_{E}ω)
form bi-Hamiltonian system (see [1]). Due to this features
non-Noether symmetries could be effectively used in construction of conservation laws
and bi-Hamiltonian structures.

g_{z} = e^{zLE}

where
g_{z}(f) = e^{zLE}(f) =
f + zL_{E}f + ½(zL_{E})²f + ⋯

Further it will be assumed that
f = {h , f}

(here
g_{z}(f) = g_{z}(f)

If in addition the generator
Y_{k} =
i_{Wk}(L_{E}ω)^{k} k = 1,2, ... n

(where [[E[E , W]]W] = 0

these conservation laws
Now let us focus on non-Noether symmetry of the Toda model –
*2n* dimensional Hamiltonian system that describes the motion
of *n* particles on the line governed by the exponential interaction.
Equations of motion of the non periodic n-particle Toda model are
*ε(k) = − ε(− k) = 1* for any natural *k* and *ε(0) = 0*)
and could be rewritten in Hamiltonian form
(3) with canonical Poisson bracket derived from symplectic form
*g*_{z} generated by the vector field
*E* will be symmetry of Toda chain if for each
*p*_{s}, q_{s} satisfying Toda equations
(7)
*g*_{z}(p_{s}), g_{z}(q_{s})
also satisfy it.
Substituting infinitesimal transformations
*[E , W] ≠ 0*
and additionally one can check that Yang-Baxter equation
*[[E[E , W]]W] = 0* is satisfied.
This symmetry could play important role in
analysis of Toda model. First let us note that calculating *L*_{E}ω
leads to the following 2-form
*ω* and *L*_{E}ω give rise to
bi-Hamiltonian structure of Toda model (compare with [2]).
The conservation laws (5)
associated with the symmetry reproduce well known
set of conservation laws of Toda chain.
*[[E[E , W]]W] = 0* satisfied by generator of the
symmetry *E* ensures that the conservation laws are in involution
i. e. *{Y*_{k},Y_{m}} = 0.
Thus the conservation laws as well as the bi-Hamiltonian structure
of the non periodic Toda chain appear to be associated with non-Noether symmetry.

q_{s} = p_{s}

p_{s} =
ε(s − 1)e^{qs − 1 − qs} −
ε(n − s)e^{qs − qs + 1}

(p

ω = dp_{s} ∧ dq_{s}

and Hamiltonian equal to
h = ½p_{s}^{2} +
e^{qs − qs + 1}

The group of transformations
g_{z}(p_{s}) = p_{s} + zE(p_{s}) + O(z²)

g_{z}(p_{s}) = q_{s} + zE(q_{s}) + O(z²)

into (7) and grouping first order terms gives rise to the
conditions
g

E(q_{s}) = E(p_{s})

E(p_{s}) =
ε(s − 1)e^{qs − 1 − qs}(E(q_{s − 1}) − E(q_{s}))

− ε(n − s)e^{qs − qs + 1}(E(q_{s}) − E(q_{s + 1}))

One can verify that the vector field defined by
E(p

− ε(n − s)e

E(p_{s}) = ½p_{s}^{2} +
ε(s − 1)(n − s + 2)e^{qs − 1 − qs} −
ε(n − s)(n − s) e^{qs − qs + 1}

+ (ε(s − 1)(p_{s − 1} + p_{s})
e^{qs − 1 − qs} −
ε(n − s)(p_{s} + p_{s + 1})
e^{qs − qs + 1}

E(q_{s}) = (n − s + 1)p_{s} −
½p_{k} +
½p_{k}

+ (p_{s}^{2} +
ε(s − 1)e^{qs − 1 − qs} +
ε(n − s)e^{qs − qs + 1})

satisfies (11) and generates symmetry of Toda chain.
It appears that this symmetry is non-Noether since it does not
preserve Poisson bracket structure + (ε(s − 1)(p

E(q

+ (p

L_{E}ω = p_{s}dp_{s} ∧ dq_{s} +
e^{qs − qs + 1}
dq_{s} ∧ q_{s + 1} + dp_{r} ∧ dp_{s}

and together
I_{1} = Y_{1} = p_{s}

I_{2} = ½Y_{1}^{2} − Y_{2} =
½p_{s}^{2} +
e^{qs − qs + 1}

I_{3} = Y_{1}^{3}
− Y_{1}Y_{2} + Y_{3} =
p_{s}^{3} +
(p_{s} + p_{s + 1})
e^{qs − qs + 1}

I_{4} = ¼Y_{1}^{4} −
Y_{1}^{2}Y_{2} + ½Y_{2}^{2} +
Y_{1}Y_{3} − Y_{4}

= ¼p_{s}^{4} +
(p_{s}^{2} + 2p_{s}p_{s + 1} +
p_{s + 1}^{2}) e^{qs − qs + 1}

+ ½e^{2(qs − qs + 1)} +
e^{qs − qs + 2}

I_{m} = (− 1)^{m}Y_{m} + m^{− 1}
(− 1)^{k}I_{m − k}Y_{k}

The condition I

I

I

= ¼p

+ ½e

I

Unlike the Toda model the dynamical systems in our next examples are
infinite dimensional and in order to ensure integrability one should construct
infinite number of conservation laws. Fortunately in several integrable models
this task could be effectively done by identifying appropriate non-Noether symmetry.
First let us consider well known infinite dimensional integrable Hamiltonian system –
nonlinear Schrödinger equation (NSE)
*u* is a smooth complex function of
*(t, x) ∈ ℝ²*. On this stage we will not specify any
boundary conditions and will just focus on symmetries of NSE. Supposing that the
vector field *E* generates the symmetry of NSE one gets the following
restriction
*u → u + zE(u) + O(z²)* generated by *E*
into NSE). It appears that NSE possesses nontrivial symmetry that is generated by the
vector field
*v* is defined by *v*_{x} = uū).
In order to construct conservation laws we also need to know Poisson bracket
structure and it appears that invariant Poisson bivector field could be defined
if *u* is subjected to either periodic
*u(t, − ∞) = u(t, + ∞)* or zero
*u(t, − ∞) = u(t, + ∞) = 0* boundary
conditions. In terms of variational derivatives the explicit form of the Poisson bivector field is
*W* is
*{ , }* defined by *W* and
*[E, W] ≠ 0*) one can construct bi-Hamiltonian structure and
conservation laws. First let us calculate Lie derivative of symplectic form along the symmetry
generator
*ω* and *L*_{E}ω
exactly reproduces the bi-Hamiltonian structure of NSE proposed by Magri
[4] while the conservation laws associated with this symmetry
are well known conservation laws of NSE
*{Y*_{k}, Y_{m}} = 0 is related to the fact that
*E* satisfies Yang-Baxter equation *[[E[E , W]]W] = 0*.

u_{t} = i(u_{xx} + 2u²ū)

where
E(u)_{t} = i[E(u)_{xx} + 2u²E(ū)
+ 4uūE(u)]

(obtained by substituting infinitesimal transformation
E(u) = i(u_{x} + u_{xx} + uv + xu²ū) −
t(u_{xxx} + 6uūu_{x})

(here
W = idx ∧

while corresponding symplectic form obtained by inverting
ω = idx δu ∧ δū

Now one can check that NSE could be rewritten in Hamiltonian form
u_{t} = {h , u}

with Poisson bracket
h = dx (u²ū² − u_{x}ū_{x})

Knowing the symmetry of NSE that appears to be non-Noether
(
L_{E}ω = [δu_{x} ∧ δū + uδv ∧ δū + ūδv ∧ δu]dx

The couple of 2-forms
I_{1} = Y_{1} = 2uū dx

I_{2} = Y_{1}^{2} − 2Y_{2} =
i(ū_{x}u − u_{x}ū) dx

I_{3} = Y_{1}^{3} − 3Y_{1}Y_{2}
+ 3Y_{3} = 2 (u²ū² − u_{x}ū_{x}) dx

I_{4} = Y_{1}^{4} − 4Y_{1}^{2}Y_{2} +
2Y_{2}^{2} + 4Y_{1}Y_{3} − 4Y_{4}

= [i(ū_{x}u_{xx} − u_{x}ū_{xx})
+ 3i(ūu²ū_{x} − uū²u_{x})] dx

I_{m} = (− 1)^{m}mY_{m} +
(− 1)^{k}I_{m − k}Y_{k}

The involutivity of the conservation laws of NSE
I

I

I

= [i(ū

I

Now let us consider other important integrable models –
Korteweg-de Vries equation (KdV) and modified Korteweg-de Vries equation (mKdV).
Here symmetries are more complicated but generator of the symmetry still can be
identified and used in construction of conservation laws. The KdV and mKdV equations
have the following form
*u* is smooth function of *(t, x) ∈ ℝ²*).
The generators of symmetries of KdV and mKdV should satisfy conditions
*u → u + zE(u) + O(z²)* into KdV and mKdV, respectively).
Further we will focus on the symmetries generated by the following vector fields
*v* and *w* are defined by *v*_{x} = u
and *w*_{x} = u²)
To construct conservation laws we need to know Poisson bracket structure
and again like in the case of NSE the Poisson bivector field is well defined
when *u* is subjected to either periodic
*u(t, − ∞) = u(t, + ∞)* or zero
*u(t, − ∞) = u(t, + ∞) = 0* boundary
conditions. For both KdV and mKdV the Poisson bivector field is
*[[E[E , W]]W] = 0*.
Thus the conservation laws and bi-Hamiltonian structures of KdV and mKdV
hierarchies are related to the non-Noether symmetries of KdV and mKdV equations.

u_{t} + u_{xxx} + uu_{x} = 0 [KdV]

and
u_{t} + u_{xxx} − 6u²u_{x} = 0 [mKdV]

(here
E(u)_{t} + E(u)_{xxx} + u_{x}E(u) + uE(u)_{x} = 0 [KdV]

and
E(u)_{t} + E(u)_{xxx} − 12uu_{x}E(u) − 6u²E(u)_{x} = 0 [mKdV]

(again this conditions are obtained by substituting infinitesimal transformation
E(u) = u_{xx} + u² +
u_{x}v + (u_{xxx} + uu_{x})

− (6u_{xxxxx} + 20u_{x}u_{xx} +
10 uu_{xxx} + 5u²u_{x}) [KdV]

and
− (6u

E(u) = − u_{xx} + 2u^{3}
+ u_{x}w − (u_{xxx} − 6u²u_{x})

− (u_{xxxxx} − 10u²u_{xxx}
− 40uu_{x}u_{xx} − 10u_{x}^{3}
+ 30u^{4}u_{x}) [mKdV]

(here − (u

W = dx ∧

with corresponding symplectic form
ω = dx δu ∧ δv

leading to Hamiltonian realization of KdV and mKdV equations
u_{t} = {h , u}

with Hamiltonians
h = (u_{x}^{2} − ) dx [KdV]

and
h = (u_{x}^{2} + u^{4}) dx [mKdV]

By taking Lie derivative of the
symplectic form along the generators of the symmetries one gets
another couple of symplectic forms
L_{E}ω = dx (δu ∧ δu_{x} + uδu ∧ δv) [KdV]

L_{E}ω = dx (δu ∧ δu_{x} − 2uδu ∧ δw) [mKdV]

involved in bi-Hamiltonian realization of KdV/mKdV hierarchies and
proposed by Magri [4]. The conservation laws associated with
the symmetries reproduce infinite sequence of conservation laws of KdV equation
I_{1} = Y_{1} =
u dx

I_{2} = Y_{1} − 2Y_{2} =
u² dx

I_{3} = Y_{1}^{3} − 3Y_{1}Y_{2} + 3Y_{3} =
( − u_{x}^{2}) dx

I_{4} = Y_{1}^{4} − 4Y_{1}^{2}Y_{2} +
2Y_{2}^{2} + 4Y_{1}Y_{3} − 4Y_{4} =

(u^{4} −
uu_{x}^{2} + u_{xx}^{2}) dx

I_{m} = (− 1)^{m}mY_{m} +
(− 1)^{k}I_{m − k}Y_{k}

and mKdV equation
I

I

I

(u

I

I_{1} = Y_{1} = − 4u² dx

I_{2} = Y_{1} − 2Y_{2} =
16(u^{4} + u_{x}^{2}) dx

I_{3} = Y_{1}^{3} − 3Y_{1}Y_{2}
+ 3Y_{3} = − 32(2u^{6} + 10 u²u_{x}^{2}
+ u_{xx}^{2}) dx

I_{4} = Y_{1}^{4} − 4Y_{1}^{2}Y_{2} +
2Y_{2}^{2} + 4Y_{1}Y_{3} − 4Y_{4} =

(5 u^{8}
+ 70u^{4}u_{x}^{2} − 7u_{x}^{4}
+ 14u²u_{xx}^{2} + u_{xxx}^{2}) dx

I_{m} = (− 1)^{m}mY_{m} +
(− 1)^{k}I_{m − k}Y_{k}

The involutivity of these conservation laws is well known and in terms of the symmetry
generators it is ensured by conditions I

I

I

(5 u

I

The purpose of the present paper was to illustrate some features of
non-Noether symmetries discussed in [1] and
to show that in several important integrable models existence of complete sets of
conservation laws could be related to the such symmetries.

- G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys. 48 (2003) 190-202,math-ph/0211014
- A. Das, Integrable models, World Scientific Lecture Notes in Physics, Vol. 30, World Scientific, Singapore, 1989
- M. Lutzky, New derivation of a conserved quantity for Lagrangian systems, J. of Phys. A: Math. Gen. 15 (1998) L721-722
- F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (5) (1978) 1156-1162