In the present paper we would like to shed more light on geometric aspects of
the concept of non-Noether symmetry and to emphasize influence of such a symmetries on the phase space geometry.
Partially the motivation for studying these issues comes from the theory of integrable models
that essentially relies on different geometric objects used for constructing conservation
laws. Among them are Frölicher-Nijenhuis
operators, bi-Hamiltonian systems, Lax pairs and bicomplexes. And it seems that the existance of these important
geometric structures could be related to the hidden non-Noether symmetries of the dynamical systems.
We would like to show how in Hamiltonian systems presence of certain non-Noether symmetries leads to the
above mentioned Lax pairs, Frölicher-Nijenhuis operators, bi-Hamiltonian structures,
bicomplexes and a number of conservation laws.

Let us first recall some basic knowledge of the Hamiltonian dynamics. The phase space of
a regular Hamiltonian system is a Poisson manifold – a smooth finite-dimensional
manifold equipped with the Poisson bivector field *W*
subjected to the following condition
*W* provides the skew symmetry of
the corresponding Poisson bracket and the condition
(1) ensures that for every triple *(f, g, h)* of smooth
functions on the phase space the Jacobi identity
*W* has maximal
rank, i. e. its *n*-th outer power, where *n* is a half-dimension of
the phase space, does not vanish *W*^{n} ≠ 0.
In this case *W* gives rise to a well known isomorphism
*Φ* between the differential 1-forms and
the vector fields defined by
*u* and could be extended to higher degree
differential forms and multivector fields by linearity and multiplicativity
*Φ(u ∧ v) = Φ(u) ∧ Φ(v)*.

[W , W] = 0

where square bracket stands for Schouten bracket or supercommutator
(for simplicity further it will be referred as commutator). In a standard manner Poisson
bivector field defines a Lie bracket on the algebra of observables
(smooth real-valued functions on phase space) called Poisson bracket:
{f , g} = W(df ∧ dg)

Skew symmetry of the bivector field
{f{g , h}} + {h{f , g}} + {g{h , f}} = 0.

is satisfied. We also assume that the dynamical system under consideration
is regular – the bivector field
Φ(u) = W(u)

for every 1-form Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equation
*h* is some fixed smooth function on the phase space called Hamiltonian.
Let us recall that each vector field *E* on the phase space generates
the one-parameter continuous group of transformations
*g*_{z} = e^{zLE} (here *L* denotes Lie derivative)
that acts on the observables as follows
*E* of the group *g*_{z} commutes with the vector field
*W(h) = {h , }*, i. e.
*E* is time dependent vector field on phase space. In this case
(8) should be replaced with
*E* does not preserve Poisson
bivector field *[E , W] ≠ 0* then *g*_{z} is called non-Noether symmetry.

Ô = {h , Ô}

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

Such a group of transformation is called symmetry of Hamilton's equation (5)
if it commutes with time evolution operator
g_{z}(Ô) = g_{z}(Ô)

in terms of the vector fields this condition means that the generator
[E , W(h)] = 0.

However we would like to consider more general
case where
E = [E , W(h)].

If in addition to (8) the vector field Now let us focus on non-Noether symmetries. We would like to show that the presence of
such a symmetry could essentially enrich the geometry of the phase space
and under the certain conditions could ensure integrability of the dynamical system.
Before we proceed let us recall that the non-Noether symmetry leads to a number of
integrals of motion
[4]. More precisely the
relationship between non-Noether symmetries and the conservation laws is described by
the following theorem.

Y^{(k)} = k = 1,2, ... n

where
Ŵ^{k} ∧ W^{n − k} = Y^{(k)}W^{n}.

(definition is correct since the space of
Ŵ^{k} ∧ W^{n − k} =
(Y^{(k)})W^{n} + Y^{(k)}[W(h) , W^{n}]

or
k(Ŵ) ∧ Ŵ^{k − 1} ∧ W^{n − k}

+ (n − k)[W(h) , W] ∧ Ŵ^{k} ∧ W^{n − k − 1} =

(Y^{(k)})W^{n} + nY^{(k)}[W(h) , W] ∧ W^{n − 1}

but according to the Liouville theorem the Hamiltonian vector field preserves + (n − k)[W(h) , W] ∧ Ŵ

(Y

W = [W(h) , W] = 0

hence, by taking into account that
E= E + [W(h) , E] = 0

we get
Ŵ = [E , W] = [E, W] + [E[W(h) , W]] = 0.

and as a result (13) yields
Y^{(k)} W^{n} = 0

but since the dynamical system is regular (
(Ŵ − cW)^{n} = 0

could be associated with the generator of symmetry.
By expanding expression (18) it is easy to verify that the conservation laws
Y^{(k)} =
c_{m[1]}c_{m[2]} ⋯ c_{m[k]}

W = D_{1} ∧ D_{3} + D_{2} ∧ D_{4}

(
h = ½z_{1}^{2} + ½z_{2}^{2} + e^{z3 − z4}

Then the vector field
E = E_{m}D_{m}

with components
E_{1} = ½z_{1}^{2} − e^{z3 − z4} −
(z_{1} + z_{2})e^{z3 − z4}

E_{2} = ½z_{2}^{2} + 2e^{z3 − z4} +
(z_{1} + z_{2})e^{z3 − z4}

E_{3} = 2z_{1} + ½z_{2} +
(z_{1}^{2} + e^{z3 − z4})

E_{4} = z_{2} − ½z_{1} + (z_{2}^{2}
+ e^{z3 − z4})

satisfies (9) condition and as a result generates symmetry of the dynamical system.
The symmetry appears to be non-Noether with Schouten bracket E

E

E

Ŵ = [E , W] = z_{1}D_{1} ∧ D_{3} +
z_{2}D_{2} ∧ D_{4} +
e^{z3 − z4}D_{1} ∧ D_{2} +
D_{3} ∧ D_{4}

calculating volume vector fields
W ∧ W = − 2D_{1} ∧ D_{2} ∧ D_{3} ∧ D_{4}

Ŵ ∧ W = − (z_{1} + z_{2})D_{1} ∧ D_{2} ∧ D_{3} ∧ D_{4}

Ŵ ∧ Ŵ = − 2(z_{1}z_{2} − e^{z3 − z4}) D_{1} ∧ D_{2} ∧ D_{3} ∧ D_{4}

and the conservation laws associated with this symmetry are just
Ŵ ∧ W = − (z

Ŵ ∧ Ŵ = − 2(z

Y^{(1)} = = ½(z_{1} + z_{2})

Y^{(2)} = = z_{1}z_{2} − e^{z3 − z4}

Y

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but also
endows the phase space with a number of interesting geometric structures and it appears that such a
symmetry is related to many important concepts used in theory of dynamical systems.
One of the such concepts is Lax pair.
Let us recall that Lax pair of Hamiltonian system on Poisson manifold *M* is
a pair *(L , P)* of smooth functions on *M* with values in some
Lie algebra *g* such that the time evolution of *L* is governed
by the following equation
*[ , ]* is a Lie bracket on *g*. It is well known that each Lax
pair leads to a number of conservation laws. When *g* is some matrix Lie algebra
the conservation laws are just traces of powers of *L*
*z*_{m}, where the bivector field
*W* and the generator of the symmetry *E* have the
following form

L = [L , P]

where
I^{(k)} = Tr(L^{k})

It is remarkable that each generator of the non-Noether
symmetry canonically leads to the Lax pair of a certain type.
In the local coordinates
W = W_{ab}D_{k} ∧ D_{m}
E = E_{m}D_{m}

corresponding Lax pair could be calculated explicitly.
Namely we have the following theorem:
L_{ab} =
(W^{−1})_{ad} (E_{c}D_{c}W_{db}
− W_{cb}D_{c}E_{d} + W_{dc}D_{c}E_{b})

P_{ab} = D_{a} (W_{bc}D_{c}h)

form the Lax pair (27) of the dynamical system P

Ŕ_{E}(u) = Φ^{− 1}([E , Φ(u)]) − L_{E}u

(here
Ŕ_{E} = L_{ab} dz_{a} ⊗ D_{b}

and the invariance condition
Ŕ_{E} = L_{W(h)}Ŕ_{E} = 0

yields
Ŕ_{E} =
L_{ab} dz_{a} ⊗ D_{b}

= (L_{ab}) dz_{a} ⊗ D_{b} +
L_{ab} (L_{W(h)}dz_{a}) ⊗ D_{b}

+ L_{ab} dz_{a} ⊗ (L_{W(h)}D_{b}) =
(L_{ab}) dz_{a} ⊗ D_{b}

+ L_{ab}D_{c}(W_{ad}D_{d}h)dz_{c} ⊗ D_{b} +
L_{ab}D_{b}(W_{cd}D_{d}h)dz_{a} ⊗ D_{c}

= (L_{ab} +
(P_{ac}L_{cb}
− L_{ac}P_{cb}))dz_{a} ⊗ D_{b} = 0

or in matrix notations
= (L

+ L

+ L

= (L

L = [L , P].

So, we have proved that the non-Noether symmetry canonically yields a Lax pair
on the algebra of linear operators on cotangent bundle over the phase space.
I^{(k)} = Tr(L^{k}) = c_{m}^{k}

This correspondence follows from the equation (18)
and the definition of the operator
L_{11} = L_{33} = z_{1}; L_{22} = L_{44} = z_{2}

L_{14} = − L_{23} = e^{z3 − z4};
L_{32} = − L_{41} = 1

The conservation laws associated with this Lax matrix are
L

I^{(1)} = Tr(L) = 2(z_{1} + z_{2})

I^{(2)} = Tr(L²) = 2z_{1}^{2} + 2z_{2}^{2}
+ 4e^{z3 − z4}

I

Now let us focus on the integrability issues. We know that
*n* integrals of motion are associated with each generator of non-Noether
symmetry and according to the Liouville-Arnold theorem Hamiltonian system is
completely integrable if it possesses *n* functionally independent integrals of
motion in involution (two functions *f* and *g* are said to be
in involution if their Poisson bracket vanishes *{f , g} = 0*).
Generally speaking the conservation laws associated with symmetry might appear to be neither
independent nor involutive.
However it is reasonable to ask the question – what condition should be satisfied
by the generator of the symmetry to ensure the involutivity
(*{Y*^{(k)} , Y^{(m)}} = 0) of conserved quantities?
In Lax theory such a condition is known as
Classical Yang-Baxter Equation (CYBE). Since involutivity of the conservation laws
is closely related to the integrability it is essential to have some analog of CYBE for the generator
of non-Noether symmetry. To address this issue we would like to propose the following theorem.

[[E[E , W]]W] = 0

and
{Y^{(k)} , Y^{(m)}} = 0

[W , W] = 0 L_{W(f)}W = [W(f) , W] = 0

By taking the Lie derivative of the expression (1)
we obtain another useful identity
L_{E}[W , W] = [E[W , W]] = [[E , W] W] + [W[E , W]] = 2[Ŵ , W] = 0.

This identity gives rise to the following relation
[Ŵ , W] = 0 ⇔ [Ŵ(f) , W] = − [Ŵ , W(f)]

and finally condition (39) ensures third identity
[Ŵ , Ŵ] = 0

yielding Liouville theorem for
[Ŵ , Ŵ] = 0 ⇔ [Ŵ(f) , Ŵ] = 0

Indeed
[Ŵ , Ŵ] = [[E , W]Ŵ] = [[Ŵ , E]W]

= − [[E , Ŵ]W] = − [[E[E , W]]W] = 0

Now let us consider two different solutions = − [[E , Ŵ]W] = − [[E[E , W]]W] = 0

(Ŵ − c_{i}W)^{n} = 0

along the vector fields
(Ŵ − c_{i}W)^{n − 1}(L_{W(cj)}Ŵ − {c_{j} , c_{i}}W) = 0,

and
(Ŵ − c_{i}W)^{n − 1}(c_{i}L_{Ŵ(cj)}W
+ {c_{j} , c_{i}}_{•}W) = 0,

where
{c_{i} , c_{j}}_{•} = Ŵ(dc_{i} ∧ dc_{j})

is the Poisson bracket calculated by means of the bivector field
({c_{i} , c_{j}}_{•}
− c_{i}{c_{i} , c_{j}})(Ŵ − c_{i}W)^{n − 1}W = 0

Thus, either
{c_{i} , c_{j}}_{•} − c_{i}{c_{i} , c_{j}} = 0

or the volume field
{c_{i} , c_{j}}_{•} − c_{j}{c_{i} , c_{j}} = 0

Finally by comparing (52) and (53) we obtain that
the functions
{c_{i} , c_{j}}_{•} = {c_{i} , c_{j}} = 0

and according to (19) the same is true for the integrals of motion
Another concept that is often used in theory of dynamical systems and could
be related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach).
Recently A. Dimakis and F. Müller-Hoissen
applied bidifferential calculi to the wide range of integrable models
including KdV hierarchy, KP equation, self-dual Yang-Mills equation,
Sine-Gordon equation, Toda models, non-linear Schrödinger
and Liouville equations. It turns out that these models can be effectively
described and analyzed using the bidifferential calculi [1], [2].

Under the bidifferential calculus we mean the graded algebra of differential forms
*Ω*^{(k)} denotes the space of *k*-degree differential forms)
equipped with a couple of differential operators
*d² = đ² = dđ + đd = 0*
conditions (see [2]).
It is interesting that if generator of the non-Noether symmetry satisfies
equation (39) then we are able to construct an invariant bidifferential calculus
of a certain type. This construction is summarized in the following theorem:

Ω = Ω^{(k)}

(
d, đ : Ω^{(k)} → Ω^{(k + 1)}

satisfying
du = Φ^{− 1}([W , Φ(u)])

đu = Φ^{− 1}([[E , W]Φ(u)])

form invariant bidifferential calculus
(
d²u = Φ^{− 1}([W , Φ(Φ^{− 1}([W , Φ(u)]))])

= Φ^{− 1}([W[W , Φ(u)]]) = 0

as a result of the property (41) and the Jacobi identity for = Φ

đ²u = Φ^{− 1}([[W , E][[W , E]Φ(u)]]) = 0

according to the property (45) of
(dđ + đd)(u) = Φ^{− 1}([[[W , E]W]Φ(u)]) = 0

as far as (43) is satisfied.
So,
(k + 1)đI^{(k)} = kdI^{(k + 1)}

coincide with the sequence of integrals of motion (36).
Proof of this correspondence lay outside the scope of present article,
but could be done in the manner similar to [1].
đz_{1} = z_{1}dz_{1} − e^{z3 − z4}dz_{4}

đz_{2} = z_{2}dz_{2} + e^{z3 − z4}dz_{3}

đz_{3} = z_{1}dz_{3} + dz_{2}

đz_{4} = z_{2}dz_{4} − dz_{1}

and is extended to whole De Rham complex by linearity, derivation property and
compatibility property đz

đz

đz

2đI^{(1)} = dI^{(2)}

Finally we would like to reveal some features of the operator
*Ŕ*_{E}
(31) and to show how Frölicher-Nijenhuis geometry could arise in
Hamiltonian system that possesses certain non-Noether symmetry.
From the geometric properties of the tangent valued forms we know
that the traces of powers of a linear operator *F*
on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion
*T(F)* vanishes, i. e. whenever for arbitrary vector fields *X,Y* the condition

T(F)(X , Y) = [FX , FY] − F([FX , Y] + [X , FY] − F[X , Y]) = 0

is satisfied.
Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used in
theory of integrable models. We would like to show
that each generator of non-Noether symmetry satisfying equation (39)
canonnically leads to invariant Frölicher-Nijenhuis operator on tangent
bundle over the phase space. Strictly speaking we have the following theorem.
R_{E}(X) = Φ(L_{E}Φ^{− 1}(X)) − [E , X]

is invariant Frölicher-Nijenhuis operator on
T(R_{E})(X , Y) = [R_{E}(X) , R_{E}(Y)]

− R_{E}([R_{E}(X) , Y] + [X , R_{E}(Y)] − R_{E}([X , Y])) = 0

First let us introduce the following auxiliary 2-forms
− R

ω = Φ^{− 1}(W), ω^{•} = Ŕ_{E}ω
ω^{••} = Ŕ_{E}ω^{•}

Using the realization (57) of the differential
dω = Φ^{− 1}([W , W]) = 0

Similarly, using the property (43) we obtain
dω^{•} = dΦ^{− 1}([E , W]) − dL_{E}ω
= Φ^{− 1}([[E , W]W]) − L_{E}dω = 0

And finally, taking into account that
dω^{••} = 2Φ^{− 1}([[E[E , W]]W]) − 2dL_{E}ω^{•}
= − 2L_{E}dω^{•} = 0

So the differential forms
dω = dω^{•} = dω^{••} = 0

Now let us consider the contraction of
i_{T(RE)(X , Y)}ω =
i_{[REX , REY]}ω −
i_{[REX , Y]}ω^{•} −
i_{[X , REY]}ω^{•} +
i_{[X , Y]}ω^{••}

= L_{REX}i_{Y}ω^{•} −
i_{REY}L_{X}ω^{•} −
L_{REX}i_{Y}ω^{•} +
i_{Y}L_{REX}ω^{•} −
L_{X}i_{REY}ω^{•}

+ i_{REY}L_{X}ω^{•} +
i_{[X , Y]}ω^{••} =
i_{Y}L_{X}ω^{••} −
L_{X}i_{Y}ω^{••} +
i_{[X , Y]}ω^{••} = 0

where we used (68) (72),
the property of the Lie derivative
= L

+ i

L_{X}i_{Y}ω =
i_{Y}L_{X}ω + i_{[X , Y]}ω

and the relations of the following type
L_{REX}ω = di_{REX}ω + i_{REX}dω
= di_{X}ω^{•} = L_{X}ω^{•} − i_{X}dω^{•}
= L_{X}ω^{•}

So we proved that for arbitrary vector fields
ω = Φ^{− 1}(W)

is also non-degenerate and vanishing of the contraction (73)
implies that the torsion
T(R_{E})(X , Y) = [R_{E}(X) , R_{E}(Y)]

− R_{E}([R_{E}(X) , Y] + [X , R_{E}(Y)] − R_{E}([X , Y])) = 0

− R

R_{E} =
z_{1}dz_{1} ⊗ D_{1} −
dz_{1} ⊗ D_{4} +
z_{2}dz_{2} ⊗ D_{2} +
dz_{2} ⊗ D_{3} +
z_{1}dz_{3} ⊗ D_{3}

+ e^{z3 − z4}dz_{3} ⊗ D_{2} +
z_{2}dz_{4} ⊗ D_{4}
− e^{z3 − z4}dz_{4} ⊗ D_{1}

(compare with [3])
+ e

In summary let us note that the non-Noether symmetries form quite interesting
class of symmetries of Hamiltonian dynamical system and lead not only to
a number of conservation laws (that under certain conditions ensure integrability),
but also enrich the geometry of the phase space by endowing it with several important
structures, such as Lax pair, bicomplex,
bi-Hamiltonian structure, Frölicher-Nijenhuis operators etc.
The present paper attempts to emphasize deep relationship between different
concepts used in construction of conservation laws and non-Noether symmetry.

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- A. Dimakis, F. Müller-Hoissen, Bicomplexes and integrable models, 2000 nlin.SI/0006029
- R. Fernandes, On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. of Phys. A: Math. Gen. 26 (1993) 3793-3803
- M. Lutzky, New derivation of a conserved quantity for Lagrangian systems, J. of Phys. A: Math. Gen. 15 (1998) L721-722