Non-Noether symmetries in Hamiltonian Dynamical Systems

George Chavchanidze

Department of Theoretical Physics, A. Razmadze Institute of Mathematics, 1 Aleksidze Street, Tbilisi 0193, Georgia

We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. Correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular Hamiltonian systems such symmetries canonically lead to Lax pairs on the algebra of linear operators on cotangent bundle over the phase space. Relationship between non-Noether symmetries and other widespread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frölicher-Nijenhuis geometry is considered. It is proved that the integrals of motion associated with a continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang-Baxter type equation. Action of one-parameter group of symmetry on algebra of integrals of motion is studied and involutivity of group orbits is discussed. Hidden non-Noether symmetries of Toda chain, Korteweg-de Vries equation, Benney system, nonlinear water wave equations and Broer-Kaup system are revealed and discussed.

Non-Noether symmetry; Conservation law; bi-Hamiltonian system; Bidifferential calculus; Lax pair; Frölicher-Nijenhuis operator; Korteweg-de Vries equation; Broer-Kaup system; Benney system; Toda chain

70H33; 70H06; 58J70; 53Z05; 35A30

Introduction

Symmetries play essential role in dynamical systems, because they usually simplify analysis of evolution equations and often provide quite elegant solution of problems that otherwise would be difficult to handle. In Lagrangian and Hamiltonian dynamical systems special role is played by Noether symmetries — an important class of symmetries that leave action invariant and have some exceptional features. In particular, Noether symmetries deserved special attention due to celebrated Noether's theorem, that established correspondence between symmetries, that leave action functional invariant, and conservation laws of Euler-Lagrange equations. This correspondence can be extended to Hamiltonian systems where it becomes more tight and evident then in Lagrangian case and gives rise to Lie algebra homomorphism between Lie algebra of Noether symmetries and algebra of conservation laws (that form Lie algebra under Poisson bracket).

Role of symmetries that are not of Noether type has been suppressed for quite a long time. However, after some publications of Hojman, Harleston, Lutzky and others (see [16], [36], [39], [40], [49]-[57]) it became clear that non-Noether symmetries also can play important role in Lagrangian and Hamiltonian dynamics. In particular, according to Lutzky [51], in Lagrangian dynamics there is definite correspondence between non-Noether symmetries and conservation laws. Moreover, each generator of non-Noether symmetry may produce whole family of conservation laws (maximal number of conservation laws that can be associated with non-Noether symmetry via Lutzky's theorem is equal to the dimension of configuration space of Lagrangian system). This fact makes non-Noether symmetries especially valuable in infinite dimensional dynamical systems, where potentially one can recover infinite sequence of conservation laws knowing single generator of non-Noether symmetry.

Existence of correspondence between non-Noether symmetries and conserved quantities raised many questions concerning relationship among this type of symmetries and other geometric structures emerging in theory of integrable models. In particular one could notice suspicious similarity between the method of constructing conservation laws from generator of non-Noether symmetry and the way conserved quantities are produced in either Lax theory, bi-Hamiltonian formalism, bicomplex approach or Lenard scheme. It also raised natural question whether set of conservation laws associated with non-Noether symmetry is involutive or not, and since it appeared that in general it may not be involutive, there emerged the need of involutivity criteria, similar to Yang-Baxter equation used in Lax theory or compatibility condition in bi-Hamiltonian formalism and bicomplex approach. It was also unclear how to construct conservation laws in case of infinite dimensional dynamical systems where volume forms used in Lutzky's construction are no longer well defined. Some of these questions were addressed in papers [11]-[14], while in the present review we would like to summarize all these issues and to provide some examples of integrable models that possess non-Noether symmetries.

Review is organized as follows. In first section we briefly recall some aspects of geometric formulation of Hamiltonian dynamics. Further, in second section, correspondence between non-Noether symmetries and integrals of motion in regular Hamiltonian systems is discussed. Lutzky's theorem is reformulated in terms of bivector fields and alternative derivation of conserved quantities suitable for computations in infinite dimensional Hamiltonian dynamical systems is suggested. Non-Noether symmetries of two and three particle Toda chains are used to illustrate general theory. In the subsequent section geometric formulation of Hojman's theorem [36] is revisited and some examples are provided. Section 4 reveals correspondence between non-Noether symmetries and Lax pairs. It is shown that non-Noether symmetry canonically gives rise to a Lax pair of certain type. Lax pair is explicitly constructed in terms of Poisson bivector field and generator of symmetry. Examples of Toda chains are discussed. Next section deals with integrability issues. An analogue of Yang-Baxter equation that, being satisfied by generator of symmetry, ensures involutivity of set of conservation laws produced by this symmetry, is introduced. Relationship between non-Noether symmetries and bi-Hamiltonian systems is considered in section 6. It is proved that under certain conditions, non-Noether symmetry endows phase space of regular Hamiltonian system with bi-Hamiltonian structure. We also discuss conditions under which non-Noether symmetry can be "recovered" from bi-Hamiltonian structure. Theory is illustrated by example of Toda chains. Next section is devoted to bicomplexes and their relationship with non-Noether symmetries. Special kind of deformation of De Rham complex induced by symmetry is constructed in terms of Poisson bivector field and generator of symmetry. Samples of two and three particle Toda chain are discussed. Section 8 deals with Frölicher-Nijenhuis recursion operators. It is shown that under certain condition non-Noether symmetry gives rise to invariant Frölicher-Nijenhuis operator on tangent bundle over phase space. The last section of theoretical part contains some remarks on action of one-parameter group of symmetry on algebra of integrals of motion. Special attention is devoted to involutivity of group orbits.

Subsequent sections of present review provide examples of integrable models that possess interesting non-Noether symmetries. In particular section 10 reveals non-Noether symmetry of

n

-particle Toda chain. Bi-Hamiltonian structure, conservation laws, bicomplex, Lax pair and Frölicher-Nijenhuis recursion operator of Toda hierarchy are constructed using this symmetry. Further we focus on infinite dimensional integrable Hamiltonian systems emerging in mathematical physics. In section 11 case of Korteweg-de Vries equation is discussed. Symmetry of this equation is identified and used in construction of infinite sequence of conservation laws and bi-Hamiltonian structure of KdV hierarchy. Next section is devoted to non-Noether symmetries of integrable systems of nonlinear water wave equations, such as dispersive water wave system, Broer-Kaup system and dispersiveless long wave system. Last section focuses on Benney system and its non-Noether symmetry, that appears to be local, gives rise to infinite sequence of conserved densities of Benney hierarchy and endows it with bi-Hamiltonian structure.

Regular Hamiltonian systems

The basic concept in geometric formulation of Hamiltonian dynamics is notion of symplectic manifold. Such a manifold plays the role of the phase space of the dynamical system and therefore many properties of the dynamical system can be quite effectively investigated in the framework of symplectic geometry. Before we consider symmetries of the Hamiltonian dynamical systems, let us briefly recall some basic notions from symplectic geometry.

The symplectic manifold is a pair

(M, ω)

where

M

is smooth even dimensional manifold and

ω

is a closed

d ω = 0

and nondegenerate 2-form on

M

. Being nondegenerate means that contraction of arbitrary non-zero vector field with

ω

does not vanish

i_{X} ω = 0 \Leftrightarrow X = 0

(here

i_{X}

denotes contraction of the vector field

X

with differential form). Otherwise one can say that

ω

is nondegenerate if its n-th outer power does not vanish (

ω^{n} \neq 0

) anywhere on

M

. In Hamiltonian dynamics

M

is usually phase space of classical dynamical system with finite numbers of degrees of freedom and the symplectic form

ω

is basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fields and the form of Hamilton's equations.

The symplectic form

ω

naturally defines isomorphism between vector fields and differential 1-forms on

M

(in other words tangent bundle

T M

of symplectic manifold can be quite naturally identified with cotangent bundle

T^{*} M

). The isomorphic map

Φ_{ω}

from

T M

into

T^{*} M

is obtained by taking contraction of the vector field with

ω

Φ_{ω} : X \to - i_{X} ω

(minus sign is the matter of convention). This isomorphism gives rise to natural classification of vector fields. Namely, vector field

X_{h}

is said to be Hamiltonian if its image is exact 1-form or in other words if it satisfies Hamilton's equation

i_{X_{h}} ω + d h = 0

for some function

h

M

. Similarly, vector field

X

is called locally Hamiltonian if it's image is closed 1-form

i_{X} ω + u = 0, d u = 0

One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem, is that these vector fields preserve symplectic form

ω

. In other words Lie derivative of the symplectic form

ω

along arbitrary locally Hamiltonian vector field vanishes

L_{X} ω = 0 \Leftrightarrow i_{X} ω + d u = 0, d u = 0

Indeed, using Cartan's formula that expresses Lie derivative in terms of contraction and exterior derivative

L_{X} = i_{X} d + d i_{X}

one gets

L_{X} ω = i_{X} d ω + d i_{X} ω = d i_{X} ω

(since

d ω = 0

) but according to the definition of locally Hamiltonian vector field

d i_{X} ω = - d u = 0

So locally Hamiltonian vector fields preserve

ω

and vise versa, if vector field preserves symplectic form

ω

then it is locally Hamiltonian.

Clearly, Hamiltonian vector fields constitute subset of locally Hamiltonian ones since every exact 1-form is also closed. Moreover one can notice that Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields. This fact can be observed as follows. First of all for arbitrary couple of locally Hamiltonian vector fields

X, Y

we have

L_{X} ω = L_{Y} ω = 0

and

L_{X} L_{Y} ω - L_{Y} L_{X} ω = L_{[X, Y]} ω = 0

so locally Hamiltonian vector fields form Lie algebra (corresponding Lie bracket is ordinary commutator of vector fields). Further it is clear that for arbitrary Hamiltonian vector field

X_{h}

and locally Hamiltonian one

Z

one has

L_{Z} ω = 0

and

i_{X_{h}} ω + d h = 0

that implies

L_{Z} (i_{X_{h}} ω + d h) = L_{[Z, X_{h}]} ω + i_{X_{h}} L_{Z} ω + d L_{Z} h = L_{[Z, X_{h}]} ω + d L_{Z} h = 0

thus commutator

[Z, X_{h}]

is Hamiltonian vector field

X_{L_{Z} h}

, or in other words Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields.

Isomorphism

Φ_{ω}

can be extended to higher order vector fields and differential forms by linearity and multiplicativity. Namely,

Φ_{ω} (X \land Y) = Φ_{ω} (X) \land Φ_{ω} (Y)

Since

Φ_{ω}

is isomorphism, the symplectic form

ω

has unique counter image

W

known as Poisson bivector field. Property

d ω = 0

together with non degeneracy implies that bivector field

W

is also nondegenerate (

W^{n} \neq 0

) and satisfies condition

[W, W] = 0

where bracket

[,]

known as Schouten bracket or supercommutator, is actually graded extension of ordinary commutator of vector fields to the case of multivector fields, and can be defined by linearity and derivation property

[C_{1} \land C_{2} \land . . . \land C_{n}, S_{1} \land S_{2} \land . . . \land S_{n}] = (- 1)^{p + q} [C_{p}, S_{q}] \land C_{1} \land C_{2} \land . . . \land Ĉ_{p} \land . . . \land C_{n} \land S_{1} \land S_{2} \land . . . \land Ŝ_{q} \land . . . \land S_{n}

where over hat denotes omission of corresponding vector field. In terms of the bivector field

W

Liouville's theorem mentioned above can be rewritten as follows

[W (u), W] = 0 \Leftrightarrow d u = 0

for each 1-form

u

. It follows from graded Jacoby identity satisfied by Schouten bracket and property

[W, W] = 0

satisfied by Poisson bivector field.

Being counter image of symplectic form,

W

gives rise to map

Φ_{W}

, transforming differential 1-forms into vector fields, which is inverted to the map

Φ_{ω}

and is defined by

Φ_{W} : u \to W (u); Φ_{W} Φ_{ω} = i d

Further we will often use these maps.

In Hamiltonian dynamical systems Poisson bivector field is geometric object that underlies definition of Poisson bracket — kind of Lie bracket on algebra of smooth real functions on phase space. In terms of bivector field

W

Poisson bracket is defined by

{f, g} = W (d f \land d g)

The condition

[W, W] = 0

satisfied by bivector field ensures that for every triple

(f, g, h)

of smooth functions on the phase space the Jacobi identity

{f {g, h}} + {h {f, g}} + {g {h, f}} = 0 .

is satisfied. Interesting property of the Poisson bracket is that map from algebra of real smooth functions on phase space into algebra of Hamiltonian vector fields, defined by Poisson bivector field

f \to X_{f} = W (d f)

appears to be homomorphism of Lie algebras. In other words commutator of two vector fields associated with two arbitrary functions reproduces vector field associated with Poisson bracket of these functions

[X_{f}, X_{g}] = X_{{f, g}}

This property is consequence of the Liouville theorem and definition of Poisson bracket. Further we also need another useful property of Hamiltonian vector fields and Poisson bracket

{f, g} = W (d f \land d g) = ω (X_{f} \land X_{g}) = L_{X_{f}} g = - L_{X_{g}} g

it also follows from Liouville theorem and definition of Hamiltonian vector fields and Poisson brackets.

To define dynamics on

M

one has to specify time evolution of observables (smooth functions on

M

). In Hamiltonian dynamical systems time evolution is governed by Hamilton's equation

\frac{d}{d t} f = {h, f}

where

h

is some fixed smooth function on the phase space called Hamiltonian. In local coordinate frame

z_{b}

bivector field

W

has the form

W = W_{b c} \frac{\partial}{\partial z_{b}} \land \frac{\partial}{\partial z_{c}}

and the Hamilton's equation rewritten in terms of local coordinates takes the form

ż_{b} = W_{b c} \frac{\partial h}{\partial z_{b}}

Non-Noether symmetries

Now let us focus on symmetries of Hamilton's equation (24). Generally speaking, symmetries play very important role in Hamiltonian dynamics due to different reasons. They not only give rise to conservation laws but also often provide very effective solutions to problems that otherwise would be difficult to solve. Here we consider special class of symmetries of Hamilton's equation called non-Noether symmetries. Such a symmetries appear to be closely related to many geometric concepts used in Hamiltonian dynamics including bi-Hamiltonian structures, Frölicher-Nijenhuis operators, Lax pairs and bicomplexes.

Before we proceed let us recall that each vector field

E

on the phase space generates the one-parameter continuous group of transformations

g_{z} = e^{z L_{E}}

(here

L

denotes Lie derivative) that acts on the observables as follows

g_{z} (f) = e^{z L_{E}} (f) = f + z L_{E} f + ½ (z L_{E})^{2} f + \dots

Such a group of transformation is called symmetry of Hamilton's equation (24) if it commutes with time evolution operator

\frac{d}{d t} g_{z} (f) = g_{z} (\frac{d}{d t} f)

in terms of the vector fields this condition means that the generator

E

of the group

g_{z}

commutes with the vector field

W (h) = {h,}

, i. e.

[E, W (h)] = 0 .

However we would like to consider more general case where

E

is time dependent vector field on phase space. In this case (30) should be replaced with

\frac{\partial}{\partial t} E = [E, W (h)] .

Further one should distinguish between groups of symmetry transformations generated by Hamiltonian, locally Hamiltonian and non-Hamiltonian vector fields. First kind of symmetries are known as Noether symmetries and are widely used in Hamiltonian dynamics due to their tight connection with conservation laws. Second group of symmetries is rarely used. While third group of symmetries that further will be referred as non-Noether symmetries seems to play important role in integrability issues due to their remarkable relationship with bi-Hamiltonian structures and Frölicher-Nijenhuis operators. Thus if in addition to (30) the vector field

E

does not preserve Poisson bivector field

[E, W] \neq 0

then

g_{z}

is called non-Noether symmetry.

Now let us focus on non-Noether symmetries. We would like to show that the presence of such a symmetry essentially enriches the geometry of the phase space and under the certain conditions can 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. More precisely the relationship between non-Noether symmetries and the conservation laws is described by the following theorem. This theorem was proposed by Lutzky in [51]. Here it is reformulated in terms of Poisson bivector field.

Let

(M, h)

be regular Hamiltonian system on the

2 n

-dimensional Poisson manifold

M

. Then, if the vector field

E

generates non-Noether symmetry, the functions

Y^{(k)} = \frac{Ŵ^{k} \land W^{n - k}}{W^{n}} k = 1, 2, . . . n

where

Ŵ = [E, W]

, are integrals of motion.

By the definition

Ŵ^{k} \land W^{n - k} = Y^{(k)} W^{n} .

(definition is correct since the space of

2 n

degree multivector fields on

2 n

degree manifold is one dimensional). Let us take time derivative of this expression along the vector field

W (h)

\frac{d}{d t} Ŵ^{k} \land W^{n - k} = (\frac{d}{d t} Y^{(k)}) W^{n} + Y^{(k)} [W (h), W^{n}]

k (\frac{d}{d t} Ŵ) \land Ŵ^{k - 1} \land W^{n - k} + (n - k) [W (h), W] \land Ŵ^{k} \land W^{n - k - 1} = (\frac{d}{d t} Y^{(k)}) W^{n} + n Y^{(k)} [W (h), W] \land W^{n - 1}

but according to the Liouville theorem the Hamiltonian vector field preserves

W

i. e.

\frac{d}{d t} W = [W (h), W] = 0

hence, by taking into account that

\frac{d}{d t} E = \frac{\partial}{\partial t} E + [W (h), E] = 0

we get

\frac{d}{d t} Ŵ = \frac{d}{d t} [E, W] = [\frac{d}{d t} E, W] + [E [W (h), W]] = 0 .

and as a result (35) yields

\frac{d}{d t} Y^{(k)} W^{n} = 0

but since the dynamical system is regular (

W^{n} \neq 0

) we obtain that the functions

Y^{(k)}

are integrals of motion.

Instead of conserved quantities

Y^{(1)} . . . Y^{(n)}

, the solutions

c_{1} . . . c_{n}

of the secular equation

(Ŵ - c W)^{n} = 0

can be associated with the generator of symmetry. By expanding expression (40) it is easy to verify that the conservation laws

Y^{(k)}

can be expressed in terms of the integrals of motion

c_{1} . . . c_{n}

in the following way

Y^{(k)} = \frac{(n - k)! k!}{n!} \sum_{m_{s} > m_{t}} c_{m_{1}} c_{m_{2}} \dots c_{m_{k}}

Note also that conservation laws

Y^{(k)}

can be also defined by means of symplectic form

ω

using the following formula

Y^{(k)} = \frac{(L_{E} ω)^{k} \land ω^{n - k}}{ω^{n}} k = 1, 2, . . . n

Conservation laws

c_{1} . . . c_{n}

can be also derived from the secular equation

(L_{E} ω - c ω)^{n} = 0

However all these expressions fail in case of infinite dimensional Hamiltonian systems where the volume form

Ω = ω^{n}

does not exist since

n = ∞

. But fortunately in these case one can define conservation laws using alternative formula

C^{(k)} = i_{W^{k}} (L_{E} ω)^{k}

as far as it involves only finite degree differential forms

(L_{E} ω)^{k}

and well defined multivector fields

W^{k}

. Note that in finite dimensional case the sequence of conservation laws

C^{(k)}

is related to families of conservation laws

Y^{(k)}

and

c_{k}

in the following way

C^{(k)} = \frac{(n - k)! k!}{n!} \sum_{m_{s} > m_{t}} c_{m_{1}} c_{m_{2}} \dots c_{m_{k}} = \frac{n!}{(n - k)! k!} Y^{(k)}

Note also that by taking Lie derivative of known conservation along the generator of symmetry

E

one can construct new conservation laws

\frac{d}{d t} Y = L_{X_{h}} Y = 0 \Rightarrow \frac{d}{d t} L_{E} Y = L_{X_{h}} L_{E} Y = L_{E} L_{X_{h}} Y = 0

since

[E, X_{h}] = 0

Besides continuous non-Noether symmetries generated by non-Hamiltonian vector fields one may encounter discrete non-Noether symmetries — noncannonical transformations that doesn't necessarily form group but commute with evolution operator

\frac{d}{d t} g (f) = g (\frac{d}{d t} f)

Such a symmetries give rise to the same conservation laws

Y^{(k)} = \frac{g (W)^{k} \land W^{n - k}}{W^{n}} k = 1, 2, . . . n

Let

M

R^{4}

with coordinates

z_{1}, z_{2}, z_{3}, z_{4}

and Poisson bivector field

W = \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{3}} + \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{4}}

and let's take the following Hamiltonian

h = \frac{1}{2} z_{1}^{2} + \frac{1}{2} z_{2}^{2} + e^{z_{3} - z_{4}}

This is so called two particle non periodic Toda model. One can check that the vector field defined as

E = \sum_{s = 1}^{4} E_{s} \frac{\partial}{\partial z_{s}}

with components

E_{1} = \frac{1}{2} z_{1}^{2} - e^{z_{3} - z_{4}} - \frac{t}{2} (z_{1} + z_{2}) e^{z_{3} - z_{4}} E_{2} = \frac{1}{2} z_{2}^{2} + 2 e^{z_{3} - z_{4}} + \frac{t}{2} (z_{1} + z_{2}) e^{z_{3} - z_{4}} E_{3} = 2 z_{1} + \frac{1}{2} z_{2} + \frac{t}{2} (z_{1}^{2} + e^{z_{3} - z_{4}}) E_{4} = z_{2} - \frac{1}{2} z_{1} + \frac{t}{2} (z_{2}^{2} + e^{z_{3} - z_{4}})

satisfies (31) condition and as a result generates symmetry of the dynamical system. The symmetry appears to be non-Noether with Schouten bracket

[E, W]

equal to

Ŵ = [E, W] = z_{1} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{3}} + z_{2} \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{4}} + e^{z_{3} - z_{4}} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} + \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{4}}

Calculation of volume vector fields

Ŵ^{k} \land W^{n - k}

gives rise to

W \land W = - 2 \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{4}} Ŵ \land W = - (z_{1} + z_{2}) \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{4}} Ŵ \land Ŵ = - 2 (z_{1} z_{2} - e^{z_{3} - z_{4}}) \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{4}}

and the conservation laws associated with this symmetry are just

Y^{(1)} = \frac{Ŵ \land W}{W \land W} = \frac{1}{2} (z_{1} + z_{2}) Y^{(2)} = \frac{Ŵ \land Ŵ}{W \land W} = z_{1} z_{2} - e^{z_{3} - z_{4}}

It is remarkable that the same symmetry is also present in higher dimensions. For example in case where

M

R^{6}

with coordinates

z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}

Poisson bivector equal to

W = \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{4}} + \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{5}} + \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{6}}

and the following Hamiltonian

h = \frac{1}{2} z_{1}^{2} + \frac{1}{2} z_{2}^{2} + \frac{1}{2} z_{3}^{2} + e^{z_{4} - z_{5}} + e^{z_{5} - z_{6}}

we still can construct symmetry similar to (53). More precisely the vector field defined for arbitrary function

F

E = \sum_{s = 1}^{6} E_{s} \frac{\partial}{\partial z_{s}}

with components specified as follows

E_{1} = \frac{1}{2} z_{1}^{2} - 2 e^{z_{4} - z_{5}} - \frac{t}{2} (z_{1} + z_{2}) e^{z_{4} - z_{5}} E_{2} = \frac{1}{2} z_{2}^{2} + 3 e^{z_{4} - z_{5}} - e^{z_{5} - z_{6}} + \frac{t}{2} (z_{1} + z_{2}) e^{z_{4} - z_{5}} E_{3} = \frac{1}{2} z_{3}^{2} + 2 e^{z_{5} - z_{6}} + \frac{t}{2} (z_{2} + z_{3}) e^{z_{5} - z_{6}}

E_{4} = 3 z_{1} + \frac{1}{2} z_{2} + \frac{1}{2} z_{3} + \frac{t}{2} (z_{1}^{2} + e^{z_{4} - z_{5}}) E_{5} = 2 z_{2} - \frac{1}{2} z_{1} + \frac{1}{2} z_{3} + \frac{t}{2} (z_{2}^{2} + e^{z_{4} - z_{5}} + e^{z_{5} - z_{6}}) E_{6} = z_{3} - \frac{1}{2} z_{1} - \frac{1}{2} z_{2} + \frac{t}{2} (z_{3}^{2} + e^{z_{5} - z_{6}})

satisfies (31) condition and generates non-Noether symmetry of the dynamical system (three particle non periodic Toda chain). Calculation of Schouten bracket

[E, W]

gives rise to expression

Ŵ = [E, W] = z_{1} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{4}} + z_{2} \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{5}} + z_{3} \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{6}} + e^{z_{4} - z_{5}} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} + e^{z_{5} - z_{6}} \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{3}} + \frac{\partial}{\partial z_{4}} \land \frac{\partial}{\partial z_{5}} + \frac{\partial}{\partial z_{5}} \land \frac{\partial}{\partial z_{6}}

Volume multivector fields

Ŵ^{k} \land W^{n - k}

can be calculated in the manner similar to

R^{4}

case and give rise to the well known conservation laws of three particle Toda chain.

Y^{(1)} = \frac{1}{6} (z_{1} + z_{2} + z_{3}) = \frac{Ŵ \land W \land W}{W \land W \land W} Y^{(2)} = \frac{1}{3} (z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} - e^{z_{4} - z_{5}} - e^{z_{5} - z_{6}}) = \frac{Ŵ \land Ŵ \land W}{W \land W \land W} Y^{(3)} = z_{1} z_{2} z_{3} - z_{3} e^{z_{4} - z_{5}} - z_{1} e^{z_{5} - z_{6}} = \frac{Ŵ \land Ŵ \land Ŵ}{W \land W \land W}

Non-Liouville symmetries

Besides Hamiltonian dynamical systems that admit invariant symplectic form

ω

, there are dynamical systems that either are not Hamiltonian or admit Hamiltonian realization but explicit form of symplectic structure

ω

is unknown or too complex. However usually such a dynamical systems possess invariant volume form

Ω

which like symplectic form can be effectively used in construction of conservation laws. Note that volume form for given manifold is arbitrary differential form of maximal degree (equal to the dimension of manifold). In case of regular Hamiltonian systems, n-th outer power of the symplectic form

ω

naturally gives rise to the invariant volume form known as Liouville form

Ω = ω^{n}

and sometimes it is easier to work with

Ω

rather then with symplectic form itself. In generic Liouville dynamical system time evolution is governed by equations of motion

\frac{d}{d t} f = X (f)

where

X

is some smooth vector field that preserves Liouville volume form

Ω

\frac{d}{d t} Ω = L_{X} Ω = 0

Symmetry of equations of motion still can be defined by condition

\frac{d}{d t} g_{z} (f) = g_{z} (\frac{d}{d t} f)

that in terms of vector fields implies that generator of symmetry

E

should commute with time evolution operator

X

[E, X] = 0

Throughout this chapter symmetry will be called non-Liouville if it is not conformal symmetry of

Ω

, or in other words if

L_{E} Ω \neq c Ω

for any constant

c

. Such a symmetries may be considered as analog of non-Noether symmetries defined in Hamiltonian systems and similarly to the Hamiltonian case one can try to construct conservation laws by means of generator of symmetry

E

and invariant differential form

Ω

. Namely we have the following theorem, which is reformulation of Hojman's theorem in terms of Liouville volume form.

Let

(M, X, Ω)

be Liouville dynamical system on the smooth manifold

M

. Then, if the vector field

E

generates non-Liouville symmetry, the function

J = \frac{L_{E} Ω}{Ω}

is conservation law.

By the definition

L_{E} Ω = J Ω .

and

J

is not just constant (again definition is correct since the space of volume forms is one dimensional). By taking Lie derivative of this expression along the vector field

X

that defines time evolution we get

L_{X} L_{E} Ω = L_{[X, E]} Ω + L_{E} L_{X} Ω = L_{X} (J Ω) = (L_{X} J) Ω + J L_{X} Ω

but since Liouville volume form is invariant

L_{X} Ω = 0

and vector field

E

is generator of symmetry satisfying

[E, X] = 0

commutation relation we obtain

(L_{X} J) Ω = 0

\frac{d}{d t} J = L_{X} J = 0

In fact theorem is valid for larger class of symmetries. Namely one can consider symmetries with time dependent generators. Note however that in this case condition

[E, X] = 0

should be replaced by

\frac{\partial}{\partial t} E = [E, X]

Note also that by calculating Lie derivative of conservation law

J

along generator of the symmetry

E

one can recover additional conservation laws

J^{(m)} = (L_{E})^{m} Ω

Let us consider symmetry of three particle non periodic Toda chain. This dynamical system with equations of motion

ż_{4} = z_{1} ż_{5} = z_{2} ż_{6} = z_{3}

ż_{1} = - e^{z_{4} - z_{5}} ż_{2} = e^{z_{4} - z_{5}} - e^{z_{5} - z_{6}} ż_{3} = e^{z_{5} - z_{6}}

possesses invariant volume form

Ω = d z_{1} \land d z_{2} \land d z_{3} \land d z_{4} \land d z_{5} \land d z_{6}

The symmetry (61) is clearly non-Liouville one as far as

L_{E} Ω = (z_{1} + z_{2} + z_{3}) Ω

and main conservation law associated with this symmetry via Theorem 2 is total momentum

J = \frac{L_{E} Ω}{Ω} = z_{1} + z_{2} + z_{3}

Other conservation laws can be recovered by taking Lie derivative of

J

along generator of symmetry

E

, in particular

J^{(1)} = L_{E} J = \frac{1}{2} z_{1}^{2} + \frac{1}{2} z_{2}^{2} + \frac{1}{2} z_{3}^{2} + e^{z_{4} - z_{5}} + e^{z_{5} - z_{6}} J^{(2)} = L_{E} J^{(1)} = \frac{1}{2} (z_{1}^{3} + z_{2}^{3} + z_{3}^{3}) + \frac{3}{2} (z_{1} + z_{2}) e^{z_{4} - z_{5}} + \frac{3}{2} (z_{2} + z_{3}) e^{z_{5} - z_{6}}

Lax Pairs

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 that plays quite important role in construction of completely integrable models. 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 given by adjoint action

\frac{d}{d t} L = [L, P] = - a d_{P} L

where

[,]

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

I^{(k)} = \frac{1}{2} T r (L^{k})

since trace is invariant under coadjoint action

\frac{d}{d t} I^{(k)} = \frac{1}{2} \frac{d}{d t} T r (L^{k}) = \frac{1}{2} T r (\frac{d}{d t} L^{k}) = \frac{k}{2} T r (L^{k - 1} \frac{d}{d t} L) = \frac{k}{2} T r (L^{k - 1} [L, P]) = \frac{1}{2} T r ([L^{k}, P]) = 0

It is remarkable that each generator of the non-Noether symmetry canonically leads to the Lax pair of a certain type. Such a Lax pairs have definite geometric origin, their Lax matrices are formed by coefficients of invariant tangent valued 1-form on the phase space. In the local coordinates

z_{s}

, where the bivector field

W

, symplectic form

ω

and the generator of the symmetry

E

have the following form

W = \sum_{r s} W_{r s} \frac{\partial}{\partial z_{r}} \land \frac{\partial}{\partial z_{r}} ω = \sum_{r s} ω_{r s} d z_{r} \land d z_{s} E = \sum_{s} E_{s} \frac{\partial}{\partial z_{s}}

corresponding Lax pair can be calculated explicitly. Namely we have the following theorem (see also [55]-[56]):

Let

(M, h)

be regular Hamiltonian system on the

2 n

-dimensional Poisson manifold

M

. Then, if the vector field

E

M

generates the non-Noether symmetry, the following

2 n \times 2 n

matrix valued functions on

M

L_{a b} = \sum_{d c} ω_{a d} (E_{c} \frac{\partial W_{d b}}{\partial z_{c}} - W_{b c} \frac{\partial E_{d}}{\partial z_{c}} + W_{d c} \frac{\partial E_{b}}{\partial z_{c}}) P_{a b} = \sum_{c} (\frac{\partial W_{b c}}{\partial z_{a}} \cdot \frac{\partial h}{\partial z_{c}} + W_{b c} \frac{\partial^{2} h}{\partial z_{a} \partial z_{c}})

form the Lax pair (84) of the dynamical system

(M, h)

Let us consider the following operator on a space of 1-forms

Ŕ_{E} (u) = Φ_{ω} ([E, Φ_{W} (u)]) - L_{E} u

(here

Φ_{W}

and

Φ_{ω}

are maps induced by Poisson bivector field and symplectic form). It is remarkable that

Ŕ_{E}

appears to be invariant linear operator. First of all let us show that

Ŕ_{E}

is really linear, or in other words, that for arbitrary 1-forms

u

and

v

and function

f

operator

Ŕ_{E}

has the following properties

Ŕ_{E} (u + v) = Ŕ_{E} (u) + Ŕ_{E} (v)

and

Ŕ_{E} (f u) = f Ŕ_{E} (u)

First property is obvious consequence of linearity of Schouten bracket, Lie derivative and maps

Φ_{W}

Φ_{ω}

. Second property can be checked directly

Ŕ_{E} (f u) = Φ_{ω} ([E, Φ_{W} (f u)]) - L_{E} (f u) = Φ_{ω} ([E, f Φ_{W} (u)]) - (L_{E} f) u - f L_{E} u = Φ_{ω} ((L_{E} f) Φ_{W} (u)) + Φ_{ω} (f [E, Φ_{W} (u)]) - (L_{E} f) u - f L_{E} u = L_{E} f Φ_{ω} Φ_{W} (u) + f Φ_{ω} ([E, Φ_{W} (u)]) - (L_{E} f) u - f L_{E} u = f (Φ_{ω} ([E, Φ_{W} (u)]) - L_{E} u) = f Ŕ_{E} (u)

as far as

Φ_{ω} Φ_{W} (u) = u

. Now let us check that

Ŕ_{E}

is invariant operator

\frac{d}{d t} Ŕ_{E} = L_{X_{h}} Ŕ_{E} = L_{X_{h}} (Φ_{ω} L_{E} Φ_{W} - L_{E}) = Φ_{ω} L_{[X_{h}, E]} Φ_{W} - L_{[X_{h}, E]} = 0

because, being Hamiltonian vector field,

X_{h}

commutes with maps

Φ_{W}

Φ_{ω}

(this is consequence of Liouville theorem) and commutes with

E

as far as

E

generates the symmetry

[X_{h}, E] = 0

. In the terms of the local coordinates

Ŕ_{E}

has the following form

Ŕ_{E} = \sum_{a b} L_{a b} d z_{a} \otimes \frac{\partial}{\partial z_{b}}

and the invariance condition

\frac{d}{d t} Ŕ_{E} = L_{W (h)} Ŕ_{E} = 0

yields

\frac{d}{d t} Ŕ_{E} = \frac{d}{d t} \sum_{a b} L_{a b} d z_{a} \otimes \frac{\partial}{\partial z_{b}} = \sum_{a b} (\frac{d}{d t} L_{a b}) d z_{a} \otimes \frac{\partial}{\partial z_{b}} + \sum_{a b} L_{a b} (L_{W (h)} d z_{a}) \otimes \frac{\partial}{\partial z_{b}} + \sum_{a b} L_{a b} d z_{a} \otimes (L_{W (h)} \frac{\partial}{\partial z_{b}}) = \sum_{a b} (\frac{d}{d t} L_{a b}) d z_{a} \otimes \frac{\partial}{\partial z_{b}} + \sum_{a b c d} L_{a b} \frac{\partial W_{a d}}{\partial z_{c}} \cdot \frac{\partial h}{\partial z_{d}} d z_{c} \otimes \frac{\partial}{\partial z_{b}} + \sum_{a b c d} L_{a b} W_{a d} \frac{\partial^{2} h}{\partial z_{c} \partial z_{d}} d z_{c} \otimes \frac{\partial}{\partial z_{b}} + \sum_{a b c d} L_{a b} \frac{\partial W_{c d}}{\partial z_{b}} \cdot \frac{\partial h}{\partial z_{d}} d z_{a} \otimes \frac{\partial}{\partial z_{c}} + \sum_{a b c d} L_{a b} W_{c d} \frac{\partial^{2} h}{\partial z_{b} \partial z_{d}} d z_{a} \otimes \frac{\partial}{\partial z_{c}} = \sum_{a b} (\frac{d}{d t} L_{a b} + \sum_{c} (P_{a c} L_{c b} - L_{a c} P_{c b})) d z_{a} \otimes \frac{\partial}{\partial z_{b}} = 0

or in matrix notations

\frac{d}{d t} 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.

The conservation laws (85) associated with the Lax pair (88) can be expressed in terms of the integrals of motion

c_{i}

in quite simple way:

I^{(k)} = \frac{1}{2} T r (L^{k}) = \sum_{s} c_{s}^{k}

This correspondence follows from the equation (40) and the definition of the operator

Ŕ_{E}

(89). One can also write down recursion relation that determines conservation laws

I^{(k)}

in terms of conservation laws

C^{(k)}

I^{(m)} + (- 1)^{m} m C^{(m)} + \sum_{k = 1}^{m - 1} (- 1)^{k} I^{(m - k)} C^{(k)} = 0

Let us calculate Lax matrix of two particle Toda chain associated with non-Noether symmetry (53). Using (88) it is easy to check that Lax matrix has eight nonzero elements

L = (\begin{matrix} z_{1} & 0 & 0 & - e^{z_{3} - z_{4}} \\ 0 & z_{2} & e^{z_{3} - z_{4}} & 0 \\ 0 & 1 & z_{1} & 0 \\ - 1 & 0 & 0 & z_{2} \end{matrix})

while matrix

P

involved in Lax pair

\frac{d}{d t} L = [L, P]

has the following form

P = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - e^{z_{3} - z_{4}} & e^{z_{3} - z_{4}} & 0 & 0 \\ e^{z_{3} - z_{4}} & - e^{z_{3} - z_{4}} & 0 & 0 \end{matrix})

The conservation laws associated with this Lax pair are total momentum and energy of two particle Toda chain

I^{(1)} = \frac{1}{2} T r (L) = z_{1} + z_{2} I^{(2)} = \frac{1}{2} T r (L^{2}) = z_{1}^{2} + z_{2}^{2} + 2 e^{z_{3} - z_{4}}

Similarly one can construct Lax matrix of three particle Toda chain, it has 16 nonzero elements

L = (\begin{matrix} z_{1} & 0 & 0 & 0 & - e^{z_{4} - z_{5}} & 0 \\ 0 & z_{2} & 0 & e^{z_{4} - z_{5}} & 0 & - e^{z_{5} - z_{6}} \\ 0 & 0 & z_{3} & 0 & e^{z_{5} - z_{6}} & 0 \\ 0 & - 1 & - 1 & z_{1} & 0 & 0 \\ 1 & 0 & - 1 & 0 & z_{2} & 0 \\ 1 & 1 & 0 & 0 & 0 & z_{3} \end{matrix})

with non-zero elements matrix

P

listed below

P = (\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ - e^{z_{4} - z_{5}} & e^{z_{4} - z_{5}} & 0 & 0 & 0 & 0 \\ e^{z_{4} - z_{5}} & - e^{z_{4} - z_{5}} - e^{z_{5} - z_{6}} & e^{z_{5} - z_{6}} & 0 & 0 & 0 \\ 0 & e^{z_{5} - z_{6}} & - e^{z_{5} - z_{6}} & 0 & 0 & 0 \end{matrix})

Corresponding conservation laws reproduce total momentum, energy and second Hamiltonian involved in bi-Hamiltonian realization of Toda chain

I^{(1)} = \frac{1}{2} T r (L) = z_{1} + z_{2} I^{(2)} = \frac{1}{2} T r (L^{2}) = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + 2 e^{z_{4} - z_{5}} + 2 e^{z_{5} - z_{6}} I^{(3)} = \frac{1}{2} T r (L^{3}) = z_{1}^{3} + z_{2}^{3} + z_{3}^{3} + 3 (z_{1} + z_{2}) e^{z_{4} - z_{5}} + 3 (z_{2} + z_{3}) e^{z_{5} - z_{6}}

Involutivity of conservation laws

Now let us focus on the integrability issues. We know that

n

integrals of motion are associated with each generator of non-Noether symmetry, in the same time we know that, according to the Liouville-Arnold theorem, regular Hamiltonian system

(M, h)

2 n

dimensional symplectic manifold

M

is completely integrable (can be solved completely) if it admits

n

functionally independent integrals of motion in involution. One can understand functional independence of set of conservation laws

c_{1}, c_{2} . . . c_{n}

as linear independence of either differentials of conservation laws

d c_{1}, d c_{2} . . . d c_{n}

or corresponding Hamiltonian vector fields

X_{c_{1}}, X_{c_{2}} . . . X_{c_{n}}

. Strictly speaking we can say that conservation laws

c_{1}, c_{2} . . . c_{n}

are functionally independent if Lesbegue measure of the set of points of phase space

M

where differentials

d c_{1}, d c_{2} . . . d c_{n}

become linearly dependent is zero. Involutivity of conservation laws means that all possible Poisson brackets of these conservation laws vanish pair wise

{c_{i}, c_{j}} = 0 i, j = 1 . . . n

In terms of the vector fields, existence of involutive family of

n

functionally independent conservation laws

c_{1}, c_{2} . . . c_{n}

implies that corresponding Hamiltonian vector fields

X_{c_{1}}, X_{c_{2}} . . . X_{c_{n}}

span Lagrangian subspace (isotropic subspace of dimension

n

) of tangent space (at each point of

M

). Indeed, due to property (23)

{c_{i}, c_{j}} = ω (X_{c_{i}}, X_{c_{j}}) = 0

thus space spanned by

X_{c_{1}}, X_{c_{2}} . . . X_{c_{n}}

is isotropic. Dimension of this space is

n

so it is Lagrangian. Note also that distribution

X_{c_{1}}, X_{c_{2}} . . . X_{c_{n}}

is integrable since due to (22)

[X_{c_{i}}, X_{c_{j}}] = X_{{c_{i}, c_{j}}} = 0

and according to Frobenius theorem there exists submanifold of

M

such that distribution

X_{c_{1}}, X_{c_{2}} . . . X_{c_{n}}

spans tangent space of this submanifold. Thus for phase space geometry existence of complete involutive set of integrals of motion implies existence of invariant Lagrangian submanifold.

Now let us look at conservation laws

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

associated with generator of non-Noether symmetry. Generally speaking these conservation laws might appear to be neither functionally independent nor involutive. However it is reasonable to ask the question – what condition should be satisfied by the generator of the non-Noether symmetry to ensure the involutivity (

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

) of conserved quantities? In Lax theory situation is very similar — each Lax matrix leads to the set of conservation laws but in general this set is not involutive, however in Lax theory there is certain condition known as Classical Yang-Baxter Equation (CYBE) that being satisfied by Lax matrix ensures that conservation laws are in involution. 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.

If the vector field

E

2 n

-dimensional Poisson manifold

M

satisfies the condition

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

and

W

bivector field has maximal rank (

W^{n} \neq 0

) then the functions (32) are in involution

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

First of all let us note that the identity (15) satisfied by the Poisson bivector field

W

is responsible for the Liouville theorem

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

that follows from the graded Jacoby identity satisfied by Schouten bracket. By taking the Lie derivative of the expression (15) 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 \Leftrightarrow [Ŵ (f), W] = - [Ŵ, W (f)]

and finally condition (110) ensures third identity

[Ŵ, Ŵ] = 0

yielding Liouville theorem for

Ŵ

[Ŵ, Ŵ] = 0 \Leftrightarrow [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

c_{i} \neq c_{j}

of the equation (40). By taking the Lie derivative of the equation

(Ŵ - c_{i} W)^{n} = 0

along the vector fields

W (c_{j})

and

Ŵ (c_{j})

and using Liouville theorem for

W

and

Ŵ

bivectors we obtain the following relations

(Ŵ - c_{i} W)^{n - 1} (L_{W (c_{j})} Ŵ - {c_{j}, c_{i}} W) = 0,

and

(Ŵ - c_{i} W)^{n - 1} (c_{i} L_{Ŵ (c_{j})} W + {c_{j}, c_{i}}_{*} W) = 0,

where

{c_{i}, c_{j}}_{*} = Ŵ (d c_{i} \land d c_{j})

is the Poisson bracket calculated by means of the bivector field

Ŵ

. Now multiplying (119) by

c_{i}

subtracting (120) and using identity (114) gives rise to

({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

(Ŵ - c_{i} W)^{n - 1} W

vanishes. In the second case we can repeat (119)-(122) procedure for the volume field

(Ŵ - c_{i} W)^{n - 1} W

yielding after

n

iterations

W^{n} = 0

that according to our assumption (that the dynamical system is regular) is not true. As a result we arrived at (123) and by the simple interchange of indices

i \leftrightarrow j

we get

{c_{i}, c_{j}}_{*} - c_{j} {c_{i}, c_{j}} = 0

Finally by comparing (123) and (124) we obtain that the functions

c_{i}

are in involution with respect to the both Poisson structures (since

c_{i} \neq c_{j}

)

{c_{i}, c_{j}}_{*} = {c_{i}, c_{j}} = 0

and according to (41) the same is true for the integrals of motion

Y^{(k)}

Theorem 4 is useful in multidimensional dynamical systems where involutivity of conservation laws can not be checked directly.

Bi-Hamiltonian systems

Further we will focus on non-Noether symmetries that satisfy condition (110). Besides yielding involutive families of conservation laws, such a symmetries appear to be related to many known geometric structures such as bi-Hamiltonian systems [53] and Frölicher-Nijenhuis operators (torsionless tangent valued differential 1-forms). The relationship between non-Noether symmetries and bi-Hamiltonian structures was already implicitly outlined in the proof of Theorem 4. Now let us pay more attention to this issue.

Originally bi-Hamiltonian structures were introduced by F. Magri in analisys of integrable infinite dimensional Hamiltonian systems such as Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies, Nonlinear Schrödinger equation and Harry Dym equation. Since that time bi-Hamiltonian formalism is effectively used in construction of involutive families of conservation laws in integrable models

Generic bi-Hamiltonian structure on

2 n

dimensional manifold consists out of two Poisson bivector fields

W

and

Ŵ

satisfying certain compatibility condition

[Ŵ, W] = 0

. If, in addition, one of these bivector fields is nondegenerate (

W^{n} \neq 0

) then bi-Hamiltonian system is called regular. Further we will discuss only regular bi-Hamiltonian systems. Note that each Poisson bivector field by definition satisfies condition (15). So we actually impose four restrictions on bivector fields

W

and

Ŵ

[W, W] = [Ŵ, W] = [Ŵ, Ŵ] = 0

and

W^{n} \neq 0

During the proof of Theorem 4 we already showed that bivector fields

W

and

Ŵ = [E, W]

satisfy conditions (126) (see (112)-(116)), thus we can formulate the following statement

Let

(M, h)

be regular Hamiltonian system on the

2 n

-dimensional manifold

M

endowed with regular Poisson bivector field

W

. Then, if the vector field

E

M

generates the non-Noether symmetry, and satisfies condition

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

the following bivector fields on

M

W, Ŵ = [E, W]

form invariant bi-Hamiltonian system (

[W, W] = [Ŵ, W] = [Ŵ, Ŵ] = 0

See proof of Theorem 4.

Bi-Hamiltonian systems obtained by taking Lie derivative of Poisson bivector along some vector field were studied in [70]

One can check that the non-Noether symmetry (53) satisfies condition (110) while bivector fields

W = \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{3}} + \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{4}}

and

Ŵ = [E, W] = z_{1} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{3}} + z_{2} \frac{\partial}{\partial z_{2}} \land \frac{\partial}{\partial z_{4}} + e^{z_{3} - z_{4}} \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{2}} + \frac{\partial}{\partial z_{3}} \land \frac{\partial}{\partial z_{4}}

form bi-Hamiltonian system

[W, W] = [W, Ŵ] = [Ŵ, Ŵ] = 0

. Similarly, one can recover bi-Hamiltonian system of three particle Toda chain associated with symmetry (61). It is formed by bivector fields

W = \frac{\partial}{\partial z_{1}} \land \frac{\partial}{\partial z_{4}} + \frac{\partial}{\partial z_{2}} \land �