Non-Noether symmetries and their influence on phase space geometry

abstract. We disscuss some geometric aspects of the concept of non-Noether symmetry.It is shown that in regular Hamiltonian systems such a symmetry canonically leadsto a Lax pair on the algebra of linear operators on cotangent bundle over the phase space.Correspondence between the non-Noether symmetries and other wide spread geometricmethods 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 thecontinuous non-Noether symmetry are in involution whenever thegenerator of the symmetry satisfies a certain Yang-Baxter type equation.

In the present paper we would like to shed more light on geometric aspects ofthe 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 modelsthat essentially relies on different geometric objects used for constructing conservationlaws. Among them are Frölicher-Nijenhuisoperators, bi-Hamiltonian systems, Lax pairs and bicomplexes. And it seems that the existance of these importantgeometric 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 theabove 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 ofa regular Hamiltonian system is a Poisson manifold – a smooth finite-dimensionalmanifold equipped with the Poisson bivector field

W

subjected to the following condition[W , W] = 0where square bracket stands for Schouten bracket or supercommutator(for simplicity further it will be referred as commutator). In a standard manner Poissonbivector 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

W

provides the skew symmetry ofthe corresponding Poisson bracket and the condition(1) ensures that for every triple

(f, g, h)

of smoothfunctions on the phase space the Jacobi identity{f{g , h}} + {h{f , g}} + {g{h , f}} = 0.is satisfied. We also assume that the dynamical system under considerationis regular – the bivector field

W

has maximalrank, i. e. its

n

-th outer power, where

n

is a half-dimension ofthe 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 andthe vector fields defined byΦ(u) = W(u)for every 1-form

u

and could be extended to higher degreedifferential forms and multivector fields by linearity and multiplicativity

Φ(u ∧ v) = Φ(u) ∧ Φ(v)

Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equationddtÔ = {h , Ô}where

h

is some fixed smooth function on the phase space called Hamiltonian.Let us recall that each vector field

E

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

g

_z = e^zL_E (here

L

denotes Lie derivative)that acts on the observables as followsg_z(Ô) = e^zL_E(Ô) = f + zL_EÔ + ½(zL_E)²Ô + ⋯Such a group of transformation is called symmetry of Hamilton's equation (5)if it commutes with time evolution operatorddt g_z(Ô) = g_z(ddtÔ)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 generalcase where

E

is time dependent vector field on phase space. In this case(8) should be replaced with∂∂tE = [E , W(h)].If in addition to (8) the vector field

E

does not preserve Poissonbivector field

[E , W] ≠ 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 ofsuch a symmetry could essentially enrich the geometry of the phase spaceand 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 ofintegrals of motion[4]. More precisely therelationship between non-Noether symmetries and the conservation laws is described bythe following theorem.

proof. By the definitionŴ^k ∧ W^{n − k} = Y^(k)Wⁿ.(definition is correct since the space of

2n

degree multivector fields on

2n

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

W(h)

,ddtŴ^k ∧ W^{n − k} = (ddtY^(k))Wⁿ + Y^(k)[W(h) , Wⁿ]ork(ddtŴ) ∧ Ŵ^{k − 1} ∧ W^{n − k}+ (n − k)[W(h) , W] ∧ Ŵ^k ∧ W^{n − k − 1} = (ddtY^(k))Wⁿ + nY^(k)[W(h) , W] ∧ W^{n − 1}but according to the Liouville theorem the Hamiltonian vector field preserves

W

i. e.ddtW = [W(h) , W] = 0hence, by taking into account thatddtE= ∂∂tE + [W(h) , E] = 0 we getddtŴ = ddt[E , W] = [ddtE, W] + [E[W(h) , W]] = 0.and as a result (13) yieldsddtY^(k) Wⁿ = 0but since the dynamical system is regular (

W

ⁿ ≠ 0)we obtain that the functions

Y

^(k) are integrals of motion.

remark. Instead of conserved quantities

Y

⁽¹⁾ ... Y⁽ⁿ⁾, thesolutions

c

₁ ... c_n of the secular equation(Ŵ − cW)ⁿ = 0could be associated with the generator of symmetry.By expanding expression (18) 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 wayY^(k) = (n − k)! k!n! ∑m[i] > m[j] c_m[1]c_m[2] ⋯ c_m[k]

example. Let

M

R

⁴ with coordinates

z

₁, z₂, z₃, z₄ and Poisson bivector fieldW = D₁ ∧ D₃ + D₂ ∧ D₄(

D

_m just denotes derivative with respect to

z

_m coordinate)and let's takeh = ½z₁² + ½z₂² + e^{z₃ − z₄}Then the vector fieldE = 4∑m = 1E_mD_mwith componentsE₁ = ½z₁² − e^{z₃ − z₄} −t2(z₁ + z₂)e^{z₃ − z₄}E₂ = ½z₂² + 2e^{z₃ − z₄} +t2(z₁ + z₂)e^{z₃ − z₄}E₃ = 2z₁ + ½z₂ + t2(z₁² + e^{z₃ − z₄})E₄ = z₂ − ½z₁ + t2(z₂² + 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 , W]

equal toŴ = [E , W] = z₁D₁ ∧ D₃ +z₂D₂ ∧ D₄ +e^{z₃ − z₄}D₁ ∧ D₂ +D₃ ∧ D₄calculating volume vector fields

Ŵ

^k ∧ W^{n − k} gives rise toW ∧ W = − 2D₁ ∧ D₂ ∧ D₃ ∧ D₄Ŵ ∧ W = − (z₁ + z₂)D₁ ∧ D₂ ∧ D₃ ∧ D₄Ŵ ∧ Ŵ = − 2(z₁z₂ − e^{z₃ − z₄}) D₁ ∧ D₂ ∧ D₃ ∧ D₄and the conservation laws associated with this symmetry are justY⁽¹⁾ = Ŵ ∧ WW ∧ W = ½(z₁ + z₂)Y⁽²⁾ = Ŵ ∧ ŴW ∧ W = z₁z₂ − e^{z₃ − z₄}

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but alsoendows the phase space with a number of interesting geometric structures and it appears that such asymmetry 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

isa pair

(L , P)

of smooth functions on

M

with values in someLie algebra

g

such that the time evolution of

L

is governedby the following equationddtL = [L , P]where

[ , ]

is a Lie bracket on

g

. It is well known that each Laxpair leads to a number of conservation laws. When

g

is some matrix Lie algebrathe conservation laws are just traces of powers of

L

I^(k) = Tr(L^k)It is remarkable that each generator of the non-Noethersymmetry canonically leads to the Lax pair of a certain type.In the local coordinates

z

_m, where the bivector field

W

and the generator of the symmetry

E

have thefollowing formW = ∑kmW_abD_k ∧ D_m E = ∑mE_mD_mcorresponding Lax pair could be calculated explicitly.Namely we have the following theorem:

proof. Let us consider the following operator on a space of 1-formsŔ_E(u) = Φ^{− 1}([E , Φ(u)]) − L_Eu(here

Φ

is the isomorphism (4)).It is obvious that

Ŕ

_Eis a linear operator and it is invariantsince time evolution commutes with both

Φ

(as far as

[W(h) , W] = 0

) and

E

(because

E

generatessymmetry). In the terms of the local coordinates

Ŕ

_E has the following formŔ_E = ∑abL_ab dz_a ⊗ D_band the invariance conditionddtŔ_E = L_W(h)Ŕ_E = 0yieldsddtŔ_E =ddt∑abL_ab dz_a ⊗ D_b= ∑ab(ddtL_ab) dz_a ⊗ D_b +∑abL_ab (L_W(h)dz_a) ⊗ D_b+ ∑abL_ab dz_a ⊗ (L_W(h)D_b) =∑ab(ddtL_ab) dz_a ⊗ D_b+ ∑abcdL_abD_c(W_adD_dh)dz_c ⊗ D_b +∑abcdL_abD_b(W_cdD_dh)dz_a ⊗ D_c= ∑ab(ddtL_ab + ∑c(P_acL_cb − L_acP_cb))dz_a ⊗ D_b = 0or in matrix notationsddtL = [L , P].So, we have proved that the non-Noether symmetry canonically yields a Lax pairon the algebra of linear operators on cotangent bundle over the phase space.

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

n

integrals of motion are associated with each generator of non-Noethersymmetry and according to the Liouville-Arnold theorem Hamiltonian system iscompletely integrable if it possesses

n

functionally independent integrals ofmotion in involution (two functions

f

and

g

are said to bein involution if their Poisson bracket vanishes

{f , g} = 0

).Generally speaking the conservation laws associated with symmetry might appear to be neitherindependent nor involutive.However it is reasonable to ask the question – what condition should be satisfiedby 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 asClassical Yang-Baxter Equation (CYBE). Since involutivity of the conservation lawsis closely related to the integrability it is essential to have some analog of CYBE for the generatorof non-Noether symmetry. To address this issue we would like to propose the following theorem.

proof. First of all let us note thatthe identity (1) satisfied by the Poissonbivector field

W

is responsible for the Liouville theorem[W , W] = 0 L_W(f)W = [W(f) , W] = 0By taking the Lie derivative of the expression (1)we obtain another useful identityL_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] = 0yielding Liouville theorem for

Ŵ

[Ŵ , Ŵ] = 0 ⇔ [Ŵ(f) , Ŵ] = 0Indeed[Ŵ , Ŵ] = [[E , W]Ŵ] = [[Ŵ , E]W]= − [[E , Ŵ]W] = − [[E[E , W]]W] = 0Now let us consider two different solutions

c

_i ≠ c_jof the equation (18). By taking the Lie derivative of the equation(Ŵ − c_iW)ⁿ = 0along 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_iW)^{n − 1}(L_{W(c_j)}Ŵ − {c_j , c_i}W) = 0,and(Ŵ − c_iW)^{n − 1}(c_iL_{Ŵ(c_j)}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

Ŵ

.Now multiplying (48) by

c

_i subtracting (49) and usingidentity (43) gives rise to({c_i , c_j}_• − c_i{c_i , c_j})(Ŵ − c_iW)^{n − 1}W = 0Thus, either{c_i , c_j}_• − c_i{c_i , c_j} = 0or the volume field

(Ŵ − c

_iW)^{n − 1}Wvanishes. In the second case we can repeat(48)-(51) procedure forthe volume field

(Ŵ − c

_iW)^{n − 1}Wyielding after

n

iterations

W

ⁿ = 0 that according to ourassumption (that the dynamical system is regular) is not true.As a result we arrived at (52) and by the simpleinterchange of indices

i ↔ j

we get{c_i , c_j}_• − c_j{c_i , c_j} = 0Finally by comparing (52) and (53) we obtain thatthe functions

c

_i are in involution with respect to the bothPoisson structures (since

c

_i ≠ c_j){c_i , c_j}_• = {c_i , c_j} = 0and according to (19) the same is true for the integrals of motion

Y

^(k).

Another concept that is often used in theory of dynamical systems and couldbe related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach).Recently A. Dimakis and F. Müller-Hoissenapplied bidifferential calculi to the wide range of integrable modelsincluding KdV hierarchy, KP equation, self-dual Yang-Mills equation,Sine-Gordon equation, Toda models, non-linear Schrödingerand Liouville equations. It turns out that these models can be effectivelydescribed and analyzed using the bidifferential calculi [1], [2].

Under the bidifferential calculus we mean the graded algebra of differential formsΩ = ∞∪k = 0 Ω^(k)(

Ω

^(k) denotes the space of

k

-degree differential forms)equipped with a couple of differential operatorsd, đ : Ω^(k) → Ω^{(k + 1)}satisfying

d

² = đ² = dđ + đd = 0conditions (see [2]).It is interesting that if generator of the non-Noether symmetry satisfiesequation (39) then we are able to construct an invariant bidifferential calculusof a certain type. This construction is summarized in the following theorem:

proof. First of all we have to show that

d

and

đ

are really differential operators , i.e., they are linear maps from

Ω

^(k) into

Ω

^{(k + 1)}, satisfy derivation property andare nilpotent (

d

² = đ² = 0).Linearity is obvious and follows from the linearity of the Schouten bracket

[ , ]

and

Φ, Φ

^{− 1}maps. Then, if

u

is a

k

-degree form

Φ

maps it on

k

-degree multivector field andthe Schouten brackets

[W , Φ(u)]

and

[[E , W]Φ(u)]

result the

k + 1

-degree multivector fields that are mapped on

k + 1

-degreedifferential forms by

Φ

^{− 1}.So,

d

and

đ

are linear maps from

Ω

^(k) into

Ω

^{(k + 1)}.Derivation property follows from the same feature of the Schouten bracket

[ , ]

and linearity of

Φ

and

Φ

^{− 1} maps.Now we have to prove the nilpotency of

d

and

đ

.Let us consider

d

²ud²u = Φ^{− 1}([W , Φ(Φ^{− 1}([W , Φ(u)]))])= Φ^{− 1}([W[W , Φ(u)]]) = 0as a result of the property (41) and the Jacobi identity for

[ , ]

bracket.In the same mannerđ²u = Φ^{− 1}([[W , E][[W , E]Φ(u)]]) = 0according to the property (45) of

[W , E] = Ŵ

and the Jacobi identity.Thus, we have proved that

d

and

đ

are differential operators(in fact

d

is ordinary exterior differential and the expression(57) is its well known representation in terms of Poisson bivector field).It remains to show that the compatibility condition

dđ + đd = 0

is fulfilled. Using definitions of

d, đ

and the Jacobi identity we get(dđ + đd)(u) = Φ^{− 1}([[[W , E]W]Φ(u)]) = 0as far as (43) is satisfied.So,

d

and

đ

form the bidifferential calculus over the gradedalgebra of differential forms.It is also clear that the bidifferential calculus

d, đ

is invariant, since both

d

and

đ

commute with time evolutionoperator

W(h) = {h, }

example. The symmetry (23) endows

R

⁴ with bicomplex structure

d, đ

where

d

is ordinary exterier derivative while

đ

is defined byđz₁ = z₁dz₁ − e^{z₃ − z₄}dz₄đz₂ = z₂dz₂ + e^{z₃ − z₄}dz₃đz₃ = z₁dz₃ + dz₂đz₄ = z₂dz₄ − dz₁and is extended to whole De Rham complex by linearity, derivation property andcompatibility property

dđ + đd = 0

. The conservation laws

I

⁽¹⁾ and

I

⁽²⁾ defined by (38)form the simpliest Lenard scheme2đI⁽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 inHamiltonian system that possesses certain non-Noether symmetry.From the geometric properties of the tangent valued forms we knowthat 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 conditionT(F)(X , Y) = [FX , FY] − F([FX , Y] + [X , FY] − F[X , Y]) = 0is satisfied.Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used intheory of integrable models. We would like to showthat each generator of non-Noether symmetry satisfying equation (39)canonnically leads to invariant Frölicher-Nijenhuis operator on tangentbundle over the phase space. Strictly speaking we have the following theorem.

proof. Invariance of

R

_E follows from the invariance of the

Ŕ

_E defined by (31)(note that for arbitrary 1-form vector field

u

and vector field

X

contraction

i

_Xu has the property

i

_{R_EX}u = i_XŔ_Eu,so

R

_E is actually transposed to

Ŕ

_E).It remains to show that the condition (39) ensures vanishing of theFrölicher-Nijenhuis torsion

T(R

_E) of

R

_E, i.e. for arbitrary vector fields

X, Y

we must getT(R_E)(X , Y) = [R_E(X) , R_E(Y)] − R_E([R_E(X) , Y] + [X , R_E(Y)] − R_E([X , Y])) = 0First let us introduce the following auxiliary 2-formsω = Φ^{− 1}(W), ω^• = Ŕ_Eω ω^•• = Ŕ_Eω^•Using the realization (57) of the differential

d

and the property (1) yieldsdω = Φ^{− 1}([W , W]) = 0Similarly, using the property (43) we obtaindω^• = dΦ^{− 1}([E , W]) − dL_Eω = Φ^{− 1}([[E , W]W]) − L_Edω = 0And finally, taking into account that

ω

^• = 2Φ^{− 1}([E , W])and using the condition (39), we getdω^•• = 2Φ^{− 1}([[E[E , W]]W]) − 2dL_Eω^• = − 2L_Edω^• = 0So the differential forms

ω, ω

^•, ω^••are closeddω = dω^• = dω^•• = 0Now let us consider the contraction of

T(R

_E) and

ω

.i_{T(R_E)(X , Y)}ω = i_{[R_EX , R_EY]}ω −i_{[R_EX , Y]}ω^• −i_{[X , R_EY]}ω^• +i_{[X , Y]}ω^••= L_{R_EX}i_Yω^• −i_{R_EY}L_Xω^• −L_{R_EX}i_Yω^• +i_YL_{R_EX}ω^• − L_Xi_{R_EY}ω^•+ i_{R_EY}L_Xω^• +i_{[X , Y]}ω^•• = i_YL_Xω^•• −L_Xi_Yω^•• +i_{[X , Y]}ω^•• = 0where we used (68) (72),the property of the Lie derivativeL_Xi_Yω =i_YL_Xω + i_{[X , Y]}ωand the relations of the following typeL_{R_EX}ω = di_{R_EX}ω + i_{R_EX}dω = di_Xω^• = L_Xω^• − i_Xdω^• = L_Xω^•So we proved that for arbitrary vector fields

X, Y

the contraction of

T(R

_E)(X , Y) and

ω

vanishes.But since

W

bivector is non-degenerate(

W

ⁿ ≠ 0), its counter imageω = Φ^{− 1}(W)is also non-degenerate and vanishing of the contraction (73)implies that the torsion

T(R

_E) itself is zero.So we getT(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

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

Non-Noether symmetries and their influence on phase space geometry

References