Non-Noether symmetries and their influence on phase space geometry
George Chavchanidze
Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia
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.
keywords. Non-Noether symmetry; Conservation law; bi-Hamiltonian system; Bidifferential calculus; Lax pair; Frölicher-Nijenhuisoperator;
msc. 70H33; 70H06; 53Z05
J. Geom. Phys. 48 (2003) 190-202
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
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
provides the skew symmetry ofthe corresponding Poisson bracket and the condition
(1) ensures that for every triple
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
has maximalrank, i. e. its
-th outer power, where
is a half-dimension ofthe phase space, does not vanish
.In this case
gives rise to a well known isomorphism
between the differential 1-forms andthe vector fields defined by
Φ(u) = W(u)for every 1-form
and could be extended to higher degreedifferential forms and multivector fields by linearity and multiplicativity
.
Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equation
ddtÔ = {h , Ô}where
is some fixed smooth function on the phase space called Hamiltonian.Let us recall that each vector field
on the phase space generatesthe one-parameter continuous group of transformations
(here
denotes Lie derivative)that acts on the observables as follows
gz(Ô) = ezLE(Ô) = f + zLEÔ + ½(zLE)2Ô + ⋯Such a group of transformation is called symmetry of Hamilton's equation
(5)if it commutes with time evolution operator
ddt gz(Ô) = gz(ddtÔ)in terms of the vector fields this condition means that the generator
of the group
commutes with the vector field
, i. e.
[E , W(h)] = 0. However we would like to consider more generalcase where
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
does not preserve Poissonbivector field
then
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.
theorem. Let be regular Hamiltonian system on the -dimensionalPoisson manifold . Then, if the vector field generatesnon-Noether symmetry, the functionsY(k) = V[k]V[0] k = 1,2, ... nwhere are multivector fields of maximal degree constructed by means of Poisson bivector and its Lie derivative , are integrals of motion.
proof. By the definition
Ŵk ∧ Wn − k = Y(k)Wn.(definition is correct since the space of
degree multivector fields on
degree manifold is one dimensional).Let us take time derivative of this expression along the vector field
,
ddtŴk ∧ Wn − k = (ddtY(k))Wn + Y(k)[W(h) , Wn]or
k(ddtŴ) ∧ Ŵk − 1 ∧ Wn − k+ (n − k)[W(h) , W] ∧ Ŵk ∧ Wn − k − 1 = (ddtY(k))Wn + nY(k)[W(h) , W] ∧ Wn − 1but according to the Liouville theorem the Hamiltonian vector field preserves
i. e.
ddtW = [W(h) , W] = 0hence, by taking into account that
ddtE= ∂∂tE + [W(h) , E] = 0 we get
ddtŴ = ddt[E , W] = [ddtE, W] + [E[W(h) , W]] = 0.and as a result
(13) yields
ddtY(k) Wn = 0but since the dynamical system is regular (
)we obtain that the functions
are integrals of motion.
example. Let
be
with coordinates
and Poisson bivector field
W = D1 ∧ D3 + D2 ∧ D4(
just denotes derivative with respect to
coordinate)and let's take
h = ½z12 + ½z22 + ez3 − z4Then the vector field
E = 4∑m = 1EmDmwith components
E1 = ½z12 − ez3 − z4 −t2(z1 + z2)ez3 − z4E2 = ½z22 + 2ez3 − z4 +t2(z1 + z2)ez3 − z4E3 = 2z1 + ½z2 + t2(z12 + ez3 − z4)E4 = z2 − ½z1 + t2(z22 + ez3 − z4)satisfies
(9) condition and as a result generates symmetry of the dynamical system.The symmetry appears to be non-Noether with Schouten bracket
equal to
Ŵ = [E , W] = z1D1 ∧ D3 +z2D2 ∧ D4 +ez3 − z4D1 ∧ D2 +D3 ∧ D4calculating volume vector fields
gives rise to
W ∧ W = − 2D1 ∧ D2 ∧ D3 ∧ D4Ŵ ∧ W = − (z1 + z2)D1 ∧ D2 ∧ D3 ∧ D4Ŵ ∧ Ŵ = − 2(z1z2 − ez3 − z4) D1 ∧ D2 ∧ D3 ∧ D4and the conservation laws associated with this symmetry are just
Y(1) = Ŵ ∧ WW ∧ W = ½(z1 + z2)Y(2) = Ŵ ∧ ŴW ∧ W = z1z2 − ez3 − 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 isa pair of smooth functions on with values in someLie algebra such that the time evolution of is governedby the following equationddtL = [L , P]where is a Lie bracket on . It is well known that each Laxpair leads to a number of conservation laws. When is some matrix Lie algebrathe conservation laws are just traces of powers of I(k) = Tr(Lk)It is remarkable that each generator of the non-Noethersymmetry canonically leads to the Lax pair of a certain type.In the local coordinates , where the bivector field and the generator of the symmetry have thefollowing formW = ∑kmWabDk ∧ Dm E = ∑mEmDmcorresponding Lax pair could be calculated explicitly.Namely we have the following theorem:
theorem. Let
be regular Hamiltonian system on the
-dimensionalPoisson manifold
.Then, if the vector field
on
generates the non-Noether symmetry,the following
matrix valued functions on
Lab = ∑dc (W−1)ad (EcDcWdb− WcbDcEd + WdcDcEb)Pab = ∑c Da (WbcDch)form the Lax pair
(27) of the dynamical system
.
proof. Let us consider the following operator on a space of 1-forms
ŔE(u) = Φ− 1([E , Φ(u)]) − LEu(here
is the isomorphism
(4)).It is obvious that
is a linear operator and it is invariantsince time evolution commutes with both
(as far as
) and
(because
generatessymmetry). In the terms of the local coordinates
has the following form
ŔE = ∑abLab dza ⊗ Dband the invariance condition
ddtŔE = LW(h)ŔE = 0yields
ddtŔE =ddt∑abLab dza ⊗ Db= ∑ab(ddtLab) dza ⊗ Db +∑abLab (LW(h)dza) ⊗ Db+ ∑abLab dza ⊗ (LW(h)Db) =∑ab(ddtLab) dza ⊗ Db+ ∑abcdLabDc(WadDdh)dzc ⊗ Db +∑abcdLabDb(WcdDdh)dza ⊗ Dc= ∑ab(ddtLab + ∑c(PacLcb − LacPcb))dza ⊗ Db = 0or in matrix notations
ddtL = [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.
example. Let us calculate Lax matrix associated with non-Noether symmetry
(23).Using
(30) it is easy to check that Lax matrix has eight nonzero elements
L11 = L33 = z1; L22 = L44 = z2L14 = − L23 = ez3 − z4; L32 = − L41 = 1The conservation laws associated with this Lax matrix are
I(1) = Tr(L) = 2(z1 + z2)I(2) = Tr(L2) = 2z12 + 2z22 + 4ez3 − z4 Now let us focus on the integrability issues. We know that 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 functionally independent integrals ofmotion in involution (two functions and are said to bein involution if their Poisson bracket vanishes ).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() 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.
theorem. If the vector field
on
-dimensionalPoisson manifold
satisfies the condition
[[E[E , W]]W] = 0and
bivector field has maximal rank (
)then the functions
(10) are in involution
{Y(k) , Y(m)} = 0 proof. First of all let us note thatthe identity
(1) satisfied by the Poissonbivector field
is responsible for the Liouville theorem
[W , W] = 0 LW(f)W = [W(f) , W] = 0By taking the Lie derivative of the expression
(1)we obtain another useful identity
LE[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] = 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
of the equation
(18). By taking the Lie derivative of the equation
(Ŵ − ciW)n = 0along the vector fields
and
and using Liouville theorem for
and
bivectors we obtain the following relations
(Ŵ − ciW)n − 1(LW(cj)Ŵ − {cj , ci}W) = 0,and
(Ŵ − ciW)n − 1(ciLŴ(cj)W + {cj , ci}•W) = 0,where
{ci , cj}• = Ŵ(dci ∧ dcj)is the Poisson bracket calculated by means of the bivector field
.Now multiplying
(48) by
subtracting
(49) and usingidentity
(43) gives rise to
({ci , cj}• − ci{ci , cj})(Ŵ − ciW)n − 1W = 0Thus, either
{ci , cj}• − ci{ci , cj} = 0or the volume field
vanishes. In the second case we can repeat
(48)-
(51) procedure forthe volume field
yielding after
iterations
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
we get
{ci , cj}• − cj{ci , cj} = 0Finally by comparing
(52) and
(53) we obtain thatthe functions
are in involution with respect to the bothPoisson structures (since
)
{ci , cj}• = {ci , cj} = 0and according to
(19) the same is true for the integrals of motion
.
corollary. Each generator of non-Noether symmetry satisfying equation
(39) endowsdynamical system with the bi-Hamiltonian structure – couple (
)of compatible (
)Poisson (
)bivector fields.
example. One can check that the non-Noether symmetry
(23) satisfiescondition
(39) and the bivector fields
and
defined by
(20) and
(24) form bi-Hamiltonian system
.
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 forms
Ω = ∞∪k = 0 Ω(k)(
denotes the space of
-degree differential forms)equipped with a couple of differential operators
d, đ : Ω(k) → Ω(k + 1)satisfying
conditions (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:
theorem. Let
be regular Hamiltonian system on the Poisson manifold
.Then, if the vector field
on
generates the non-Noether symmetryand satisfies the equation
(39), the differential operators
du = Φ− 1([W , Φ(u)])đu = Φ− 1([[E , W]Φ(u)])form invariant bidifferential calculus(
)over the graded algebra of differential forms on
.
proof. First of all we have to show that
and
are really differential operators , i.e., they are linear maps from
into
, satisfy derivation property andare nilpotent (
).Linearity is obvious and follows from the linearity of the Schouten bracket
and
maps. Then, if
is a
-degree form
maps it on
-degree multivector field andthe Schouten brackets
and
result the
-degree multivector fields that are mapped on
-degreedifferential forms by
.So,
and
are linear maps from
into
.Derivation property follows from the same feature of the Schouten bracket
and linearity of
and
maps.Now we have to prove the nilpotency of
and
.Let us consider
d2u = 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 manner
đ2u = Φ− 1([[W , E][[W , E]Φ(u)]]) = 0according to the property
(45) of
and the Jacobi identity.Thus, we have proved that
and
are differential operators(in fact
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
is fulfilled. Using definitions of
and the Jacobi identity we get
(dđ + đd)(u) = Φ− 1([[[W , E]W]Φ(u)]) = 0as far as
(43) is satisfied.So,
and
form the bidifferential calculus over the gradedalgebra of differential forms.It is also clear that the bidifferential calculus
is invariant, since both
and
commute with time evolutionoperator
.
example. The symmetry
(23) endows
with bicomplex structure
where
is ordinary exterier derivative while
is defined by
đz1 = z1dz1 − ez3 − z4dz4đz2 = z2dz2 + ez3 − z4dz3đz3 = z1dz3 + dz2đz4 = z2dz4 − dz1and is extended to whole De Rham complex by linearity, derivation property andcompatibility property
. The conservation laws
and
defined by
(38)form the simpliest Lenard scheme
2đI(1) = dI(2) Finally we would like to reveal some features of the operator
(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
on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion
vanishes, i. e. whenever for arbitrary vector fields
the condition
T(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.
theorem. Let
be regular Hamiltonian system on the Poisson manifold
.If the vector field
on
generates the non-Noether symmetryand satisfies the equation
(39) then the linear operator, defined forevery vector field
by equation
RE(X) = Φ(LEΦ− 1(X)) − [E , X]is invariant Frölicher-Nijenhuis operator on
.
proof. Invariance of
follows from the invariance of the
defined by
(31)(note that for arbitrary 1-form vector field
and vector field
contraction
has the property
,so
is actually transposed to
).It remains to show that the condition
(39) ensures vanishing of theFrölicher-Nijenhuis torsion
of
, i.e. for arbitrary vector fields
we must get
T(RE)(X , Y) = [RE(X) , RE(Y)] − RE([RE(X) , Y] + [X , RE(Y)] − RE([X , Y])) = 0First let us introduce the following auxiliary 2-forms
ω = Φ− 1(W), ω• = ŔEω ω•• = ŔEω•Using the realization
(57) of the differential
and the property
(1) yields
dω = Φ− 1([W , W]) = 0Similarly, using the property
(43) we obtain
dω• = dΦ− 1([E , W]) − dLEω = Φ− 1([[E , W]W]) − LEdω = 0And finally, taking into account that
and using the condition
(39), we get
dω•• = 2Φ− 1([[E[E , W]]W]) − 2dLEω• = − 2LEdω• = 0So the differential forms
are closed
dω = dω• = dω•• = 0Now let us consider the contraction of
and
.
iT(RE)(X , Y)ω = i[REX , REY]ω −i[REX , Y]ω• −i[X , REY]ω• +i[X , Y]ω••= LREXiYω• −iREYLXω• −LREXiYω• +iYLREXω• − LXiREYω•+ iREYLXω• +i[X , Y]ω•• = iYLXω•• −LXiYω•• +i[X , Y]ω•• = 0where we used
(68) (72),the property of the Lie derivative
LXiYω =iYLXω + i[X , Y]ωand the relations of the following type
LREXω = diREXω + iREXdω = diXω• = LXω• − iXdω• = LXω•So we proved that for arbitrary vector fields
the contraction of
and
vanishes.But since
bivector is non-degenerate(
), its counter image
ω = Φ− 1(W)is also non-degenerate and vanishing of the contraction
(73)implies that the torsion
itself is zero.So we get
T(RE)(X , Y) = [RE(X) , RE(Y)]− RE([RE(X) , Y] + [X , RE(Y)] − RE([X , Y])) = 0 example. Note that operator
associated with non-Noethersymmetry
(23) reproduces well known Frölicher-Nijenhuis operator
RE =z1dz1 ⊗ D1 −dz1 ⊗ D4 +z2dz2 ⊗ D2 +dz2 ⊗ D3 + z1dz3 ⊗ D3+ ez3 − z4dz3 ⊗ D2 +z2dz4 ⊗ D4 − ez3 − z4dz4 ⊗ D1(compare with
[3])
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.
acknowledgements. Author is grateful to Zakaria Giunashvili, George Jorjadze andMichael Maziashvili for constructive discussions and help.This work was supported by INTAS (00-00561).
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