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

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 $W$subjected to the following condition[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 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 $Wn\ne \; 0$.In this case $W$ gives rise to a well known isomorphism$\Phi $ 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$\Phi (u\; \wedge \; v)\; =\; \Phi (u)\; \wedge \; \Phi (v)$.

Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equationd dt Ô = {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^{zLE} (here $L$ denotes Lie derivative)that acts on the observables as followsg_{z}(Ô) = e^{zLE}(Ô) = f + zL_{E}Ô + ½(zL_{E})^{2}Ô + ⋯ Such a group of transformation is called symmetry of Hamilton's equation (5)if it commutes with time evolution operatord dt g_{z}(Ô) = g_{z}(d dt Ô) 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∂ ∂t E = [E , W(h)]. If in addition to (8) the vector field $E$ does not preserve Poissonbivector field $[E\; ,\; W]\; \ne \; 0$ 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.

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 equationd dt L = [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 = ∑ km W_{ab}D_{k} ∧ D_{m} E = ∑ m E_{m}D_{m} corresponding Lax pair could be calculated explicitly.Namely we have the following theorem:

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.

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)} ($\Omega (k)$ denotes the space of $k$-degree differential forms)equipped with a couple of differential operatorsd, đ : Ω^{(k)} → Ω^{(k + 1)} satisfying$d2=\; \u01112=\; d\u0111\; +\; \u0111d\; =\; 0$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:

Finally we would like to reveal some features of the operator$\u0154$_{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]) = 0 is 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.

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