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 $(M,\; \omega )$ where $M$ is smooth even dimensional manifold and $\omega$ is a closed dω = 0 and nondegenerate 2-form on $M$. Being nondegenerate means that contraction of arbitrary non-zero vector field with $\omega$ does not vanish i_Xω = 0 ⇔ X = 0 (here $i$_X denotes contraction of the vector field $X$ with differential form). Otherwise one can say that $\omega$ is nondegenerate if its n-th outer power does not vanish ($\omega n\ne \; 0$) 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 $\omega$ is basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fields and the form of Hamilton's equations. The symplectic form $\omega$ naturally defines isomorphism between vector fields and differential 1-forms on $M$ (in other words tangent bundle $TM$ of symplectic manifold can be quite naturally identified with cotangent bundle $T*M$). The isomorphic map $\Phi$_ω from $TM$ into $T*M$ is obtained by taking contraction of the vector field with $\omega$ Φ_ω: X → − i_Xω (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_Xhω + dh = 0 for some function $h$ on $M$. Similarly, vector field $X$ is called locally Hamiltonian if it's image is closed 1-form i_Xω + u = 0,        du = 0 One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem, is that these vector fields preserve symplectic form $\omega$. In other words Lie derivative of the symplectic form $\omega$ along arbitrary locally Hamiltonian vector field vanishes L_Xω = 0 ⇔ i_Xω + du = 0,       du = 0 Indeed, using Cartan's formula that expresses Lie derivative in terms of contraction and exterior derivative L_X = i_Xd + di_X one gets L_Xω = i_Xdω + di_Xω = di_Xω (since $d\omega \; =\; 0$) but according to the definition of locally Hamiltonian vector field di_Xω = − du = 0 So locally Hamiltonian vector fields preserve $\omega$ and vise versa, if vector field preserves symplectic form $\omega$ 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 $L$_Xω = L_Yω = 0 and L_XL_Yω − L_YL_Xω = 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_Zω = 0 and i_Xhω + dh = 0 that implies L_Z(i_Xhω + dh) = L_{[Z , Xh]}ω + i_XhL_Zω + dL_Zh = L_{[Z , Xh]}ω + dL_Zh = 0 thus commutator $[Z\; ,\; X$_h] is Hamiltonian vector field $X$_LZh, or in other words Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields. Isomorphism $\Phi$_ω can be extended to higher order vector fields and differential forms by linearity and multiplicativity. Namely, Φ_ω(X ∧ Y) = Φ_ω(X) ∧ Φ_ω(Y) Since $\Phi$_ω is isomorphism, the symplectic form $\omega$ has unique counter image $W$ known as Poisson bivector field. Property $d\omega \; =\; 0$ together with non degeneracy implies that bivector field $W$ is also nondegenerate ($Wn\ne \; 0$) 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 [C₁ ∧ C₂ ∧ ... ∧ C_n , S₁ ∧ S₂ ∧ ... ∧ S_n] = (− 1)^{p + q}[C_p , S_q] ∧ C₁ ∧ C₂ ∧ ... ∧ Ĉ_p ∧ ... ∧ C_n ∧ S₁ ∧ S₂ ∧ ... ∧ Ŝ_q ∧ ...∧ 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 ⇔ du = 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 $\Phi$_W, transforming differential 1-forms into vector fields, which is inverted to the map $\Phi$_ω and is defined by Φ_W: u → W(u);       Φ_WΦ_ω = id 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(df ∧ dg) 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 → X_f = W(df) 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(df ∧ dg) = ω(X_f ∧ X_g) = L_Xfg = − L_Xgg 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 ddtf = {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_bc ∂∂z_b ∧ ∂∂z_c and the Hamilton's equation rewritten in terms of local coordinates takes the form ż_b = W_bc∂h∂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^zL_E (here $L$ denotes Lie derivative) that acts on the observables as follows g_z(f) = e^zL_E(f) = f + zL_Ef + ½(zL_E)²f + ⋯ Such a group of transformation is called symmetry of Hamilton's equation (24) if it commutes with time evolution operator ddt g_z(f) = g_z(ddtf) 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 ∂∂tE = [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 $[E\; ,\; W]\; \ne \; 0$ then $g$_z is called non-Noether symmetry. Now let us focus on non-Noether symmetries. We would like to show that the presence 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 $2n$-dimensional Poisson manifold $M$. Then, if the vector field $E$ generates non-Noether symmetry, the functions Y^(k) = Ŵ^k ∧ W^{n − k}Wⁿ           k = 1,2, ... n where $Ŵ\; =\; [E\; ,\; W]$, are integrals of motion. By the definition Ŵ^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] or k(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] = 0 hence, 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 (35) yields ddtY^(k)Wⁿ = 0 but since the dynamical system is regular ($Wn\ne \; 0$) 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 − cW)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) = (n − k)! k!n! ∑m_s > m_t c_m1c_m2 ⋯ c_mk Note also that conservation laws $Y(k)$ can be also defined by means of symplectic form $\omega$ using the following formula Y^(k) = (L_Eω)^k ∧ ω^{n − k}ωⁿ       k = 1,2, ... n Conservation laws $c$₁ ... c_n can be also derived from the secular equation (L_Eω − cω)ⁿ = 0 However all these expressions fail in case of infinite dimensional Hamiltonian systems where the volume form Ω = ωⁿ does not exist since $n\; =\; \infty$. But fortunately in these case one can define conservation laws using alternative formula C^(k) = i_W^k(L_Eω)^k as far as it involves only finite degree differential forms $(L$_Eω)^k and well defined multivector fields $Wk$. 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) = (n − k)! k!n! ∑m_s > m_t c_m1c_m2 ⋯ c_mk = 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 ddtY = L_XhY = 0 ⇒ ddtL_EY = L_XhL_EY = L_EL_XhY = 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 ddt g(f) = g(ddtf) Such a symmetries give rise to the same conservation laws Y^(k) = g(W)^k ∧ W^{n − k}Wⁿ       k = 1,2, ... n Let $M$ be $R4$ with coordinates $z$₁, z₂, z₃, z₄ and Poisson bivector field W = ∂∂z₁ ∧ ∂∂z₃ + ∂∂z₂ ∧ ∂∂z₄ and let's take the following Hamiltonian h = 12z₁² + 12z₂² + e^{z₃ − z₄} This is so called two particle non periodic Toda model. One can check that the vector field defined as E = 4∑s = 1 E_s∂∂z_s with components E₁ = 12z₁² − e^{z₃ − z₄} − t2(z₁ + z₂)e^{z₃ − z₄} E₂ = 12z₂² + 2e^{z₃ − z₄} + t2(z₁ + z₂)e^{z₃ − z₄} E₃ = 2z₁ + 12z₂ + t2(z₁² + e^{z₃ − z₄}) E₄ = z₂ − 12z₁ + t2(z₂² + e^{z₃ − z₄}) 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] = z₁∂∂z₁ ∧ ∂∂z₃ + z₂∂∂z₂ ∧ ∂∂z₄ + e^{z₃ − z₄} ∂∂z₁ ∧ ∂∂z₂ + ∂∂z₃ ∧ ∂∂z₄ Calculation of volume vector fields $Ŵk\wedge \; Wn\; -\; k$ gives rise to W ∧ W = − 2 ∂∂z₁ ∧ ∂∂z₂ ∧ ∂∂z₃ ∧ ∂∂z₄ Ŵ ∧ W = − (z₁ + z₂) ∂∂z₁ ∧ ∂∂z₂ ∧ ∂∂z₃ ∧ ∂∂z₄ Ŵ ∧ Ŵ = − 2(z₁z₂ − e^{z₃ − z₄}) ∂∂z₁ ∧ ∂∂z₂ ∧ ∂∂z₃ ∧ ∂∂z₄ and the conservation laws associated with this symmetry are just Y⁽¹⁾ = Ŵ ∧ WW ∧ W = 12(z₁ + z₂) Y⁽²⁾ = Ŵ ∧ ŴW ∧ W = z₁z₂ − e^{z₃ − z₄} It is remarkable that the same symmetry is also present in higher dimensions. For example in case where $M$ is $R6$ with coordinates z₁, z₂, z₃, z₄, z₅, z₆ Poisson bivector equal to W = ∂∂z₁ ∧ ∂∂z₄ + ∂∂z₂ ∧ ∂∂z₅ + ∂∂z₃ ∧ ∂∂z₆ and the following Hamiltonian h = 12z₁² + 12z₂² + 12z₃² + e^{z₄ − z₅} + e^{z₅ − z₆} we still can construct symmetry similar to (53). More precisely the vector field defined for arbitrary function $F$ as E = 6∑s = 1 E_s∂∂z_s with components specified as follows E₁ = 12z₁² − 2e^{z₄ − z₅} − t2(z₁ + z₂)e^{z₄ − z₅} E₂ = 12z₂² + 3e^{z₄ − z₅} − e^{z₅ − z₆} + t2(z₁ + z₂)e^{z₄ − z₅} E₃ = 12z₃² + 2e^{z₅ − z₆} + t2(z₂ + z₃)e^{z₅ − z₆} E₄ = 3z₁ + 12z₂ + 12z₃ + t2(z₁² + e^{z₄ − z₅}) E₅ = 2z₂ − 12z₁ + 12z₃ + t2(z₂² + e^{z₄ − z₅} + e^{z₅ − z₆}) E₆ = z₃ − 12z₁ − 12z₂ + t2(z₃² + e^{z₅ − z₆}) 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] = z₁∂∂z₁ ∧ ∂∂z₄ + z₂∂∂z₂ ∧ ∂∂z₅ + z₃∂∂z₃ ∧ ∂∂z₆ + e^{z₄ − z₅} ∂∂z₁ ∧ ∂∂z₂ + e^{z₅ − z₆} ∂∂z₂ ∧ ∂∂z₃ + ∂∂z₄ ∧ ∂∂z₅ + ∂∂z₅ ∧ ∂∂z₆ Volume multivector fields $Ŵk\wedge \; Wn\; -\; k$ can be calculated in the manner similar to $R4$ case and give rise to the well known conservation laws of three particle Toda chain. Y⁽¹⁾ = 16(z₁ + z₂ + z₃) = Ŵ ∧ W ∧ WW ∧ W ∧ W Y⁽²⁾ = 13 (z₁z₂ + z₁z₃ + z₂z₃ − e^{z₄ − z₅} − e^{z₅ − z₆}) = Ŵ ∧ Ŵ ∧ WW ∧ W ∧ W Y⁽³⁾ = z₁z₂z₃ − z₃e^{z₄ − z₅} − z₁e^{z₅ − z₆} = Ŵ ∧ Ŵ ∧ ŴW ∧ W ∧ W

Non-Liouville symmetries

Besides Hamiltonian dynamical systems that admit invariant symplectic form $\omega$, there are dynamical systems that either are not Hamiltonian or admit Hamiltonian realization but explicit form of symplectic structure $\omega$ is unknown or too complex. However usually such a dynamical systems possess invariant volume form $\Omega$ 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 $\omega$ naturally gives rise to the invariant volume form known as Liouville form Ω = ωⁿ and sometimes it is easier to work with $\Omega$ rather then with symplectic form itself. In generic Liouville dynamical system time evolution is governed by equations of motion ddtf = X(f) where $X$ is some smooth vector field that preserves Liouville volume form $\Omega$ ddtΩ = L_XΩ = 0 Symmetry of equations of motion still can be defined by condition ddt g_z(f) = g_z(ddtf) 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 $\Omega$, or in other words if L_EΩ ≠ cΩ 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 $\Omega$. Namely we have the following theorem, which is reformulation of Hojman's theorem in terms of Liouville volume form. Let $(M,\; X,\; \Omega )$ be Liouville dynamical system on the smooth manifold $M$. Then, if the vector field $E$ generates non-Liouville symmetry, the function J = L_EΩΩ is conservation law. By the definition L_EΩ = JΩ. 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_XL_EΩ = L_{[X , E]}Ω + L_EL_XΩ = L_X(JΩ) = (L_XJ)Ω + JL_XΩ but since Liouville volume form is invariant $L$_XΩ = 0 and vector field $E$ is generator of symmetry satisfying $[E\; ,\; X]\; =\; 0$ commutation relation we obtain (L_XJ)Ω = 0 or ddtJ = L_XJ = 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 ∂∂tE = [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)^mΩ Let us consider symmetry of three particle non periodic Toda chain. This dynamical system with equations of motion ż₄ = z₁ ż₅ = z₂ ż₆ = z₃ ż₁ = − e^{z₄ − z₅} ż₂ = e^{z₄ − z₅} − e^{z₅ − z₆} ż₃ = e^{z₅ − z₆} possesses invariant volume form Ω = dz₁ ∧ dz₂ ∧ dz₃ ∧ dz₄ ∧ dz₅ ∧ dz₆ The symmetry (61) is clearly non-Liouville one as far as L_EΩ = (z₁ + z₂ + z₃) Ω and main conservation law associated with this symmetry via Theorem 2 is total momentum J = L_EΩΩ = z₁ + z₂ + z₃ Other conservation laws can be recovered by taking Lie derivative of $J$ along generator of symmetry $E$, in particular J⁽¹⁾ = L_EJ = 12z₁² + 12z₂² + 12z₃² + e^{z₄ − z₅} + e^{z₅ − z₆} J⁽²⁾ = L_EJ⁽¹⁾ = 12 (z₁³ + z₂³ + z₃³) + 32 (z₁ + z₂)e^{z₄ − z₅} + 32 (z₂ + z₃)e^{z₅ − z₆}

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 ddtL = [L , P] = − ad_PL 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) = 12 Tr(L^k) since trace is invariant under coadjoint action ddtI^(k) = 12 ddt Tr(L^k) = 12 Tr(ddtL^k) = k2 Tr(L^{k − 1}ddtL) = k2 Tr(L^{k − 1}[L , P]) = 12 Tr([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 $\omega$ and the generator of the symmetry $E$ have the following form W = ∑rs W_rs∂∂z_r ∧ ∂∂z_r       ω = ∑rs ω_rsdz_r ∧ dz_s       E = ∑s E_s∂∂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 $2n$-dimensional Poisson manifold $M$. Then, if the vector field $E$ on $M$ generates the non-Noether symmetry, the following $2n\times 2n$ matrix valued functions on $M$ L_ab = ∑dc ω_ad E_c∂W_db∂z_c − W_bc∂E_d∂z_c + W_dc∂E_b∂z_c P_ab = ∑c ∂W_bc∂z_a·∂h∂z_c + W_bc∂²h∂z_a∂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_Eu (here $\Phi$_W and $\Phi$_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(fu) = fŔ_E(u) First property is obvious consequence of linearity of Schouten bracket, Lie derivative and maps $\Phi$_W, $\Phi$_ω. Second property can be checked directly Ŕ_E(fu) = Φ_ω([E , Φ_W(fu)]) − L_E(fu) = Φ_ω([E , fΦ_W(u)]) − (L_Ef)u − fL_Eu = Φ_ω((L_Ef)Φ_W(u)) + Φ_ω(f[E , Φ_W(u)]) − (L_Ef)u − fL_Eu = L_EfΦ_ωΦ_W(u) + fΦ_ω([E , Φ_W(u)]) − (L_Ef)u − fL_Eu = f(Φ_ω([E , Φ_W(u)]) − L_Eu) = fŔ_E(u) as far as $\Phi$_ωΦ_W(u) = u. Now let us check that $Ŕ$_E is invariant operator ddtŔ_E = L_XhŔ_E = L_Xh(Φ_ωL_EΦ_W − L_E) = Φ_ωL_{[Xh , E]}Φ_W − L_{[Xh, E]} = 0 because, being Hamiltonian vector field, $X$_h commutes with maps $\Phi$_W, $\Phi$_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 = ∑ab L_ab dz_a ⊗ ∂∂z_b and the invariance condition ddtŔ_E = L_W(h)Ŕ_E = 0 yields ddtŔ_E = ddt∑ab L_ab dz_a ⊗ ∂∂z_b = ∑ab (ddtL_ab) dz_a ⊗ ∂∂z_b + ∑ab L_ab (L_W(h)dz_a) ⊗ ∂∂z_b + ∑ab L_ab dz_a ⊗ (L_W(h)∂∂z_b) = ∑ab (ddtL_ab) dz_a ⊗ ∂∂z_b + ∑abcd L_ab∂W_ad∂z_c·∂h∂z_d dz_c ⊗ ∂∂z_b + ∑abcd L_abW_ad∂²h∂z_c∂z_d dz_c ⊗ ∂∂z_b + ∑abcdL_ab∂W_cd∂z_b·∂h∂z_d dz_a ⊗ ∂∂z_c + ∑abcd L_abW_cd ∂²h∂z_b∂z_d dz_a ⊗ ∂∂z_c = ∑ab ddtL_ab + ∑c(P_acL_cb − L_acP_cb) dz_a ⊗ ∂∂z_b = 0 or in matrix notations ddtL = [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) = 12 Tr(L^k) = ∑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)^mmC^(m) + m − 1∑k = 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 = z₁ 0 0 − e^{z₃ − z₄} 0 z₂ e^{z₃ − z₄} 0 0 1 z₁ 0 − 1 0 0 z₂ while matrix $P$ involved in Lax pair ddtL = [L , P] has the following form P = 0 0 1 0 0 0 0 1 − e^{z₃ − z₄} e^{z₃ − z₄} 0 0 e^{z₃ − z₄} − e^{z₃ − z₄} 0 0 The conservation laws associated with this Lax pair are total momentum and energy of two particle Toda chain I⁽¹⁾ = 12 Tr(L) = z₁ + z₂ I⁽²⁾ = 12 Tr(L²) = z₁² + z₂² + 2e^{z₃ − z₄} Similarly one can construct Lax matrix of three particle Toda chain, it has 16 nonzero elements L = z₁ 0 0 0 − e^{z₄ − z₅} 0 0 z₂ 0 e^{z₄ − z₅} 0 − e^{z₅ − z₆} 0 0 z₃ 0 e^{z₅ − z₆} 0 0 − 1 − 1 z₁ 0 0 1 0 − 1 0 z₂ 0 1 1 0 0 0 z₃ with non-zero elements matrix $P$ listed below P = 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 − e^{z₄ − z₅} e^{z₄ − z₅} 0 0 0 0 e^{z₄ − z₅} − e^{z₄ − z₅} − e^{z₅ − z₆} e^{z₅ − z₆} 0 0 0 0 e^{z₅ − z₆} − e^{z₅ − z₆} 0 0 0 Corresponding conservation laws reproduce total momentum, energy and second Hamiltonian involved in bi-Hamiltonian realization of Toda chain I⁽¹⁾ = 12 Tr(L) = z₁ + z₂ I⁽²⁾ = 12 Tr(L²) = z₁² + z₂² + z₃² + 2e^{z₄ − z₅} + 2e^{z₅ − z₆} I⁽³⁾ = 12 Tr(L³) = z₁³ + z₂³ + z₃³ + 3(z₁ + z₂)e^{z₄ − z₅} + 3(z₂ + z₃)e^{z₅ − z₆}

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 $(M,\; h)$ on $2n$ 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, c₂ ... c_n as linear independence of either differentials of conservation laws $dc$₁, dc₂ ... dc_n or corresponding Hamiltonian vector fields $X$_c1, X_c2 ... X_cn. Strictly speaking we can say that conservation laws $c$₁, c₂ ... c_n are functionally independent if Lesbegue measure of the set of points of phase space $M$ where differentials $dc$₁, dc₂ ... dc_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, c₂ ... c_n implies that corresponding Hamiltonian vector fields $X$_c1, X_c2 ... X_cn 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_ci , X_cj) = 0 thus space spanned by $X$_c1, X_c2 ... X_cn is isotropic. Dimension of this space is $n$ so it is Lagrangian. Note also that distribution $X$_c1, X_c2 ... X_cn is integrable since due to (22) [X_ci , X_cj] = X_{{ci , cj}} = 0 and according to Frobenius theorem there exists submanifold of $M$ such that distribution $X$_c1, X_c2 ... X_cn 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 $E$ on $2n$-dimensional Poisson manifold $M$ satisfies the condition [[E[E , W]]W] = 0 and $W$ bivector field has maximal rank ($Wn\ne \; 0$) 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       ⇔       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      ⇔      [Ŵ(f) , W] = − [Ŵ , W(f)] and finally condition (110) ensures third identity [Ŵ , Ŵ] = 0 yielding Liouville theorem for $Ŵ$ [Ŵ , Ŵ] = 0      ⇔      [Ŵ(f) , Ŵ] = 0 Indeed [Ŵ , Ŵ] = [[E , W]Ŵ] = [[Ŵ , E]W] = − [[E , Ŵ]W] = − [[E[E , W]]W] = 0 Now let us consider two different solutions $c$_i ≠ c_j of the equation (40). By taking the Lie derivative of the equation (Ŵ − c_iW)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_iW)^{n − 1}(L_W(cj)Ŵ − {c_j , c_i}W) = 0, and (Ŵ − c_iW)^{n − 1}(c_iL_Ŵ(cj)W + {c_j , c_i}_∗W) = 0, where {c_i , c_j}_∗ = Ŵ(dc_i ∧ dc_j) is the Poisson bracket calculated by means of the bivector field $Ŵ$. 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_iW)^{n − 1}W = 0 Thus, either {c_i , c_j}_∗ − c_i{c_i , c_j} = 0 or the volume field $(Ŵ\; -\; c$_iW)^{n − 1}W vanishes. In the second case we can repeat (119)-(122) procedure for the volume field $(Ŵ\; -\; c$_iW)^{n − 1}W yielding after $n$ iterations $Wn=\; 0$ 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 ≠ 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 $2n$ 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 ($Wn\ne \; 0$) 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 Wⁿ ≠ 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 $2n$-dimensional manifold $M$ endowed with regular Poisson bivector field $W$. Then, if the vector field $E$ on $M$ 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 = ∂∂z₁ ∧ ∂∂z₃ + ∂∂z₂ ∧ ∂∂z₄ and Ŵ = [E , W] = z₁∂∂z₁ ∧ ∂∂z₃ + z₂∂∂z₂ ∧ ∂∂z₄ + e^{z₃ − z₄} ∂∂z₁ ∧ ∂∂z₂ + ∂∂z₃ ∧ ∂∂z₄ 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 = ∂∂z₁ ∧ ∂∂z₄ + ∂∂z₂ ∧ ∂∂z₅ + ∂∂z₃ ∧ ∂∂z₆ and Ŵ = [E , W] = z₁∂∂z₁ ∧ ∂∂z₄ + z₂∂∂z₂ ∧ ∂∂z₅ + z₃∂∂z₃ ∧ ∂∂z₆ + e^{z₄ − z₅} ∂∂z₁ ∧ ∂∂z₂ + e^{z₅ − z₆} ∂∂z₂ ∧ ∂∂z₃ + ∂∂z₄ ∧ ∂∂z₅ + ∂∂z₅ ∧ ∂∂z₆ In terms of differential forms bi-Hamiltonian structure is formed by couple of closed differential 2-forms: symplectic form $\omega$ (such that $d\omega \; =\; 0$ and $\omega n\ne \; 0$) and $\omega \ast =\; L$_Eω (clearly $d\omega \ast =\; dL$_Eω = L_Edω = 0). It is important that by taking Lie derivative of Hamilton's equation i_Xhω + dh = 0 along the generator $E$ of symmetry L_E(i_Xhω + dh) = i_{[E , Xh]}ω + i_XhL_Eω + L_Edh = i_Xhω^∗ + dL_Eh = 0 one obtains another Hamilton's equation i_Xhω^∗ + dh^∗ = 0 where $h\ast =\; L$_Eh. This is actually second Hamiltonian realization of equations of motion and thus under certain conditions existence of non-Noether symmetry gives rise to additional presymplectic structure $\omega \ast$ and additional Hamiltonian realization of the dynamical system. In many integrable models admitting bi-Hamiltonian realization (including Toda chain, Korteweg-de Vries hierarchy, Nonlinear Schrödinger equation, Broer-Kaup system and Benney system) non-Noether symmetries that are responsible for existence of bi-Hamiltonian structures has been found and motivated further investigation of relationship between symmetries and bi-Hamiltonian structures. Namely it seems to be interesting to know whether in general case existence of bi-Hamiltonian structure is related to non-Noether symmetry. Let us consider more general case and suppose that we have couple of differential 2-forms $\omega$ and $\omega \ast$ such that dω = dω^∗ = 0,       ωⁿ ≠ 0 i_Xhω + dh = 0 and i_Xhω^∗ + dh^∗ = 0 The question is whether there exists vector field $E$ (generator of non-Noether symmetry) such that $[E\; ,\; X$_h] = 0 and $\omega \ast =\; L$_Eω. The answer depends on $\omega \ast$. Namely if $\omega \ast$ is exact form (there exists 1-form $\theta \ast$ such that $\omega \ast =\; d\theta \ast$) then one can argue that such a vector field exists and thus any exact bi-Hamiltonian structure is related to hidden non-Noether symmetry. To outline proof of this statement let us introduce vector field $E\ast$ defined by i_E^∗ω = θ^∗ (such a vector field always exist because $\omega$ is nondegenerate 2-form). By construction L_E^∗ ω = ω^∗ Indeed L_E^∗ω = di_E^∗ω + i_E^∗dω = dθ^∗ = ω^∗ And i_{[E^∗, Xh]}ω = L_E^∗(i_Xhω) − i_XhL_E^∗ω = − d(E^∗(h) − h^∗) = − dh' In other words $[X$_h , E^∗] is Hamiltonian vector field [X_h , E] = X_h' One can also construct locally Hamiltonian vector field $X$_g, that satisfies the same commutation relation. Namely let us define function (in general case this can be done only locally) g(z) = t∫0 h'dt where integration along solution of Hamilton's equation, with fixed origin and end point in $z(t)\; =\; z$, is assumed. And then it is easy to verify that locally Hamiltonian vector field associated with $g(z)$, by construction, satisfies the same commutation relations as $E\ast$ (namely $[X$_h , X_g] = X_h'). Using $E\ast$ and $X$_h' one can construct generator of non-Noether symmetry — non-Hamiltonian vector field $E\; =\; E\ast -\; X$_g commuting with $X$_h and satisfying L_Eω = L_E^∗ω − L_Xgω = L_E^∗ω = ω^∗ (thanks to Liouville's theorem $L$_Xgω = 0). So in case of regular Hamiltonian system every exact bi-Hamiltonian structure is naturally associated with some (non-Noether) symmetry of space of solutions. In case where bi-Hamiltonian structure is not exact ($\omega \ast$ is closed but not exact) then due to ω^∗ = L_Eω = di_Eω + i_Edω = di_Eω it is clear that such a bi-Hamiltonian system is not related to symmetry. However in all known cases bi-Hamiltonian structures seem to be exact.

Bidifferential calculi

Another important concept that is often used in theory of dynamical systems and may be related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach). Recently A. Dimakis and F. Müller-Hoissen applied bidifferential calculi to the wide range of integrable models including KdV hierarchy, KP equation, self-dual Yang-Mills equation, Sine-Gordon equation, Toda models, non-linear Schrödinger and Liouville equations. It turns out that these models can be effectively described and analyzed using the bidifferential calculi [17], [24]. Here we would like to show that each generator of non-Noether symmetry satisfying condition $[[E[E\; ,\; W]]W]\; =\; 0$ gives rise to certain bidifferential calculus. Before we proceed let us specify what kind of bidifferential calculi we plan to consider. Under the bidifferential calculus we mean the graded algebra of differential forms over the phase space Ω = ∞∪k = 0 Ω^(k) ($\Omega (k)$ denotes the space of $k$-degree differential forms) equipped with a couple of differential operators d, đ : Ω^(k) → Ω^{(k + 1)} satisfying conditions $d2=\; đ2=\; dđ\; +\; đd\; =\; 0$ (see [24]). In other words we have two De Rham complexes $M,\; \Omega ,\; d$ and $M,\; \Omega ,\; đ$ on algebra of differential forms over the phase space. And these complexes satisfy certain compatibility condition — their differentials anticommute with each other $dđ\; +\; đd\; =\; 0$. Now let us focus on non-Noether symmetries. It is interesting that if generator of the non-Noether symmetry satisfies equation $[[E[E\; ,\; W]]W]\; =\; 0$ then we are able to construct an invariant bidifferential calculus of a certain type. This construction is summarized in the following theorem: Let $(M\; ,\; h)$ be regular Hamiltonian system on the Poisson manifold $M$. Then, if the vector field $E$ on $M$ generates the non-Noether symmetry and satisfies the equation [[E[E , W]]W] = 0, the differential operators du = Φ_ω([W , Φ_W(u)]) đu = Φ_ω([[E , W]Φ_W(u)]) form invariant bidifferential calculus ($d2=\; đ2=\; dđ\; +\; đd\; =\; 0$) over the graded algebra of differential forms on $M$. First of all we have to show that $d$ and $đ$ are really differential operators , i.e., they are linear maps from $\Omega (k)$ into $\Omega (k\; +\; 1)$, satisfy derivation property and are nilpotent ($d2=\; đ2=\; 0$). Linearity is obvious and follows from the linearity of the Schouten bracket $[\; ,\; ]$ and $\Phi$_W, Φ_ω maps. Then, if $u$ is a $k$-degree form $\Phi$_W maps it on $k$-degree multivector field and the Schouten brackets $[W\; ,\; \Phi$_W(u)] and $[[E\; ,\; W]\Phi$_W(u)] result the $k\; +\; 1$-degree multivector fields that are mapped on $k\; +\; 1$-degree differential forms by $\Phi$_ω. So, $d$ and $đ$ are linear maps from $\Omega (k)$ into $\Omega (k\; +\; 1)$. Derivation property follows from the same feature of the Schouten bracket $[\; ,\; ]$ and linearity of $\Phi$_W and $\Phi$_ω maps. Now we have to prove the nilpotency of $d$ and $đ$. Let us consider $d2u$ d²u = Φ_ω([W , Φ_W(Φ_ω([W , Φ_W(u)]))]) = Φ_ω([W[W , Φ_W(u)]]) = 0 as a result of the property (112) and the Jacoby identity for $[\; ,\; ]$ bracket. In the same manner đ²u = Φ_ω([[W , E][[W , E]Φ_W(u)]]) = 0 according to the property (116) of $[W\; ,\; E]\; =\; Ŵ$ and the Jacoby identity. Thus, we have proved that $d$ and $đ$ are differential operators (in fact $d$ is ordinary exterior differential and the expression (151) 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 Jacoby identity we get (dđ + đd)(u) = Φ_ω([[[W , E]W]Φ_W(u)]) = 0 as far as (114) is satisfied. So, $d$ and $đ$ form the bidifferential calculus over the graded algebra of differential forms. It is also clear that the bidifferential calculus $d,\; đ$ is invariant, since both $d$ and $đ$ commute with time evolution operator $W(h)\; =\; \{h,\; \}$. Conservation laws that are associated with the bidifferential calculus (151) (152) and form Lenard scheme (see [24]): (k + 1)đI^(k) = kdI^{(k + 1)} coincide with the sequence of integrals of motion (98). Proof of this correspondence lays outside the scope of present manuscript, but can be done in the manner similar to [17]. The symmetry (53) endows $R4$ with bicomplex structure $d,\; đ$ where $d$ is ordinary exterior derivative while $đ$ is defined by đz₁ = z₁dz₁ − e^{z₃ − z₄}dz₄ đz₂ = z₂dz₂ + e^{z₃ − z₄}dz₃ đz₃ = z₁dz₃ + dz₂ đz₄ = z₂dz₄ − dz₁ and is extended to whole De Rham complex by linearity, derivation property and compatibility property $dđ\; +\; đd\; =\; 0$. By direct calculations one can verify that calculus constructed in this way is consistent and satisfies $đ2=\; 0$ property. To illustrate technique let us explicitly check that $đ2z$₁ = 0. Indeed đ²z₁ = đđz₁ = đ(z₁dz₁ − e^{z₃ − z₄}dz₄) = đz₁ ∧ dz₁ + z₁đdz₁ − e^{z₃ − z₄}đz₃ ∧ dz₄ + e^{z₃ − z₄}đz₄ ∧ dz₄ − e^{z₃ − z₄}đdz₄ = đz₁ ∧ dz₁ − z₁dđz₁ − e^{z₃ − z₄}đz₃ ∧ dz₄ + e^{z₃ − z₄}đz₄ ∧ dz₄ + e^{z₃ − z₄}dđz₄ = 0 Because of properties đz₁ ∧ dz₁ = e^{z₃ − z₄}dz₁ ∧ dz₄, − z₁dđz₁ = z₁e^{z₃ − z₄}dz₃ ∧ dz₄, − e^{z₃ − z₄}đz₃ ∧ dz₄ = − z₁e^{z₃ − z₄}dz₁ ∧ dz₄ − e^{z₃ − z₄}dz₂ ∧ dz₄, e^{z₃ − z₄}đz₄ ∧ dz₄ = e^{z₃ − z₄}dz₂ ∧ dz₄ and e^{z₃ − z₄}dđz₄ = − e^{z₃ − z₄}dz₁ ∧ dz₄ Similarly one can show that đ²z₂ = đ²z₃ = đ²z₄ = 0 and thus $đ$ is nilpotent operator $đ2=\; 0$. Note also that conservation laws I⁽¹⁾ = z₁ + z₂ I⁽²⁾ = z₁² + z₂² + 2e^{z₃ − z₄} form the simplest Lenard scheme 2đI⁽¹⁾ = dI⁽²⁾ Similarly one can construct bidifferential calculus associated with non-Noether symmetry (61) of three particle Toda chain. In this case $đ$ can be defined by đz₁ = z₁dz₁ − e^{z₄ − z₅}dz₅ đz₂ = z₂dz₂ + e^{z₄ − z₅}dz₄ − e^{z₅ − z₆}dz₆ đz₃ = z₃dz₃ + e^{z₅ − z₆}dz₅ đz₄ = z₁dz₄ − dz₂ − dz₃ đz₅ = z₂dz₅ + dz₁ − dz₃ đz₆ = z₃dz₆ + dz₁ + dz₂ and as in case of two particle Toda it can be extended to whole De Rham complex by linearity, derivation property and compatibility property $dđ\; +\; đd\; =\; 0$. One can check that conservation laws of Toda chain I⁽¹⁾ = z₁ + z₂ I⁽²⁾ = z₁² + z₂² + z₃² + 2e^{z₄ − z₅} + 2e^{z₅ − z₆} I⁽³⁾ = z₁³ + z₂³ + z₃³ + 3(z₁ + z₂)e^{z₄ − z₅} + 3(z₂ + z₃)e^{z₅ − z₆} form Lenard scheme 2đI⁽¹⁾ = dI⁽²⁾ 3đI⁽²⁾ = 2dI⁽³⁾

Frölicher-Nijenhuis geometry

Finally we would like to reveal some features of the operator $Ŕ$_E (89) and to show how Frölicher-Nijenhuis geometry arises in Hamiltonian system that possesses certain non-Noether symmetry. From the geometric properties of the tangent valued forms we know that the traces of powers of a linear operator $F$ on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion $T(F)$ vanishes, i. e. whenever for arbitrary vector fields $X,Y$ the condition T(F)(X , Y) = [FX , FY] − F([FX , Y] + [X , FY] − F[X , Y]) = 0 is satisfied. Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used in theory of integrable models, where they play role of recursion operators and are used in construction of involutive family of conservation laws. We would like to show that each generator of non-Noether symmetry satisfying equation $[[E[E\; ,\; W]]W]\; =\; 0$ canonically leads to invariant Frölicher-Nijenhuis operator on tangent bundle over the phase space. This operator can be expressed in terms of generator of symmetry and isomorphism defined by Poisson bivector field. Strictly speaking we have the following theorem. Let $(M\; ,\; h)$ be regular Hamiltonian system on the Poisson manifold $M$. If the vector field $E$ on $M$ generates the non-Noether symmetry and satisfies the equation [[E[E , W]]W] = 0 then the linear operator, defined for every vector field $X$ by equation R_E(X) = Φ_W(L_EΦ_ω(X)) − [E , X] is invariant Frölicher-Nijenhuis operator on $M$. Invariance of $R$_E follows from the invariance of the $Ŕ$_E defined by (89) (note that for arbitrary 1-form vector field $u$ and vector field $X$ contraction $i$_Xu has the property $i$_REXu = i_XŔ_Eu, so $R$_E is actually transposed to $Ŕ$_E). It remains to show that the condition (110) ensures vanishing of the Frölicher-Nijenhuis torsion $T(R$_E) of $R$_E, i.e. for arbitrary vector fields $X,\; Y$ we must get T(R_E)(X , Y) = [R_E(X) , R_E(Y)] − R_E([R_E(X) , Y] + [X , R_E(Y)] − R_E([X , Y])) = 0 First let us introduce the following auxiliary 2-forms ω = Φ_ω(W),       ω^∗ = Ŕ_Eω       ω^∗∗ = Ŕ_Eω^∗ Using the realization (151) of the differential $d$ and the property (15) yields dω = Φ_ω([W , W]) = 0 Similarly, using the property (114) we obtain dω^∗ = dΦ_ω([E , W]) − dL_Eω = Φ_ω([[E , W]W]) − L_Edω = 0 And finally, taking into account that $\omega \ast =\; 2\Phi$_ω([E , W]) and using the condition (110), we get dω^∗∗ = 2Φ_ω([[E[E , W]]W]) − 2dL_Eω^∗ = − 2L_Edω^∗ = 0 So the differential forms $\omega ,\; \omega \ast ,\; \omega \ast \ast$ are closed dω = dω^∗ = dω^∗∗ = 0 Now let us consider the contraction of $T(R$_E) and $\omega$. i_{T(RE)(X , Y)}ω = i_{[REX , REY]}ω − i_{[REX , Y]}ω^∗ − i_{[X , REY]}ω^∗ + i_{[X , Y]}ω^∗∗ =L_REXi_Yω^∗ − i_REYL_Xω^∗ − L_REXi_Yω^∗ + i_YL_REXω^∗ − L_Xi_REYω^∗ + i_REYL_Xω^∗ + i_{[X , Y]}ω^∗∗ = i_YL_Xω^∗∗ − L_Xi_Yω^∗∗ + i_{[X , Y]}ω^∗∗ = 0 where we used (175) (179), the property L_Xi_Yω = i_YL_Xω + i_{[X , Y]}ω of the Lie derivative and the relations of the following type L_REXω = di_REXω + i_REXdω = 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 $\omega$ vanishes. But since $W$ bivector is non-degenerate ($Wn\ne \; 0$), its counter image ω = Φ_ω(W) is also non-degenerate and vanishing of the contraction (180) implies that the torsion $T(R$_E) itself is zero. So we get T(R_E)(X , Y) = [R_E(X) , R_E(Y)] − R_E([R_E(X) , Y] + [X , R_E(Y)] − R_E([X , Y])) = 0 The operator $R$_E associated with non-Noether symmetry (53) reproduces well known Frölicher-Nijenhuis operator R_E = z₁dz₁ ⊗ ∂∂z₁ − dz₁ ⊗ ∂∂z₄ + z₂dz₂ ⊗ ∂∂z₂ + dz₂ ⊗ ∂∂z₃ + z₁dz₃ ⊗ ∂∂z₃ + e^{z₃ − z₄}dz₃ ⊗ ∂∂z₂ + z₂dz₄ ⊗ ∂∂z₄ − e^{z₃ − z₄}dz₄ ⊗ ∂∂z₁ (compare with [30]). The operator $Ŕ$_E plays the role of recursion operator for conservation laws I⁽¹⁾ = z₁ + z₂ I⁽²⁾ = z₁² + z₂² + 2e^{z₃ − z₄} Indeed one can check that 2Ŕ_E(dI⁽¹⁾) = dI⁽²⁾ Similarly using non-Noether symmetry (61) one can construct recursion operator of three particle Toda chain R_E = z₁dz₁ ⊗ ∂∂z₁ − e^{z₄ − z₅}dz₅ ⊗ ∂∂z₁ + z₂dz₂ ⊗ ∂∂z₂ + e^{z₄ − z₅}dz₄ ⊗ ∂∂z₂ − e^{z₅ − z₆}dz₆ ⊗ ∂∂z₂+ z₃dz₃ ⊗ ∂∂z₃ + e^{z₅ − z₆}dz₅ ⊗ ∂∂z₃ + z₁dz₄ ⊗ ∂∂z₄ − dz₂ ⊗ ∂∂z₄ − dz₃ ⊗ ∂∂z₄ + z₂dz₅ ⊗ ∂∂z₅ + dz₁ ⊗ ∂∂z₅ − dz₃ ⊗ ∂∂z₅ + z₃dz₆ ⊗ ∂∂z₆ + dz₁ ⊗ ∂∂z₆ + dz₂ ⊗ ∂∂z₆ and as in case of two particle Toda chain, operator $Ŕ$_E appears to be recursion operator for conservation laws I⁽¹⁾ = z₁ + z₂ I⁽²⁾ = z₁² + z₂² + z₃² + 2e^{z₄ − z₅} + 2e^{z₅ − z₆} I⁽³⁾ = z₁³ + z₂³ + z₃³ + 3(z₁ + z₂)e^{z₄ − z₅} + 3(z₂ + z₃)e^{z₅ − z₆} and fulfills the following recursion condition dI⁽³⁾ = 3Ŕ_E(dI⁽²⁾) = 6(Ŕ_E)²(dI⁽¹⁾)

One-parameter families of conservation laws

One-parameter group of transformations $g$_z defined by (28) naturally acts on algebra of integrals of motion. Namely for each conservation law ddtJ = 0 one can define one-parameter family of conserved quantities $J(z)$ by applying group of transformations $g$_z to $J$ J(z) = g_z(J) = e^zL_EJ = J + zL_EJ + ½(zL_E)²J + ... Property (29) ensures that $J(z)$ is conserved for arbitrary values of parameter $z$ ddtJ(z) = ddtg_z(J) = g_z (ddtJ) = 0 and thus each conservation law gives rise to whole family of conserved quantities that form orbit of group of transformations $g$_a. Such an orbit $J(z)$ is called involutive if conservation laws that form it are in involution {J(z₁) , J(z₂)} = 0 (for arbitrary values of parameters $z$₁, z₂). On $2n$ dimensional symplectic manifold each involutive family that contains $n$ functionally independent integrals of motion naturally gives rise to integrable system (due to Liouville-Arnold theorem). So in order to identify those orbits that may be related to integrable models it is important to know how involutivity of family of conserved quantities $J(z)$ is related to properties of initial conserved quantity $J(0)\; =\; J$ and nature of generator $E$ of group $g$_z = e^zL_E. In other words we would like to know what condition must be satisfied by generator of symmetry $E$ and integral of motion $J$ to ensure that $\{J(z$₁) , J(z₂)} = 0. To address this issue and to describe class of vector fields that possess nontrivial involutive orbits we would like to propose the following theorem Let $M$ be Poisson manifold endowed with 1-form $s$ such that [W[W(s),W](s)] = c₀[W(s)[W(s) ,W]]       (c₀ ≠ − 1) Then each function $J$ satisfying property W(L_W(s)dJ) = c₁[W(s),W](dJ)      (c₁ ≠ 0) ($c$_0,1 are some constants) gives rise to involutive set of functions J^(m) = (L_W(s))^mJ       {J^(m), J^(k)} = 0 First let us inroduce linear operator $R$ on bundle of multivector fields and define it for arbitrary multivector field $V$ by condition R(V) = ½ ([W(s),V] − Φ_W(L_W(s)Φ_ω(V))) Proof of linearity of this operator is identical to proof given for (89) so we will skip it. Further it is clear that R(W) = [W(s),W] and R²(W) = R([W(s),W]) = ½([W(s)[W(s),W]] − Φ_W((L_W(s))²ω)) = ½(1 + c₀)[W(s)[W(s),W]] where we used property Φ_W((L_W(s))²ω) = Φ_W(L_W(s)L_W(s)ω) = Φ_W(i_W(s)dL_W(s)ω) + Φ_W(di_W(s)L_W(s)ω) = [W,Φ_W(i_W(s)L_W(s)ω)] = [W[W(s),W](s)] = c₀[W(s)[W(s),W]] At the same time by taking Lie derivative of (199) along the vector field $W(s)$ one gets [W[W(s),W](s)] = (L_W(s)R + R²)(W) comparing (200) and (202) yields (1 + c₀)(L_W(s)R + R²) = 2R² and thus (1 + c₀)L_W(s)R = (1 − c₀)R² Further let us rewrite condition (196) as follows W(L_W(s)dJ) = c₁R(W)(dJ) due to linearity of operator $R$ this condition can be extended to R^m(W)(L_W(s)dJ) = c₁R^{m + 1}(W)(dJ) Now assuming that the following condition is true W((L_W(s))^mdJ) = c_mR^m(W)(dJ) let us take its Lie derivative along vector field $W(s)$. We get R(W)((L_W(s))^mdJ) + W((L_W(s))^{m + 1}dJ) = mc_m1 − c₀1 + c₀R^{m + 1}(W)(dJ) + c_mR^{m + 1}(W)(dJ) where we used properties (199) and (204). Note also that (207) together with linearity of operator $R$ imply that R^kW((L_W(s))^mdJ) = c_mR^{k + m}(W)(dJ) and thus (208) reduces to W((L_W(s))^{m + 1}dJ) = c_{m + 1}R^{m + 1}(W)(dJ) where $c$_{m + 1} is defined by (1 + c₀)c_{m + 1} = mc_n(1 − c₀) So we proved that if assumtion (207) is valid for $m$ then it is also valid for $m\; +\; 1$, we also know that for $m\; =\; 1$ it matches (205) and thus by induction we proved that condition (207) is valid for arbitrary $m$ while $c$_n can be determined by c_m(1 + c₀)^{m − 1} = c₀(m − 1)!(1 − c₀)^{m − 1} Now using (207) and (209) it is easy to show that functions $(L$_W(s))^mJ are in involution. Indeed {(L_W(s))^mJ, (L_W(s))^kJ} = W(d(L_W(s))^mJ ∧ d(L_W(s))^kJ) = W((L_W(s))^mdJ ∧ (L_W(s))^kdJ) = c_mc_kW(dJ ∧ dJ) = 0 So we have proved that the functions (197) are in involution. Further we will use this theorem to prove involutivity of family of conservation laws constructed using non-Noether symmetry of Toda chain.

Toda Model

To illustrate features of non-Noether symmetries we often refer to two and three particle non-periodic Toda systems. However it turns out that non-Noether symmetries are present in generic n-particle non-periodic Toda chains as well, moreover they preserve basic features of symmetries (53), (61). In case of n-particle Toda model symmetry yields $n$ functionally independent conservation laws in involution, gives rise to bi-Hamiltonian structure of Toda hierarchy, reproduces Lax pair of Toda system, endows phase space with Frölicher-Nijenhuis operator and leads to invariant bidifferential calculus on algebra of differential forms over phase space of Toda system. First of all let us remind that Toda model is $2n$ dimensional Hamiltonian system that describes the motion of $n$ particles on the line governed by the exponential interaction. Equations of motion of the non periodic n-particle Toda model are ddtq_s = p_s ddtp_s = ε(s − 1)e^{q_{s − 1} − q_s} − ε(n − s)e^{q_s − q_{s + 1}} ($\epsilon (k)\; =\; -\; \epsilon (-\; k)\; =\; 1$ for any natural $k$ and $\epsilon (0)\; =\; 0$) and can be rewritten in Hamiltonian form (24) with canonical Poisson bracket defined by Poisson bivector W = n∑s = 1 ∂∂p_s ∧ ∂∂q_s and Hamiltonian equal to h = ½n∑s = 1p_s² + n − 1∑s = 1e^{q_s − q_{s + 1}} Note that in two and three particle case we have used slightly different notations z_s = p_s z_{n + s} = q_s        s = 1, 2, (3); n = 2(3) for local coordinates. The group of transformations $g$_z generated by the vector field $E$ will be symmetry of Toda chain if for each $p$_s, q_s satisfying Toda equations (214) $g$_z(p_s), g_z(q_s) also satisfy it. Substituting infinitesimal transformations g_z(p_s) = p_s + zE(p_s) + O(z²) g_z(p_s) = q_s + zE(q_s) + O(z²) into (214) and grouping first order terms gives rise to the conditions ddtE(q_s) = E(p_s) ddtE(p_s) = ε(s − 1)e^{q_{s − 1} − q_s} (E(q_{s − 1}) − E(q_s)) − ε(n − s)e^{q_s − q_{s + 1}} (E(q_s) − E(q_{s + 1})) One can verify that the vector field defined by E(p_s) = ½p_s² + ε(s − 1)(n − s + 2)e^{q_{s − 1} − q_s} − ε(n − s)(n − s) e^{q_s − q_{s + 1}} + t2(ε(s − 1)(p_{s − 1} + p_s) e^{q_{s − 1} − q_s} − ε(n − s)(p_s + p_{s + 1}) e^{q_s − q_{s + 1}}) E(q_s) = (n − s + 1)p_s − ½s − 1∑k = 1 p_k + ½n∑k = s + 1 p_k + t2(p_s² + ε(s − 1)e^{q_{s − 1} − q_s} + ε(n − s)e^{q_s − q_{s + 1}}) satisfies (31) and generates symmetry of Toda chain. It appears that this symmetry is non-Noether since it does not preserve Poisson bracket structure $[E\; ,\; W]\; \ne \; 0$ and additionally one can check that Yang-Baxter equation $[[E[E\; ,\; W]]W]\; =\; 0$ is satisfied. This symmetry may play important role in analysis of Toda model. First let us note that calculating $L$_EW leads to the following Poisson bivector field Ŵ = [E , W] = n∑s = 1 p_s∂∂p_s ∧ ∂∂q_s + n − 1∑s = 1 e^{q_s − q_{s + 1}} ∂∂p_s ∧ ∂∂q_{s + 1} + ∑r > s ∂∂q_s ∧ ∂∂q_r and together $W$ and $L$_EW give rise to bi-Hamiltonian structure of Toda model (compare with [30]). Thus bi-Hamiltonian realization of Toda chain can be considered as manifestation of hidden symmetry. In terms of bivector fields these bi-Hamiltonian system is formed by The conservation laws (45) associated with the symmetry reproduce well known set of conservation laws of Toda chain. I⁽¹⁾ = C⁽¹⁾ = n∑s = 1p_s I⁽²⁾ = (C⁽¹⁾)² − 2C⁽²⁾ = n∑s = 1 p_s² + 2n − 1∑s = 1 e^{q_s − q_{s + 1}} I⁽³⁾ = C⁽¹⁾)³ − 3C⁽¹⁾C⁽²⁾ + 3C⁽³⁾ = n∑s = 1 p_s³ + 3n − 1∑s = 1 (p_s + p_{s + 1}) e^{q_s − q_{s + 1}} I⁽⁴⁾ = C⁽¹⁾)⁴ − 4(C⁽¹⁾)²C⁽²⁾ + 2(C⁽²⁾)² + 4C⁽¹⁾C⁽³⁾ − 4C⁽⁴⁾ = n∑s = 1 p_s⁴ + 4n − 1∑s = 1 (p_s² + 2p_sp_{s + 1} + p_{s + 1}²) e^{q_s − q_{s + 1}} + 2n − 1∑s = 1 e^{2(q_s − q_{s + 1})} + 4n − 2∑s = 1 e^{q_s − q_{s + 2}} I^(m) = (− 1)^{m + 1}mC^(m) + m − 1∑k = 1 (− 1)^{k + 1}I^{(m − k)}C^(k) The condition $[[E[E\; ,\; W]]W]\; =\; 0$ satisfied by generator of the symmetry $E$ ensures that the conservation laws are in involution i. e. $\{C(k),\; C(m)\}\; =\; 0$. Thus the conservation laws as well as the bi-Hamiltonian structure of the non periodic Toda chain appear to be associated with non-Noether symmetry. Using formula (88) one can calculate Lax pair associated with symmetry (220). Lax matrix calculated in this way has the following non-zero entries (note that in case of $n\; =\; 2$ and $n\; =\; 3$ this formula yields matrices (102)-(105)) L_{k, k} = L_{n + k, n + k} = p_k L_{n + k, k + 1} = − L_{n + k + 1, k} = ε(n − k)e^{q_k − q_{k + 1}} L_{k, n + m} = ε(m − k) m, k = 1, 2, ... , n while non-zero entries of $P$ matrix involved in Lax pair are P_{k, n + k} = 1 P_{n + k, k} = − ε(k − 1)e^{q_{k − 1} − q_k} − ε(n − k)e^{q_k − q_{k + 1}} P_{n + k, k + 1} = ε(n − k)e^{q_k − q_{k + 1}} P_{n + k, k − 1} = ε(k − 1)e^{q_{k − 1} − q_k} k = 1, 2, ... , n This Lax pair constructed from generator of non-Noether symmetry exactly reproduces known Lax pair of Toda chain. Like two and three particle Toda chain, n-particle Toda model also admits invariant bidifferential calculus on algebra of differential forms over the phase space. This bidifferential calculus can be constructed using non-Noether symmetry (see (152)), it consists out of two differential operators $d,\; đ$ where $d$ is ordinary exterior derivative while $đ$ can be defined by đq_s = p_sdq_s + ∑r > sdp_r − ∑s > rdp_r đp_s = p_sdp_s − e^{q_s − q_{s + 1}}dq_{s + 1} + e^{q_{s − 1} − q_s}dq_s and is extended to whole De Rham complex by linearity, derivation property and compatibility property $dđ\; +\; đd\; =\; 0$. By direct calculations one can verify that calculus constructed in this way is consistent and satisfies $đ2=\; 0$ property. One can also check that conservation laws (222) form Lenard scheme (k + 1)đI^(k) = kdI^{(k + 1)} Further let us focus on Frölicher-Nijenhuis geometry. Using formula (173) one can construct invariant Frölicher-Nijenhuis operator, out of generator of non-Noether symmetry of Toda chain. Operator constructed in this way has the form Ŕ_E = n∑s = 1p_s(dp_s ⊗ ∂∂q_s + dq_s ⊗ ∂∂p_s) − n − 1∑s = 1 e^{q_s − q_{s + 1}}dq_{s + 1} ⊗ ∂∂p_s + n − 1∑s = 1 e^{q_{s − 1} − q_s}dq_s ⊗ ∂∂p_s − ∑s > r (dp_s ⊗ ∂∂q_r − dp_r ⊗ ∂∂q_s) One can check that Frölicher-Nijenhuis torsion of this operator vanishes and it plays role of recursion operator for n-particle Toda chain in sense that conservation laws $I(k)$ satisfy recursion relation (k + 1)R_E(dI^(k)) = kdI^{(k + 1)} Thus non-Noether symmetry of Toda chain not only leads to n functionally independent conservation laws in involution, but also essentially enriches phase space geometry by endowing it with invariant Frölicher-Nijenhuis operator, bi-Hamiltonian system, bicomplex structure and Lax pair. Finally, in order to outline possible applications of Theorem 8 let us study action of non-Noether symmetry (220) on conserved quantities of Toda chain. Vector field $E$ defined by (220) generates one-parameter group of transformations (28) that maps arbitrary conserved quantity $J$ to J(z) = J + zJ⁽¹⁾ + z²2!J⁽²⁾ + z³3!J⁽³⁾ + ⋯ where J^(m) = (L_E)^mJ In particular let us focus on family of conserved quantities obtained by action of $g$_a = e^aL_E on total momenta of Toda chain J = n∑s = 1 p_s By direct calculations one can check that family $J(z)$, that forms orbit of non-Noether symmetry generated by $E$, reproduces entire involutive family of integrals of motion (222). Namely J⁽¹⁾ = L_EJ = ½ n∑s = 1 p_s² + n − 1∑s = 1 e^{q_s − q_{s + 1}} J⁽²⁾ = L_EJ⁽¹⁾ = (L_E)²J = 12 n∑s = 1p_s³ + 32n − 1∑s = 1 (p_s + p_{s + 1}) e^{q_s − q_{s + 1}} J⁽³⁾ = L_EJ⁽²⁾ = (L_E)³J = ¾ n∑s = 1p_s⁴ + 3n − 1∑s = 1(p_s² + 2p_sp_{s + 1} + p_{s + 1}²)e^{q_s − q_{s + 1}} + 32 n − 1∑s = 1 e^{2(q_s − q_{s + 1})} + 3n − 2∑s = 1 e^{q_s − q_{s + 2}} J^(m) = L_EJ^{(m − 1)} = (L_E)^mJ Involutivity of this set of conservation laws can be verified using Theorem 8. In particular one can notice that differential 1-form $s$ defined by E = W(s) (where $E$ is generator of non-Noether symmetry (220)) satisfies condition [W[W(s),W](s)] = 3[W(s)[W(s) ,W]] while conservation law $J$ defined by (231) has property W(L_W(s)dJ) = − [W(s),W](dJ) and thus according to Theorem 8 conservation laws (232) are in involution.

Korteweg-de Vries equation

Toda model provided good example of finite dimensional integrable Hamiltonian system that possesses non-Noether symmetry. However there are many infinite dimensional integrable Hamiltonian systems and in this case in order to ensure integrability one should construct infinite number of conservation laws. Fortunately in several integrable models this task can be effectively simplified by identifying appropriate non-Noether symmetry. First let us consider well known infinite dimensional integrable Hamiltonian system – Korteweg-de Vries equation (KdV). The KdV equation has the following form u_t + u_xxx + uu_x = 0 (here $u$ is smooth function of $(t,\; x)\; \in \; R2$). The generators of symmetries of KdV should satisfy condition E(u)_t + E(u)_xxx + u_xE(u) + uE(u)_x = 0 which is obtained by substituting infinitesimal transformation $u\; \to \; u\; +\; zE(u)\; +\; O(z2)$ into KdV equation and grouping first order terms. Later we will focus on the symmetry generated by the following vector field E(u) = 2u_xx + 23u² + 16u_xv + x2(u_xxx + uu_x) − t4(6u_xxxxx + 20u_xu_xx + 10 uu_xxx + 5u²u_x) (here $v$ is defined by $v$_x = u). If $u$ is subjected to zero boundary conditions $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )\; =\; 0$ then KdV equation can be rewritten in Hamiltonian form u_t = {h , u} with Poisson bivector field equal to W = + ∞∫− ∞dx δδu∧ {δδu}_x and Hamiltonian defined by h = + ∞∫− ∞ (u_x² − u³3) dx By taking Lie derivative of the symplectic form along the generator of the symmetry one gets second Poisson bivector [E , W] = W = + ∞∫− ∞dx ({δδu}_xx ∧ {δδu}_x + 23uδδu ∧ {δδu}_x) involved in bi-Hamiltonian structure of KdV hierarchy and proposed by Magri [58]. Now let us show how non-Noether symmetry can be used to construct conservation laws of KdV hierarchy. By integrating KdV it is easy to show that J⁽⁰⁾ = + ∞∫− ∞u dx is conserved quantity. At the same time Lie derivative of any conserved quantity along generator of symmetry is conserved as well, while taking Lie derivative of $J(0)$ along $E$ gives rise to infinite sequence of conservation laws $J(m)=\; (L$_E)^mJ⁽⁰⁾ that reproduce well known conservation laws of KdV equation J⁽⁰⁾ = + ∞∫− ∞u dx J⁽¹⁾ = L_EJ⁽⁰⁾ = ¼+ ∞∫− ∞u² dx J⁽²⁾ = (L_E)²J⁽⁰⁾ = 58+ ∞∫− ∞ (u³3 − u_x²) dx J⁽³⁾ = (L_E)³J⁽⁰⁾ = 3516+ ∞∫− ∞ (536u⁴ − 53uu_x² + u_xx²) dx J^(m) = (L_E)^mJ⁽⁰⁾ Thus the conservation laws and bi-Hamiltonian structures of KdV hierarchy are related to the non-Noether symmetry of KdV equation.

Nonlinear water wave equations

Among nonlinear partial differential equations that describe propagation of waves in shallow water there are many remarkable integrable systems. We have already discussed case of KdV equation, that possess non-Noether symmetries leading to the infinite sequence of conservation laws and bi-Hamiltonian realization of these equations, now let us consider other important water wave systems. It is reasonable to start with dispersive water wave system [73],[74], since many other models can be obtained from it by reduction. Evolution of dispersive water wave system is governed by the following set of equations u_t = u_xw + uw_x v_t = uu_x − v_xx + 2v_xw + 2vw_x w_t = w_xx − 2v_x + 2ww_x Each symmetry of this system must satisfy linear equation E(u)_t = (wE(u))_x + (uE(w))_x E(v)_t = (uE(u))_x − E(v)_xx + 2(wE(v))_x + 2(vE(w))_x E(w)_t = E(w)_xx − 2E(v)_x + 2(wE(w))_x obtained by substituting infinitesimal transformations u → u + zE(u) + O(z²) v → v + zE(v) + O(z²) w → w + zE(w) + O(z²) into equations of motion (245) and grouping first order (in $a$) terms. One of the solutions of this equation yields the following symmetry of dispersive water wave system E(u) = uw + x(uw)_x + 2t(uw² − 2uv + uw_x)_x E(v) = 32u² + 4vw − 3v_x + x(uu_x + 2(vw)_x − v_xx) + 2t(u²w − uu_x − 3v² + 3vw² − 3v_xw + v_xx)_x E(w) = w² + 2w_x − 4v + x(2ww_x + w_xx − 2v_x) − 2t(u² + 6vw − w³ − 3ww_x − w_xx)_x and it is remarkable that this symmetry is local in sense that $E(u)$ in point $x$ depends only on $u$ and its derivatives evaluated in the same point, (this is not the case in KdV where symmetry is non local due to presence of non local field $v$ defined by $v$_x = u). Before we proceed let us note that dispersive water wave system is actually infinite dimensional Hamiltonian dynamical system. Assuming that $u,\; v$ and $w$ fields are subjected to zero boundary c