On integrable systems outside Nijenhuis and Haantjes geometry
aa r X i v : . [ n li n . S I] F e b On integrable systems outside Nijenhuis and Haantjesgeometry
A. V. Tsiganov,
St.Petersburg State University, St.Petersburg, Russiae-mail: [email protected]
Abstract
We present three new examples of the integrable systems related to the Killing tensorswith non-zero Nijenhuis and Haantjes tensors on the four-dimensional Euclidean space R . The corresponding four integrals of motion are polynomials of the first, second, andfourth-order in momenta. Keywords:
Killing tensors, integrable systems, separation of variables
In 1951-55 Nijenhuis and Haantjes proposed construction of the integrable distributions as thecomplements to eigenvector fields X i of a (1,1) tensor field A on some n -dimensional manifold[6, 9]. This construction is applicable if • all n eigenvalues of A are real, distinct, and functionally independent; • either Nijenhuis tensor N A is equal to zero or Haantjes tensor H A is equal to zero.Now Nijenhuis and Haantjes tensors can be found in various parts of mathematics, mathematicalphysics, and classical mechanics, but the overwhelming majority of applications is related tothe vanishing of one of these tensors N A ( u, v ) = 0 or H A ( u, v ) = 0 (1.1)for any vector fields u and v , which guarantees integrability of all the distributions spannedby the n − X i , i.e. with geometry of the so-called Nijenhuis and Haantjesmanifolds.We want to study (1,1) tensor fields A with the simultaneously non-zero Nijenhuis andHaantjes tensors N A ( u, v ) = 0 , and H A ( u, v ) = 0 (1.2)and the corresponding integrable systems outside Nijenhuis and Haantjes geometry. Examplesof such (1,1) Killing tensors in R and the corresponding integrable Hamiltonian systems arediscussed in [10, 11].In this note, we discuss a few generalizations of these Killing tensors in configurationalspace R and present new integrable Hamiltonian systems on the phase space T ∗ R outside ofthe Nijenhuis and Haantjes geometry. In this section, we review basic notions concerning the Killing, Nijenhuis and Haantjes tensors,following the original papers [5, 6, 9] and the related ones [1, 3, 4, 8].1et us consider a Riemannian or pseudo-Riemannian manifold M , dim M = n endowedwith coordinates q = ( q , . . . , q n ) and metric g( q ), which defines Hamiltonian on the cotangentbundle T ∗ M T = n X i,j =1 g i,j ( q ) p i p j . Metric establishes an isomorphism between the tangent space and its dual. This identifiesco- and contravariant tensor components via lowering or rising indices using the metric. Inparticular a tensor A of valency two can be identified with (0, 2), (2,0), or (1, 1) tensor field.Any tensor field A of valency two defines the polynomial of second order in momenta T = n X i,j =1 A ij p i p j . (1.3)If these polynomials T and T are in the involution { T , T } = 0 (1.4)with respect to the canonical Poisson bracket on the phase space T ∗ M { p i , p j } = { q i , q j } = 0 , { q i , p j } = δ ij , i, j = 1 , . . . , n, (1.5)then A is a Killing tensor. Equation (1.4) is equivalent to the Killing equation on the configu-rational space M ∇ i A jk + ∇ j A ki + ∇ k A ij = 0 , (1.6)where ∇ is compatible with the metric Levi-Civita connection.Integrability of dynamical systems on the phase space T ∗ M can be related with integrabilityof the eigen-distributions of A [1, 5, 8], which is described by Nijenhuis and Haantjes tensorson the configurational space M : N A ( u, v ) = A [ u, v ] + [ Au, Av ] − A ([ Au, v ] + [ u, Av ]) , and H A ( u, v ) = A N ( u, v ) + N A ( Au, Av ) − A ( N A ( Au, v ) + N A ( u, Av )) , where u, v are arbitrary vector fields and [ ., . ] denotes the commutator of two vector fields.On the local coordinate chart q = ( q , . . . , q n ) the alternating (1, 2) Nijenhuis tensor takesthe form ( N A ) ijk = n X α =1 ∂A ik ∂q α A αj − ∂A ij ∂q α A αk + (cid:18) ∂A αj ∂q k − ∂A αk ∂q j (cid:19) A iα ! The corresponding Haantjes tensor looks like( H A ) ijk = n X α,β =1 (cid:16) A iα A αβ ( N A ) βjk + ( N A ) iαβ A αj A βk − A iα (cid:16) ( N A ) αβk A βj + ( N A ) αjβ A βk (cid:17)(cid:17) . Properties of these tensors are discussed in [3, 4].
Let us consider Euclidean space R with Cartesian coordinates q , q , q and metricg = . (2.1)2n this case Killing equation (1.6) has a partial solution depending on the parameter aA = − q q q − aq − aq aq , det A = aq ( aq − q )4 . (2.2)It is easy to prove the following statement. Proposition 1
Metric g (2.1) and Killing tensor A (2.2) define integrable Hamiltonian systemwith functionally independent integrals of motion T = n X i,j =1 g ij p i p j = p + p + p ,T = n X i,j =1 A ij p i p j = p ( p q − p q ) + ap ( p q − p q ) ,T = p a p , (2.3) in the involution with respect to the Poisson bracket { q i , q j } = 0 , { q i , p j } = g ij , { p i , p j } = 0 . (2.4) Here a is an arbitrary number. Thus, in this case properties of the Nejenhuis and Haantjes tensors do not affect on the Liouvilleintegrability of the superintegrabile or degenerate Hamiltonian T similar to [2, 7].Let us now consider non-degenerate Hamiltonians H = T + V ( q , q , q ) , H = T + U ( q , q , q ) , (2.5)and try to solve equation { H , H } = 0 (2.6)for functions V and U on the configurational space. Proposition 2
At generic a , equation (2.6) has one separable in Cartesian coordinates solution V a ( q , q , q ) = f ( q ) + f ( q ) + f ( q ) ≡ c ( q + 4 q + q ) + 4 c q + c q + c aq , whereas the second function is equal to U a ( q , q , q ) = c q ( q − aq ) + c ( q − aq ) − c q q + c q q . Non-separable solutions exist if and only if parameter a is the solution of one of the equations ( H A ) kij = − ( H A ) kji = 0 , i = j = k , i, j, k = 1 , , , which have the following four solutions . a = − , ( H A ) ijk = 0 , i, j, k = 1 , , , . a = − , ( H A ) = − ax x x a + 1)( a + 2) = 0 , . a = 1 , ( H A ) = ax x x a + 1)( a −
1) = 0 , . a = − / H A ) = ax x x a + 1)(2 a + 1) = 0 . H , which are in the involution with Hamiltonians H , (2.5) { H , H } = { H , H } = 0 . • If a = −
1, then H A = 0, but eigenvalues of A are functionally dependent λ , = − q ± q q + q + q , λ = − q = λ + λ , the and, therefore, the standard Nijenhuis-Haantjes construction of the integrable distri-butions does not work. Nevertheless, it is easy to construct a family of integrable systemsusing the standard Eisenhart method [5], see details in [10, 11]. Indeed, if V ( q ) is one ofthe potentials separable in coordinates v , and angular coordinate ϕ defined by q = v v sin ϕ , q = v − v , q = v v cos ϕ , (2.7)then third integral of motion H = p ϕ is in involution with H , (2.5). • If a = 1, then non-separable in Cartesian coordinates part of solution is equal to V ( q ) = α (cid:16) q + 6 q q + q + 12 q ( q + q ) + 16 q (cid:17) , (2.8)whereas potential U ( q ) has the form U ( q ) = 2 αq ( q − q )( q + 2 q + q )) . The third integral of motion is the polynomial of the fourth-order in momenta. H = p p + 2 α n X i,j S ij ( q ) p i p j + 4 α W ( q )where S ( q ) = q q − q q q q q ( q + 4 q + q ) − q q q q q − q q q q q ( q + 4 q + q ) − q q q q q and W ( q ) = q q ( q + 2 q + q ) . In [10, 11] this third integral is related to a pair of the Killing tensors, which are conformaltensors concerning A . • If a = −
2, then V ( q ) = α ( q + 4 q + 4 q ) q , U ( q ) = − αq ( q + 2 q + 2 q ) q (2.9)and third integral of motion is polynomial of fourth order in momenta H = p (cid:16) ( p q − p q ) + ( p q − p q ) (cid:17) + 2 αq n X i,j S ij ( q ) p i p j + 4 α q W ( q )4here S = q + q )( q + 2 q + 2 q ) − q q ( q + 2 q + 2 q ) − q q ( q + 2 q + 2 q ) − q q ( q + 2 q + 2 q ) 2 q q − q q q − q q ( q + 2 q + 2 q ) − q q q q q and W ( q ) = ( q + q )( q + 2 q + 2 q ) , see details in [10, 11]. • If a = − /
2, then non-separable in Cartesian coordinates part of solution is equal to V = α ( q + 4 q + 4 q ) q . This solution coincides with solution at a = − q ↔ q .Thus, Killing tensor (2.2) generate three integrable systems outside Nijenhuis and Haantjesgeometry. In all these cases we have a pair of eigenvector fields X i and X j which are orthogonalwith respect to Haantjes operator because H A ( X i , X j ) = ( H A ) kij X k = 0 , i = j = k . Below we discuss similar conditions for n = 4 case, when we have to study triads of eigenvectorfields X , . . . , X n − according to Haantjes [6] and Nijenhuis [9]. Let us consider Euclidean space R with Cartesian coordinates q , q , q , q and metricg = Deformation of the Killing tensor (2.2) A = − q q b q + b q + b q + d q − aq b q − b q + b q + d − aq aq − b q − b q + b q + d b q + b q + b q + d b q − b q + b q + d − b q − b q + b q + d − b q − b q − b q + d is a partial solution of the corresponding Killing equation (1.6) depending on eleven parameters a , b , . . . , b and d , . . . d .We want to study relations between Haantjes tensor H A on the configurational space andintegrable system on the phase space with fixed two Hamiltonians H = n X i,j =1 g ij p i p j + V ( q ) , H = n X i,j =1 A ij p i p j + U ( q ) , (3.1)and two unknown independent integrals of motion H , in the involution with H , . In a genericcase, integrals of motion H , can not be polynomials of the first or second order of momenta.5elow, for brevity, we consider a special solution which is a’priori associated with integralof motion H = p q − p q : A = − q q q − aq bq − aq aq bq − bq (3.2)At V ( q ) = 0 we have degenerate or superintegrable Hamiltonian H (3.1) and it is easy to provethe following statement. Proposition 3
Metric g (2.1) and Killing tensor A (2.2) define integrable Hamiltonian systemwith functionally independent integrals of motion T = n X i,j =1 g ij p i p j = p + p + p + p ,T = n X i,j =1 A ij p i p j = p ( p q − p q ) + ap ( p q − p q ) + 2 p ( p q − p q ) ,T = p p a , T = 4 b p ln p in the involution with respect to the Poisson bracket (2.4). Here a and b are arbitrary numbers. In this case properties of the Nejenhuis and Haantjes tensors do not affect the Liouville inte-grability of the degenerate Hamiltonian T in a suitable domain of the phase space T ∗ R .Let us consider non-degenerate Hamiltonians H and H (3.1) which Poisson commute toeach other. Proposition 4
At generic a and b equation { H , H } = 0 (2.6) has one separable in Cartesiancoordinates solution V ab = c ( q + 4 q + q + q ) b + c q + c q + c aq + c bq whereas the second function is equal to U ab = c q ( q − aq − bq ) b + c ( q − aq + 2 bq )4 − c q q − c q q − c q q . Non-separable solutions exist if and only if parameters a and b are solution of a suitable subsetof the algebraic equations ( H A ) kij = − ( H A ) kji = 0 , i = j = k ∈ { , , , } , (3.3) similar to the 3D case. Let us write out all the entries of Haantjes tensor from (3.3)( H A ) = − a ( a + 1)( a + 2) q q q , ( H A ) = a ( a − a + 1) q q q , ( H A ) = b ( b − b − q q q , ( H A ) = − b (2 b − b + 1) q q q , ( H A ) = 3 ab ( a + 2 b ) q q q , ( H A ) = − ab ( a − b )( a + 2 b ) q q q , (3.4)6nd ( H A ) = a ( a + 1)(2 a + 1) q q q , ( H A ) = − b (4 b − b − q q q , ( H A ) = 3 ab (2 b − q q q , ( H A ) = − ab ( a + 1) q q q , ( H A ) = − ab ( a + 2 b )( a + b ) q q q , ( H A ) = ab ( a + 4 b )( a + 2 b ) q q q . (3.5)Solving these equations we obtain pairs of parameters ( a, b ) and the corresponding separableor non-separable solutions of the equation { H , H } = 0 (2.6).Below we present only four solutions associated with non-separable potentials V ( q ). If a = − b = 1 /
2, then Haantjes tensor is equal to zero H A = 0, but its eigenvalues of A are functionally dependent to each other λ , = − q ± q q + q + q + q , λ = λ = − q = λ + λ . Similar to (2.7) we can add two angular coordinates ϕ = arctan q q , ψ = arctan q q to separated variables λ , and take one of the potentials V ( q ) separable in these coordinates.As a result, we obtain a family of integrable system with two additional integrals of motion H = p ϕ , H = p ψ in the involution with Hamiltonians H , (3.1). This family of integrable systems can be ob-tained in the framework of the standard Eisenhart approach [5]. If a = 1 and b = 1 /
2, then we have seven zero and five non-zero entries in (3.4-3.5) andnon-separable part of potential reads as V ( q ) = α (cid:16) q + 12 q q + 6 q q + 2 q q + 16 q + 12 q q + 12 q q + q + 6 q q + q (cid:17) . In this case U ( q ) = 2 αq ( q − q + q )( q + 2 q + q + q )and Hamiltonians H , (3.1) Poisson commute with integrals of motion H = p q − p q and H = p ( p + p ) + 2 α X i,j =1 S ij ( q ) p i p j + α W ( q ) , which are polynomials of first and fourth-order in momenta, respectively. Here S ( q ) = q ( q + q ) − q q q q q ( q +4 q + q + q ) − q q q − q q q q ( q + q ) − q q ( q + q ) − q q q q q ( q +4 q + q + q ) − q q ( q + q ) − q q ( q + q ) q q ( q +4 q + q + q ) − q q q − q q q q q ( q +4 q + q + q ) 2 q ( q + q ) and W ( q ) = 4 q ( q + q )( q + 2 q + q + q ) . This integrable system is a natural generalization of the 3D system (2.8).7 .3 Third solution If a = − b = 1 /
2, then we have eight zero and four non-zero entries in (3.4-3.5) and V ( q ) = α ( q + 4 q + 4 q + q )( q + q ) , U ( q ) = − αq ( q + 2 q + 2 q + q )2( q + q ) ) . In this case Hamiltonians H , (3.1) Poisson commute with integrals of motion H = L and H = ( p + p ) (cid:16) L + L + L + L (cid:17) − ( p + p ) L + α q + q ) X i,j =1 S ij ( q ) p i p j + α W ( q )which are polynomials of first and fourth-order in momenta, respectively. Here L ij = p q j − p j q i , W ( q ) = α ( q + q )( q + 2 q + 2 q + q ) q + q ) and symmetric matrix S is equal to S = s − q q ( q +2 q +2 q + q ) − q q ( q +2 q +2 q + q ) − q q ( q +4 q +4 q + q ) − q q ( q +2 q +2 q + q ) 4 q ( q + q ) − q q ( q + q ) − q q ( q +2 q +2 q + q ) − q q ( q +2 q +2 q + q ) − q q ( q + q ) 4 q ( q + q ) − q q ( q +2 q +2 q + q ) − q q ( q +4 q +4 q + q ) − q q ( q +2 q +2 q + q ) − q q ( q +2 q +2 q + q ) s , where s = 4 q q + 4 q q + q q + 8 q + 16 q q + 8 q q + 8 q + 8 q q + q ,s = q + 8 q q + 8 q q + q q + 8 q + 16 q q + 4 q q + 8 q + 4 q q . This integrable system is the first non-trivial generalization of the 3D system (2.9). If a = − b = 1, then we have nine zero and three non-zero entries in (3.4-3.5) and V ( q ) = α ( q + 4 q + 4 q + 4 q ) q , U ( q ) = − αq ( q + 2 q + 2 q + 2 q )2 q . In this case Hamiltonians H , (3.1) Poisson commute with integrals of motion H = L and H = p (cid:16) L + L + L (cid:17) + 2 αq
61 4 X i,j =1 S ij ( q ) p i p j + α W ( q )Here W ( q ) = ( q + q + q )( q + 2 q + 2 q + 2 q ) q , and symmetric matrix S is equal to S = s − q q ( q +2 q +2 q +2 q ) − q q ( q +2 q +2 q +2 q ) − q q ( q +2 q +2 q +2 q ) − q q ( q +2 q +2 q +2 q ) 2 q ( q + q ) − q q q − q q q − q q ( q +2 q +2 q +2 q ) − q q q q ( q + q ) − q q q − q q ( q +2 q +2 q +2 q ) − q q q − q q q q ( q + q ) where s = 2( q + q + q )( q + 2 q + 2 q + 2 q ) . This integrable system is the second more trivial generalization of the 3D system (2.9).Summing up, we obtain four non-trivial integrable systems associated with Haantjes tensorswith the following properties of the entries ( H A ) kij , i = k = k :8 all entries vanishing, a = − b = 1 / • five entries non vanishing, a = 1 and b = 1 / • four entries non vanishing, a = − b = 1 / • three entries non vanishing, a = − b = 1;Other suitable values of ( a, b ) generates the same non-separable in Cartesian coordinates solu-tions V ( q ) up to changes of q ↔ q , q ↔ q and q ↔ q .Relation between a number of non-vanishing entries of the Haantjes tensor on the configu-rational space and the existence of non-trivial integrable systems on the phase space is an openquestion. There are many curvilinear orthogonal coordinate systems with so-called angular coordinateswhich can be obtained from elliptic coordinate system1 + n X i =1 q i z − δ i = Q ni =1 ( z − u i ) Q nj =1 ( z − δ i ) , δ < δ < · · · < δ n (4.6)or parabolic coordinate system in R n z − q n + n − X i =1 q i z − δ i = Q ni =1 ( z − u i ) Q n − j =1 ( z − δ i ) , δ < δ < · · · < δ n − by letting two or more of the parameters δ i coincide [8]. For instance, if n = 3 and δ = δ in(4.6) we have 1 + r z − δ + q z − δ = ( z − u )( z − u )( z − δ )( z − δ ) , r = q + q . This determines a mapping r = f ( u , u ) and q = f ( u , u ) that defines elliptic coordinatesin R = { r, q } . In order to get an orthogonal coordinate system in R , we have to complement r with an angular coordinate ϕ in the ( q , q ) - plane, for instance through q = f ( u , u ) cos ϕ , q = f ( u , u ) sin ϕ , q = f ( u , u ) . These equations define the prolate spherical coordinate system ( u , u , ϕ ). When δ = δ , weget in a similar manner the oblate spherical coordinate system, and when δ = δ = δ , weobtain spherical coordinate system in R .For all these orthogonal coordinate systems we can construct integrable Hamiltonian sys-tems in the framework of the Eisenhart construction with a complete set of integrals of motion,which are polynomials of second order in momenta, using metric g, characteristic Killing tensor A and momenta associated with the angular coordinates. The corresponding Killing tensor A satisfies one of the Nijenhuis-Haantjes conditions N A ( u, v ) = 0 or H A ( u, v ) = 0 , but it has functionally dependent eigenvalues and, therefore, the Nijenhuis-Haantjes construc-tion of the integrable distributions as the complements to eigenvector fields does not work inthis case.Starting with the Killing tensor with dependent eigenvalues we can construct Killing tensor A with non-vanishing Nijenhuis and Haantjes tensors [10, 11] N A ( u, v ) = 0 , and H A ( u, v ) = 0 .
9e suppose that metric g and this Killing tensor A also determine integrable Hamiltoniansystems with integrals of motion, which are polynomials of the higher-order in momenta, if andonly if a suitable number of the entries ( H A ) kij are equal to zero.In this note, we obtain integrable Hamiltonian systems outside of the Nijenhuis and Haan-tjes geometry by using a brute force method. It will be interesting to formulate a theory thatallows us to directly get integrals of motion similar to the Eisenhart or Lax constructions. Theseparation of variables for these systems is also an open question.The work was supported by the Russian Science Foundation. References [1] I. Benn,
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