Transforming Stäckel Hamiltonians of Benenti type to polynomial form
Jean de Dieu Maniraguha, Krzysztof Marciniak, Célestin Kurujyibwami
aa r X i v : . [ n li n . S I] J a n Transforming St¨ackel Hamiltonians of Benentitype to polynomial form
Jean de Dieu Maniraguha † , Krzysztof Marciniak ‡ and C´elestin Kurujyibwami † † College of Science and Technology, University of Rwanda, P.O. Box: 3900, Kigali,Rwanda
E-mail: [email protected]: [email protected] ‡ Department of Science and Technology, Campus Norrk¨oping, Link¨oping University,601-74 Norrk¨oping, Sweden,
E-mail: [email protected] January 2, 2021
Abstract
In this paper we discuss two canonical transformations that turn St¨ackel sepa-rable Hamiltonians of Benenti type into polynomial form: transformation to Vi`etecoordinates and transformation to Newton coordinates. Transformation to New-ton coordinates has been applied to these systems only very recently and in thispaper we present a new proof that this transformation indeed leads to polynomialform of St¨ackel Hamiltonians of Benenti type. Moreover we present all geometricingredients of these Hamiltonians in both Vi`ete and Newton coordinates.
Keywords and phrases: Hamiltonian systems, Hamilton-Jacobi theory, St¨ackel sys-tems, Benenti systems, polynomial form, Vi`ete coordinates, Newton coordinates
The aim of this paper is to investigate two canonical transformations of the phase spaceto coordinates in which the so called St¨ackel separable systems of Benenti type attain apolynomial form, as well as to present all geometric objects, related with such systems(the pseudo-Riemannian metric tensor and its Killing tensors as well as the conformalKilling tensor, present in the Hamiltonians of the system) in these new coordinates.St¨ackel systems constitute an important family of quadratic in momenta Hamiltoniansystems that are separable, in the sense of Hamilton-Jacobi theory, in orthogonal coor-dinates. These systems were introduced by Paul St¨ackel in [9], where he presented theconditions for separability of Hamilton-Jacobi equation of a natural Hamiltonian system(that is a system of the form H = K + V where K is a quadratic in momenta formand V is a potential defined on the underlying configurational space of the system) inorthogonal coordinates, see for example [11] for a comprehensive review of this subject.St¨ackel systems can most conveniently be obtained from the separation relations [10] thatare linear in the Hamiltonians H i and quadratic in momenta µ i . Further specification ofingredients in these separation relations lead to so called Benenti systems (see the nextsection for all the necessary definitions and details).The obtained St¨ackel (or Benenti) Hamiltonians H j , as well as their geometric com-ponents, are usually given by complicated rational functions, if written in the canonicalcoordinates in which they were originally created through separation relations. In litera-ture, two maps turning Benenti systems into polynomial form are known: the map to the1o called Vi`ete coordinates [2] and the map to the so called Newton coordinates [7], thesecond map discovered only recently. In this paper we improve the results obtained in [7]by presenting an alternative, much simpler, proof of its main result using the direct mapbetween Vi`ete coordinates and Newton coordinates. We also present the explicit formof all the geometric structures that are present in the Benenti Hamiltonians in Newtoncoordinates. These results are new. Consider a 2 n -dimensional manifold M equipped with a Poisson bracket π . Supposealso that ( λ, µ ) = ( λ , . . . , λ n , µ , . . . , µ n ) are global Darboux coordinates on M (i.e, { λ i , λ j } = { µ i , µ j } = 0 for all i, j = 1 , . . . , n while { λ i , µ j } = δ ij ). A set of algebraicequations of the form ϕ i ( λ i , µ i , a , . . . , a n ) = 0, i = 1 , . . . , n (1)is called separation relations if it is globally solvable (except possibly for a union of lowerdimensional submanifolds) with respect to the parameters a j ∈ R .Among all possible separations relations (1), a natural subclass consists of the sepa-ration relations that are linear in the Hamiltonians H k : n X k =1 S ik ( λ i , µ i ) H k = ψ i ( λ i , µ i ) , i = 1 , . . . , n. (2)Here S ik and ψ i are arbitrary smooth functions of two arguments ( λ i , µ i ). The relations (2)are called the generalized St¨ackel separation relations and the related dynamical systems,obtained by solving (2) with respect to H k , are called the generalized St¨ackel systems . Thematrix S = [ S ik ( λ i , µ i )] is called a generalized St¨ackel matrix . Although the restriction toseparation relations linear in H k seems to be very strong, it appears that an overwhelmingmajority of all separable systems considered in the literature falls into various subclassesof this class. The most important class of systems in (2) is the class of classical St¨ackelsystems , that is systems with the matrix S being a St¨ackel matrix (so that S ik = S ik ( λ i ))and with ψ i being quadratic in momenta µ : S ik ( λ i , µ i ) = S ik ( λ i ) , ψ i ( λ i , µ i ) = 12 f i ( λ i ) µ i − ϕ i ( λ i ) , so that the separation relations (2) attain the form ϕ i ( λ i ) + n X k =1 S ik ( λ i ) H k = f i ( λ i ) µ i , i = 1 , . . . , n. (3)The relations (3) are called St¨ackel separation relations . A particular St¨ackel system isthus defined by a choice of the St¨ackel matrix S ik ( λ i ) and by a choice of 2 n functions f i and ϕ i . Solving the relations (3) with respect to H k we obtain n quadratic in momentafunctions (Hamiltonians) on M H r = 12 µ T A r µ + V r ( λ ), r = 1 , . . . , n, (4)where A r are n × n matrices given by A r = diag (cid:0) f ( λ ) (cid:0) S − (cid:1) r , . . . , f n ( λ n ) (cid:0) S − (cid:1) rn (cid:1) , r = 1 , . . . , n.
2s the Hamiltonians (4) are defined through separation relations, they are in involutionwith respect to the canonical Poisson bracket on M .There is a natural geometric interpretation of St¨ackel systems given by (4). If wefactorize A r as A r = K r G , where G = A = diag (cid:0) f ( λ ) (cid:0) S − (cid:1) , . . . , f n ( λ n ) (cid:0) S − (cid:1) n (cid:1) and K r = diag (cid:18) ( S − ) r ( S − ) , . . . , ( S − ) rn ( S − ) n (cid:19) , r = 1 , . . . , n (so that K = I ) then we can interpret the matrix G as a contravariant form of a metrictensor on a manifold Q such that M = T ∗ Q is the cotangent bundle to Q . The corre-sponding covariant metric tensor will be denoted by g so that gG = I . It can be shownthat the matrices K r are then (1 , G . For a fixed St¨ackelmatrix S we have thus the whole family of metrics G parametrized by n arbitrary func-tions f i of one variable λ i . The tensors K r are then Killing tensors for any metric from thisfamily. Thus, the St¨ackel Hamiltonians H r in (4) are geodesic Hamiltonians of a Liouvilleintegrable system in the Riemannian space ( M , g ). Further, due to the linearity of theseparation relations (3), the functions V r ( λ ) on Q are defined by the following separationrelations n X k =1 S ik ( λ i ) V k = − ϕ i ( λ i ) , i = 1 , . . . , n, and are called in literature separable potentials on Q . From now on we restrict ourselves to the case the St¨ackel matrix S in (3) is of the veryparticular form S ij = λ n − ji or explicitly: S = λ n − λ n − . . . λ n − n λ n − n . . . (5)thus being a Vandermonde matrix. The corresponding St¨ackel systems are thus definedby separation relations of the form ϕ i ( λ ) + n X j =1 λ n − ji H j = 12 f i ( λ i ) µ i i = 1 , . . . , n, (6)and are called in literature Benenti systems . Benenti systems have been studied much inliterature recently, see for example [1, 3] and references therein.The inverse of S as given by (5) is given by the following lemma. Lemma 1. If S is the n × n Vandermonde matrix given by S ij = λ n − ji then (cid:2) S − (cid:3) ij = − j ∂ρ i ∂λ j , where ρ i = ( − i σ i ( λ ) , ∆ j = Q k = j ( λ j − λ k ) and where σ r ( λ ) are elementary symmetric polynomials.
3y definition σ i ( λ ) = X ≤ j <... Suppose that we change the position coordinates on the base manifold Q through themap q i = ρ i ( λ ) i = 1 , . . . , n, (19)where, as in Lemma 1, ρ i ( λ ) = ( − i σ i ( λ ). This map induces the map (point transfor-mation) on T ∗ Q : p = (cid:0) J − V (cid:1) T µ, (20)where J V is the Jacobian of the map (19):( J V ) ij = ∂ρ i ∂λ j . (21)Let us find an explicit form of (20). To do this we need the following lemma. Lemma 2. Denote by k i the i -th column of an n × n nondegenerate matrix A : A = ( k | k | . . . | k n ) and by r j the j -th row of its inverse A − = r r ... r n . Then, if α i ∈ R for i = 1 , . . . , n ( α k | α k | . . . | α n k n ) − = r /α r /α ... r n /α n . r i k j = δ ij . An analogous lemma isof course true if we consider rows of A instead of its columns. Combining lemmas 1 and2 we obtain that (cid:0) J − V (cid:1) ij = − λ n − ji ∆ i (22)and thus the map (20) can be written as p i = − n X k =1 λ n − ik ∆ k µ k , i = 1 , . . . , n. (23)The coordinates ( q, p ) defined by (19) and (23) are called Vi`ete coordinates . To summarize,the map ( λ, µ ) → ( q, p ) from separation coordinates to Vi`ete coordinates is given by q i = ρ i ( λ ), p i = − n X k =1 λ n − ik ∆ k µ k , i = 1 , . . . , n. (24)Being a point transformation, the map (24) is a canonical map which means that Vi`etecoordinates are Darboux (canonical) coordinates as well: { q i , q j } = { p i , p j } = 0 , { q i , p j } = δ ij . Let us now investigate the structure of Benenti Hamiltonians (10) in Vi`ete coordinates( q, p ). The Hamiltonians (10) are of course written in tensor form so that in Vi`ete coor-dinates H r ( q, p ) = 12 p T K r ( q ) G ( q ) p + V r ( q ) , r = 1 , . . . , n (25)where, by transformation laws for tensors, K r ( q ) = J V K r ( J V ) − , G ( q ) = J V G ( J V ) T . (26)The first formula in (26) yields, after some calculation( K r ( q )) ij = q i − j + r − , i ≤ j and r ≤ j − q i − j + r − , i > j and r > j . (27)Here and throughout the whole section we use the convention that q = 1 and q k = 0for k < k > n . Thus, all the K r ( q ) are linear in q -variables. Further, forthe monomial case f ( λ i ) = λ mi with m ∈ { , . . . , n + 1 } we can obtain from the secondformula in (26) that G ijm ( q ) = q i + j + m − n − , i, j = 1 , . . . , n − m − q i + j + m − n − , i, j = n − m + 1 , . . . , n m = 0 , . . . , n (28) G ijm ( q ) = q i q j − q i + j , i, j = 1 , . . . , n , m = n + 1 . The formulas (27) and (28) can alternatively be obtained with the help of the special con-formal Killing tensor L by using the formulas (14) and (13), respectively, and the fact that7he tensor L can be easily calculated in Vi`ete coordinates through tensor transformationlaw L ( q ) = J V L ( J V ) − . We obtain L ij ( q ) = − δ j q i + δ i +1 j that is L ( q ) = − q − q n . (29)Note therefore that L happens to have the same form in q -coordinates as the recursionmatrix (18). This seems to be a pure coincidence without any deeper meaning; we stressagain that R in (18) is not a tensor. In any case, due to the fact that all the entries in L are linear in q i we see that all the entries in G m are linear in q i for m = 0 , . . . , n + 1,quadratic in q i for m = n + 1 and higher order polynomials for higher m . Moreover, by(27), all entries in K r ( q ) are linear in q i . Using all these facts and the formula (12) weobtain the following important corollary: Corollary 3. If f is a polynomial in (11), then the geodesic parts of Benenti Hamiltonians(25) have a polynomial form. Moreover, if the right hand side of (15) is a polynomial, thenby the recursive relations (17)-(18) also the potentials V r in the Benenti Hamiltonians (10)are in this case polynomials in q i . Thus, in such a case, the whole Hamiltonians H r ( q, p ) (and not just their geodesic parts) are polynomials. Example 4. Consider the case n = 2, f ( λ ) = 1 (i.e. a purely monomial situation with m = 0 in (28), so that G = G ) and ϕ ( λ ) = λ . Then the separation curve (16) becomes λ + λH + H = 12 µ and yields the Hamiltonians H i in the explicit form H = 12( λ − λ ) ( µ − µ ) − ( λ + λ λ + λ ) H = 12( λ − λ ) ( λ µ − λ µ ) + λ λ ( λ + λ )so both Hamiltonians are rational functions of separation coordinates ( λ, µ ). The aboveHamiltonians have exactly the form (10) with the metric G = G = diag (cid:18) , (cid:19) ,and with the Killing tensors (8) given explicitly by: K = I , K = − diag( λ , λ ) . The map (24) to Vi`ete coordinates has the explicit form: q = − ( λ + λ ), q = λ λ ,p = 1 λ − λ ( λ µ − λ µ ), p = 1 λ − λ ( µ − µ ) . 8n elementary calculations shows that H i in these variables attain the form H ( q, p ) = 12 q p + p p − q + q H ( q, p ) = 12 p + q p p + 12 q p − p q − q q which is in agreement with (28) and (27). Explicitly: G ( q ) = (cid:18) q (cid:19) , K ( q ) = I , K ( q ) = (cid:18) − q q (cid:19) . Thus, the Hamiltonians H r become polynomial in Vi`ete coordinates ( q, p ). Example 5. Consider the case n = 3, f ( λ ) = λ (so that m = 1 in (28) and thus G = G )and ϕ ( λ ) = λ . Then the separation curve (16) becomes λ + λ H + λH + H = 12 λµ . Solving the corresponding separation coordinates yields the Benenti Hamiltonians (10)with the metric G = LG with L = diag ( λ , λ , λ )so that G = LG = diag (cid:18) λ ∆ , λ ∆ , λ ∆ (cid:19) and with the Killing tensors (8) given explicitly by K = I , K = diag ( λ + λ , λ + λ , λ + λ ) ,K = − diag ( λ λ , λ λ , λ λ ) , while the potentials V r = V (5) r have the form V (5)1 = λ + λ + λ + λ λ + λ λ + λ λ + λ λ + λ λ + λ λ + λ λ λ ,V (5)2 = λ λ + λ λ + λ λ + 2 λ λ λ + λ λ + λ λ + 2 λ λ λ + 2 λ λ λ + λ λ + λ λ + λ λ + λ λ ,V (5)3 = λ λ λ (cid:0) λ + λ + λ + λ λ + λ λ + λ λ (cid:1) . The map (24) to Vi`ete coordinates has now the form q = − ( λ + λ + λ ), q = λ λ + λ λ + λ λ , q = − λ λ λ and p p p = (cid:0) J − V (cid:1) T µ µ µ with J V and J − V given by (4.1) and (22) respectively. Explicitly J V = − − − λ + λ λ + λ λ + λ − λ λ − λ λ − λ λ J − V = − λ ∆ λ ∆ λ ∆ λ ∆ λ ∆ λ ∆ . An elementary calculation shows that H i in these variables attain the form H ( q, p ) = 12 q p + p p − q p + q − q q + q ,H ( q, p ) = 12 p − q p − q q p + q p p − q p p + q q − q q − q ,H ( q, p ) = − q p − q q p − q p p − q q p p + q q − q q , which is in agreement with (28) and (27). Explicitly: G ( q ) = q q q , K ( q ) = I , K ( q ) = − q q − q q ,K ( q ) = − q q − q q , while the tensor L attains the form as in (29): L ( q ) = − q − q − q . Note again that the Hamiltonians H r become polynomial in Vi`ete coordinates ( q, p ). The second method of turning Benenti Hamiltonian systems (10) into a polynomial formis by using Newton coordinates. This method has been discovered by V. M. Buchstaberand A. V. Mikhailov [7] only quite recently. In this section we present our own proofof this result, independent of the work [7]. We also investigate in detail the structure ofBenenti Hamiltonians (10) in Newton coordinates.Consider the following map (consisting of a sequence of Newton polynomials) on thebase manifold Q : Q i = 1 i n X s =1 λ is . (30)This map induces the map on T ∗ Q : P = (cid:0) J − N (cid:1) T µ, (31)where P = ( P , . . . , P n ) T and J N is the Jacobian of the map (30),( J N ) ij = ∂Q i ∂λ j = λ i − j . J N = V T , where V is the Vandermonde matrix, but different from S : V = λ . . . λ n − ... ... . . . ...1 λ n λ n − n . (32)This also means that (31) leads to P = V − µ . Lemma 6. In the above notation (cid:0) V − (cid:1) ij = − j ∂ρ n − i +1 ∂λ j . The reader should compare this lemma with Lemma 1. Thus, the map (30) inducesthe following map on T ∗ Q Q i = 1 i n X s =1 λ is , P i = − n X j =1 j ∂ρ n − i +1 ∂λ j µ j , i = 1 , . . . , n (33)and we call the coordinates ( Q, P ) Newton coordinates on M . The reader should comparethis map with the map (24). Again, since the map ( λ, µ ) → ( Q, P ) is a point transforma-tion map on T ∗ Q , the Newton coordinates ( Q, P ) are Darboux (canonical) coordinates,that is { Q i , Q j } = { P i , P j } = 0 , { Q i , P j } = δ ij . Let us now investigate the structure of Benenti Hamiltonians (10) in ( Q, P )-coordinates.The Hamiltonians (10) are written in tensor form and thus H r ( Q, P ) = 12 P T K r ( Q ) G ( Q ) P + V r ( Q ) , r = 1 , . . . , n. (34)In the monomial case, i.e., when f ( λ ) = λ m we have H r ( Q, P ) = 12 P T K r ( Q ) L m ( Q ) G ( Q ) P + V r ( Q ) , r = 1 , . . . , n. (35)Let us now investigate the structure of (34) and in particular (35), in Newton coordinates.Due to tensor transformation laws, L ( Q ) , K r ( Q ) and G ( Q ) are given by L ( Q ) = J N L ( J N ) − , K r ( Q ) = J N K r ( J N ) − (36)and by G ( Q ) = J N G ( J N ) T . (37)In order to express explicitly the right hand sides of (36 ) and (37) we need to invert themap λ → Q given by (30), which is in general not algebraically invertible. Let us thusconsider the map q → Q between the Vi`ete coordinates (24) and the Newton coordinates.In the recent paper [4] this map is given by Q r = − r r X k =1 V ( n + r − k ) k ( q ) , r = 1 , . . . , n, (38)where V ( α ) k ( q ) are the basic separable potentials as given by (17)-(18). Below we presenta theorem in which we considerably extend the understanding of the above formula.11 heorem 7. The map q → Q as given by (38) has the following form: Q r ( q ) = − q r + τ ( r − r ( q , . . . , q r − ) , r = 1 , . . . , n, (39) where τ ( α ) r denotes a polynomial of order α and where τ (0)1 = 0 . The map q → Q isalgebraically invertible, with the inverse map of the form q r ( Q ) = − Q r + η ( r − r ( Q , . . . , Q r − ) , r = 1 , . . . , n, (40) where η ( α ) r denotes a polynomial of order α with η (0)1 = 0 . Moreover, neither τ ( α ) r nor η ( α ) r depends on n . One proves this theorem by direct calculations, using the properties of basic separablepotentials V ( α ) k . This theorem means that both the map q → Q and its inverse Q → q arepolynomial maps and moreover that the transformation between the first n variables, i.e.between q , . . . , q n and Q , . . . , Q n , does not change after increasing n to n + 1. Expicitly,the first few expressions in both maps are Q = − q ,Q = − q + 12 q ,Q = − q − q + q q ,Q = − q + 14 q − q q + q q + 12 q ...for the map Q → q and q = − Q ,q = − Q + 12 Q ,q = − Q − Q + Q Q ,q = − Q + 124 Q − Q Q + Q Q + 12 Q ...for the inverse map q → Q . It is now possible to calculate the tensor L in the Newtoncoordinates Q . After some calculations we obtain: L ( Q ) = J N L ( J N ) − = . . . 00 0 1 . . . − q n ( Q ) − q n − ( Q ) − q n − ( Q ) . . . − q ( Q ) , (41)or, equivalently L ( Q ) ij = − q n − j +1 ( Q ) δ in + δ ij − , i, j = 1 , . . . , n, where the functions q i ( Q ) are given by (40). Thus, the entries of L ( Q ) are polynomials,and the same is of course true for any positive power L m ( Q ) of L ( Q ).12et us now calculate the Killing tensors K r in Newton coordinates Q . We will doit by transforming K r ( q ), as given by (27), to Q variables, by the formula K r ( Q ) = J V N K r ( q ) ( J V N ) − , where J V N is the Jacobian transformation from Vi`ete coordinates toNewton coordinates. First we find that( J V N ) i,j = n X s =0 q s ( J V N ) i − s,j − q q i − s + q i , i, j = 1 , . . . , n, with ( J V N ) ,j = − , ( J V N ) ,j = q , ( J V N ) ,j = − q + q for any fixed j . This also yieldsthat ( J V N ) − i,j = − q i − j , i, j = 1 , . . . , n. Note that this last result also means that the map (39) can now be extended to the wholemanifold M = T ∗ Q by completing it with the map between the canonical momenta: P i = (cid:2) ( J V N ) − (cid:3) Tij p j = − n X j =1 q j − i p j , i = 1 , ..., n. (42)After some calculations we obtain that( K r ( Q )) ij = q i − j + r − ( Q ) , i − j ≤ r ≤ n − i + 1 − q i − j + r − ( Q ) , i − j > r > n − i + 10 otherwise , (43)cf. (27). Thus, since all q i ( Q ) by (40) are polynomials then all the entries of K r ( Q )are polynomials in Q i as well. Finally, let us consider G ( Q ), i.e. the metric G inNewton coordinates, by transforming G ( q ), as given by (28), into Q variables, by thetransformation formula G ( Q ) = J V N G ( q ) ( J V N ) T . Lemma 8. The metric G in Newton coordinates (30) attains the form of a lower-triangular Hankel matrix given by the recursive formulas G ( Q ) i,j = − k P s =1 q s ( Q ) ( G ) i − s,j + q ( Q ) q i − ( Q ) − q i ( Q ) , i ≥ j , i < j for i, j = 3 , . . . , n, (44) with G ( Q ) ,j = 1 , G ( Q ) ,j = − q , and G ( Q ) ,j = q − q for arbitrary fixed j . As a consequence, the metric G m ( Q ) also attains the form of a lower-triangular Hankelmatrix. This can be verified using induction with respect to m in G m ( Q ) i,j = L ( Q ) ij G m − ( Q ) i,j . Taking into account the formulas (41), (43) and Lemma 8 we obtain a corollary that isan analogue of Corollary 3 for Newton coordinates. Corollary 9. If f is a polynomial in (11), then the geodesic parts of Benenti Hamilto-nians H r ( Q, P ) in (34) have in Newton coordinates (33) a polynomial form. Moreover,if the right hand side of (15) is a pure polynomial, then the potentials V r ( Q ) in the Be-nenti Hamiltonians (34) are in this case also polynomials. Thus, in such a case, all theHamiltonians H r ( Q, P ) (and not just their geodesic parts) are polynomials. Example 10. We proceed in the same setting as in Example 4, i.e. we consider the case n = 2, f ( λ ) = 1 (so that m = 0) and ϕ ( λ ) = λ , but in Newton coordinates. The map(39)-(42) reads now Q = − q , Q = 12 q − q ,P = − p − q p , P = − p and it transforms the Hamiltonians from Example 24 to the form H ( Q, P ) = 12 P Q + P P − Q − Q ,H ( Q, P ) = − P Q + 12 P Q + 12 P + 12 Q − Q Q , which is in agreement with (44) and (43). Explicitly: G ( Q ) = (cid:18) Q (cid:19) , K ( Q ) = I , K ( Q ) = (cid:18) − Q Q − Q (cid:19) . Moreover, L becomes L ( Q ) = (cid:18) Q − Q Q (cid:19) . Example 11. We now consider Example 5 in Newton coordinates i.e. the case n = 3, m = 1 and ϕ ( λ ) = λ . As n = 3 the map (39) is now Q = − q , Q = 12 q − q , Q = − q + q q − q , (45)and its inverse (40) is q = − Q , q = 12 Q − Q , q = − Q + Q Q − Q . The map (42) between momenta is P P P = − − q − q − − q − p p p , (46)with the inverse p p p = − − Q − Q − Q − − Q − P P P . The map (45)-(46) transforms the Hamiltonians H r ( q, p ) in Example 5 to the form H ( Q, P ) = 12 P T Q Q Q + Q Q Q + Q Q + Q Q + Q P + V (5)1 ( Q ) H ( Q, P ) = 12 P T Q − Q Q − Q Q − Q Q − Q Q − Q − Q − Q Q + Q Q + Q P + V (5)2 ( Q ) H ( Q, P ) = 12 P T Q − Q Q + Q Q − Q Q + Q Q − Q Q + Q Q Q − Q Q + Q Q − Q Q + Q Q Q − Q Q + Q Q − Q Q + Q Q P + V (5)3 ( Q ) , V (5)1 ( Q ) = − Q − Q Q − Q ,V (5)2 ( Q ) = Q Q − Q Q − Q + 112 Q ,V (5)3 ( Q ) = − Q + 13 Q Q − Q Q + Q Q − Q Q , which is again in agreement with (44) and (43). Explicitly: G ( Q ) = Q Q Q + Q , K ( Q ) = I,K ( Q ) = − Q − Q Q − Q Q + Q Q − Q ,K ( Q ) = Q − Q − Q Q − Q Q + Q Q − Q Q + Q , and L ( Q ) = Q − Q Q + Q Q − Q Q . In this paper we have considered Benenti Hamiltonian systems generated by a singleseparation curve. These systems turn out to have a rational form when expressed intheir separation coordinates. Under certain additional conditions they can be cast intopolynomial form using either Vi`ete coordinates (24) or Newton coordinates (33), thelast result due to Buchstaber and Mikhailov [7]. We have presented a new version ofBuchstaber and Mikhailov results: not only have we proven their result in a more explicitway but we also presented the explicit form of all the geometric objects, associated withBenenti Hamiltonians, in Newton coordinates. This has been done by constructing andanalysing the map between the Vi`ete and Newton coordinates.A natural questions that arises is whether it is possible to extend our construction tothe case that H i are not generated by a single separation curve but by the more generalseparation relations (6) i.e. with different f i and ϕ i . This will be a subject of anotherresearch paper. Acknowledgements The research of J.D. Maniraguha and C. Kurujyibwami was supported by InternationalScience Programme (ISP, Uppsala University) in collaboration with Eastern Africa Uni-versities Mathematics Programme (EAUMP). The research of K. Marciniak was partiallysupported by the Swedish International Development Cooperation Agency (Sida) throughthe Rwanda-Sweden bilateral research cooperation.15 eferences [1] B laszak M., Marciniak K., From St¨ackel systems to integrable hierarchies of PDE’s:Benenti class of separation relations, J. Math. Phys. , 2006, , 3, p.032904.[2] B laszak, M., Quantum versus Classical Mechanics and Integrability Problems: to-wards a unification of approaches and tools , Springer, 2019.[3] B laszak M., Marciniak K., St¨ackel transform of Lax equations, Stud. Appl. Math. ,2020.[4] B laszak M., Marciniak K., Domanski Z., Systematic construction of non-autonomous Hamiltonian equations of Painlev´e-type. I. 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