This is a preprint of: “Integrability by separation of variables”, Jaume Llibre, Rafael Orlando Ramı́rez, Phys. D, vol. 409, 132523:1–22, 2020. DOI: [10.1016/j.physd.2020.132523] INTEGRABILITY BY SEPARATION OF VARIABLES JAUME LLIBRE1 AND RAFAEL RAMÍREZ2 Abstract. We study the integrability in the Jacobi sense (integrability by separation of variables), of the Hamiltonian differential systems using the Levi- Civita Theorem. 1. Introduction The study of the integrability by separation of variables of the Hamilton-Jacobi equations is a classical problem in Mechanics, dating back to the foundational works of Jacobi, Sta̋ckel, Levi-Civita and others. In 1904 and in a letter addressed to P. Stackel and published in the Matematische Annalen [19], Levi-Civita deals with the problem of the integration by separation of variables. In the introduction of this letter he writtes: Ho notato che si possono facilmente assegnare (sotto forma esplicita di equazioni a derivate parziali . . . ) le condizioni necessarie e sufficienti cui deve soddisfare una H affinche lequazione ( ) ∂W ∂W (1) H (z1 , . . . , zM , ,..., = h, ∂z1 ∂zM ammetta un integrale completo della forma (2) W = W1 (z1 , α1 , . . . , αM ) + . . . + WM (zM , α1 , . . . , αM ), dove α1 , . . . , αM and h sono le costanti arbitrarie. Da queste condizioni scatur- iscono alcune conseguenze di indole generale, che mi sembrano abbastanza interes- santi, per quanto il dedurre da esse la completa risoluzione del problema apparisca ancora laborioso, e non vi sia nemmeno - oserei affermare - grande speranza di trovare tipi essenzialmente nuovi, oltre a quelli da Lei Sta̋ckel scoperti. Indeed, Levi-Civita shows that Theorem 1 (Levi-Civita Theorem). Hamilton-Jacobi equation (1) has a first in- tegral of the form (2), if and only if the Hamiltonian H satisfy the M (M − 1)/2 second-order partial differential equations ∂H ∂H ∂ 2 H ∂H ∂H ∂ 2 H Ljk (z, P ) := + ∂Pj ∂Pk ∂zj ∂zk ∂zj ∂zk ∂Pj ∂Pk (3) ∂H ∂H ∂ 2 H ∂H ∂H ∂ 2 H − − = 0, ∂Pj ∂zk ∂zj ∂Pk ∂Pk ∂zj ∂Pj ∂zk 2010 Mathematics Subject Classification. 34C05, 34C07. Key words and phrases. separation of variables, Hamiltonian system, Levi-Civita conditions, Hamilton-Jacobi equation, Lie algebra, Frobenius integrability, Stark problem, integrability in the Jacobi sense, integrability in the Liouville sense, Lax equation. 1 2 J. LLIBRE AND RAFAEL RAMÍREZ for j, k = 1, . . . , M with j ̸= k. The application of this criterion to the investigation of the integrability of the Hamiltonian systems is a non-trivial problem. Levi-Civita‘s result provides a crite- rion for deciding when a given Hamiltonian H independent of the time is separable or not, it does not give an effective method for finding the separable coordinates for a given Hamiltonian. To find such coordinates it is in general a difficult problem which has only been solved for particular Hamiltonians. The integrability by separation of variables of the Hamilton-Jacobi equations has recently received a big attention due to its applications to the theory of integrable partial differential equations of Korteweg de Vries type and to the theory of quan- tum integrable systems (see for instance [8, 21, 25]). In the review Separation of Variables. New Trends see [21] Sklyanin argued that separation of variables could be the most universal tool to solve integrable models of the classical and quantum mechanics. The question of separation of variables for Hamiltonian systems was studied intensively in the second half of the last century (see for instance [1, 4, 5, 7, 9, 13, 12, 26, 21]). For an outline of the theory of separation of variables we refer to the book of Kalnins [15]. There exists an equivalent definition of separability, originally due to Jacobi and recently widely used by Sklyanin and his collaborators (see for instance [21]). Consider a Hamiltonian mechanical system defined by the Hamiltonian H = H1 with M degrees of freedom and integrable in the Liouvilles sense, i.e. there exist a family of M − 1 first integrals H2 , . . . , HM such that (a) they are in involution with respect to the Poisson bracket, that is ∑M ( ) ∂Hj ∂Hk ∂Hj ∂Hk {Hj , Hk } = − = 0, j=1 ∂Pj ∂zj ∂zj ∂Pj for j, k = 1, . . . , M, (a) they are independent, i.e. the rank of the matrix formed by the gradients of H1 , . . . , HM is M except (perhaps) in a set of zero Lebesgue measure. An integrable Liouville Hamiltonian system with M degree of freedom and M first integral H1 = H, H2 , . . . , HM is separable in the canonical coordinates (z1 , . . . , zM , P1 , . . . , PM ) if there exist n non-trivial relations (4) Φj (zj , Pj , H1 , . . . , HM ) = 0, for j = 1, . . . , M, connecting single pairs (zj , Pj ) of canonical coordinates with the M first integrals H1 , . . . , HM . Note that the knowledge of the separation relations (4) allows to reduce the problem of finding a separated solution of the Hamilton-Jacobi equation to quadratures. Indeed, one can solve the relations (4) with respect to Pj then we get that Pj = fj (zj , H1 , ..., HM ) and then we can define the generatrix function M ∫ zj ∑ S(z1 , . . . , zM , α1 , . . . , αM ) = fj (uj , H1 , . . . , HM )|H1 =α1 ,...,HM =αM duj , j=1 zj0 of the Hamilton-Jacobi equation (see for instance [11, 22]). INTEGRABILITY BY SEPARATION OF VARIABLES 3 The aim of this paper is to study the problem of separation of variables by using the classical approach, i.e. the Levi-Civita approach. In particular we give new properties of the Levi-Civita conditions (see section 3), and we establish the relations between the integrability in Jacobi and Frobienius sense (see Theorem 10).We determine a new equivalent expression for the Levi-Civita conditions (see Theorem 13). We obtain all the Hamiltonian vector fields admitting a two or three dimensional Lie algebra (see Theorem 18 and Propositions 24 and 25). Finally we prove the integrability of some new Hamiltonian vector fields (see Propositions 26 and 31). 2. Preliminary results. On the Hamilton-Jacobi equation Let H = H(z1 , . . . , zM , P1 , . . . , PM ) be a Hamiltonian. We study the Hamilton- ian vector field ∑M ∑M ( ) ∂H ∂ ∂H ∂ (5) ΓH := Γj = − . j=1 j=1 ∂Pj ∂zj ∂zj ∂Pj associated to the Hamiltonian system dzk dPk (6) = {H, zk }, = {H, Pk }, for k = 1, . . . , M dt dt The transformation of the R2M space (7) (z1 , . . . , zM , P1 , . . . , PM ) −→ (z1∗ , . . . , zM ∗ , P1∗ , . . . , Pm ∗ ) under the condition ( ) ∂ (z1∗ , . . . , zM ∗ , P1∗ , . . . , PM ∗ ) (8) det ̸= 0, ∂ (z1 , . . . , zM , P1 , . . . , PM ) is called canonical transformation if the Hamiltonian system (6) is transformed into the Hamiltonian system dzk∗ dPk∗ (9) = {H ∗ , zk∗ }, = {H ∗ , Pk∗ }, for k = 1, . . . , M dt dt The Hamilton-Jacobi theory wants to find the canonical transformations which writes system (6) in its simplest form. The following theorem is well known (see for instance [2, 10]). Theorem 2. Transformation (7) satisfying (8) is canonical if and only if there exists a function F and a constant c such that M ∑ (10) (Pk∗ dzk∗ − Pk dzk ) − (H ∗ − cH)dt = −dF (t, z1 , . . . , zM , P1 , . . . , PM ), k=1 ( ) ∂2F where F is such that det ̸= 0. ∂zj ∂Pk Among the group of canonical transformations there exists a subgroup which is determined by the condition ( ) ∂ (z1∗ , . . . , zM ∗ ) det ̸= 0, ∂ (P1 , . . . , PM ) 4 J. LLIBRE AND RAFAEL RAMÍREZ Under this condition it is possible to choose (t, z1 , . . . , zM , z1∗ , . . . , zM ∗ ) as Hamil- tonian variables. In these variables the function F is usually denoted by S, i.e. S(t, z1 , . . . , zM , z1∗ , . . . , zM ∗ ) = F (t, z1 , . . . , zM , P1 , . . . , PM )|P −→z∗ Under these conditions from (10) we get that ∂S ∂S ∂S = cPk , = −Pk∗ , = H ∗ − cH. ∂zk ∂zk∗ ∂t The most interesting subcase of canonical transformations is when H ∗ = 0. Clearly that in this case from (9) it follows that zk∗ = αk , Pk∗ = βk . Hence to construct a canonical transformation it is necessary and sufficient to de- termine S as a solution of the so called Hamilton-Jacobi equation ( ) ∂S ∂S ∂S (11) + cH t, z1 , . . . , zM , ,..., = 0, ∂t ∂z1 ∂zM ( 2 ) ∂ S with det ̸= 0. ∂zj ∂αk Theorem 3 (Jacobi Theorem). The integration of (6) is equivalent to solve (11). For more details of Jacobi Theorem see for instance [2, 10]. A solution S of the Hamilton-Jacobi equation, contains M + 1 undetermined constants, the first M of them denoted as α1 , α2 , ..., αM , and the last one coming ∂S from the integration of . If the Hamiltonian does not depend on the time ex- ∂t ∂S plicitly, then the derivative in the Hamilton-Jacobi equation must be constant, ∂t usually denoted by −h, consequently (12) S = W (z1 , z2 , . . . , zM ) − ht. If the function W can be separated completely into M functions of the form Wm (zm , α1 , α2 ...αM ) for m = 1, . . . , M , i.e. S = W1 (z1 , α1 , α2 ...αM ) + . . . + WM (zM , α1 , α2 ...αM ) − ht, then we say that the Hamiltonian system is integrable by separation of variables or integrable in the Jacobi sense. When the Hamiltonian H does not depend on t from (11) and (12) the Hamilton- Jacobi equation reduces to the equation (1). 3. Properties of the Hamiltonian systems satisfying Levi-Civita conditions The aim of this section is to prove some properties of the Hamiltonian systems which admit a separation of variables. Differential system (3) has the following properties. (I) Partial differential system (3) can be written in the following equivalent forms INTEGRABILITY BY SEPARATION OF VARIABLES 5 (i)   ∂H  ∂2H ∂2H  0 0 −  ∂Pk   ∂zj ∂zk ∂Pj ∂zk      ∂H   ∂2H   ( ) 0 −   ∂H ∂H ∂H ∂H  0 0  ∂Pj   ∂zj ∂Pk   = 0,   ∂H  ∂Pk ∂Pj ∂zk ∂zj  2 ∂ H    0 0 0   ∂zk   ∂Pj ∂Pk       ∂H  0 0 0 0 ∂zj (ii) By considering that (3) can be rewritten as ( ) ( ) ∂H ∂ 2 H ∂H ∂ 2 H ∂H ∂H ∂ 2 H ∂H ∂ 2 H1 ∂H − − − = 0, ∂Pk ∂zk ∂Pj ∂zk ∂Pk ∂Pj ∂zj ∂Pk ∂zk ∂zj ∂zk ∂Pk ∂zj ∂Pj ∂H ∂ ∂H ∂ and introducing the vector field Γk = − (see formula ∂Pk ∂zk ∂zk ∂Pk (5)) we easily deduced that equations (3) can be written as ( ) ( ) ∂H ∂H ∂H ∂H (13) Γk − Γk = 0. ∂zj ∂Pj ∂Pj ∂zj (iii) Denote by ∂H / ∂H (14) Rj = − , ∂zj ∂Pj then differential system (3) can be written as (15) Γk Rj = 0, where k, j = 1, . . . , M, j ̸= k. Indeed, after some computations we can show that ∂H ∂ ∂H ∂ Γk Rj = (Rj ) − (Rj ) ∂Pk ∂zk ∂zk ∂Pk ( )−2 ( ( ) ( ) ) ∂H ∂H ∂H ∂H ∂H =− Γk − Γk = 0. ∂Pj ∂zj ∂Pj ∂Pj ∂zj (II) Proposition 4. Let H = H(z1 , . . . , zM , P1 , . . . , PM ) be a solution of (3). Then H = H(P1 , . . . , PM , z1 , . . . , zM ) is also a solution of (3). Proof. It is evident.  We observe that this property is important because if we have a Hamil- tonian which is integrable in the Jacobi sense, then the same Hamiltonian under the change zj ←→ Pj is integrable in the Jacobi sense. (III) Proposition 5. Let H be a solution of (3). Then F (H) satisfies (3) for arbitrary C 2 function F. 6 J. LLIBRE AND RAFAEL RAMÍREZ Proof. After some computations it is easy to show that ∂F (H) ∂F (H) ∂H ∂F (H) ∂F (H) ∂H = , = , ∂Pj ∂H ∂Pj ∂zj ∂H ∂zj ∂ 2 F (H) ∂F (H) ∂ 2 H ∂ 2 F (H) ∂H ∂H = + , ∂Pk ∂Pj ∂H ∂Pk ∂Pj ∂H∂H ∂Pj ∂Pk ∂ 2 F (H) ∂F (H) ∂ 2 H ∂ 2 F (H) ∂H ∂H = + , ∂Pk ∂zj ∂H ∂Pk ∂zj ∂H∂H ∂zj ∂Pk ∂ 2 F (H) ∂F (H) ∂ 2 H ∂ 2 F (H) ∂H ∂H = + , ∂zk ∂zj ∂H ∂zk ∂zj ∂H∂H ∂zj ∂zk Hence after some computations we get that ∂F (H) ∂F (H) ∂ 2 F (H) ∂F (H) ∂F (H) ∂ 2 F (H) + ∂Pj ∂Pk ∂zj ∂zk ∂zj ∂zk ∂Pj ∂Pk ∂F (H) ∂F (H) ∂ 2 H ∂F (H) ∂F (H) ∂ 2 F (H) − − ∂Pj ∂zk ∂zj ∂Pk ∂Pk ∂zj ∂Pj ∂zk ( )3 ( ∂F (H) ∂H ∂H ∂ 2 H ∂H ∂H ∂ 2 H = + ∂H ∂Pj ∂Pk ∂zj ∂zk ∂zj ∂zk ∂Pj ∂Pk ∂H ∂H ∂ H 2 ∂H ∂H ∂ H ) 2 − − . ∂Pj ∂zk ∂zj ∂Pk ∂Pk ∂zj ∂Pj ∂zk Consequently in view of (3) we obtain that ∂F (H) ∂F (H) ∂ 2 F (H) ∂F (H) ∂F (H) ∂ 2 F (H) + ∂Pj ∂Pk ∂zj ∂zk ∂zj ∂zk ∂Pj ∂Pk ∂F (H) ∂F (H) ∂ 2 F (H) ∂F (H) ∂F (H) ∂ 2 F (H) − − = 0, ∂Pj ∂zk ∂zj ∂Pk ∂Pk ∂zj ∂Pj ∂zk i.e. since the function H satisfies the Levi-Civita conditions, then the new Hamiltonian F (H) is integrable in the Jacobi sense.  (IV) Proposition 6. Consider the functions ∑m m ∑ U= αj (zj , Pj ), V = βj (zj , Pj ), j=1 j=1 where αj (zj , Pj ) and βj (zj , Pj ) are C 2 functions which satisfy the conditions ∂αj ∂βj ∂αj ∂βj {αj , βj } := − ̸ 0. = ∂Pj ∂zj ∂zj ∂Pj Let F and G be C 2 functions. (i) Then the Hamiltonian vector ΓH with H = F (U ) + G(V ) is integrable in the Jacobi sense if and only if (16) F (U ) = −λ log |U + a| , G(V ) = λ log |V + b| , where a, b and λ are are arbitrary constants. Consequently λ V +b H(U, V ) = log U +a INTEGRABILITY BY SEPARATION OF VARIABLES 7 (ii) Then the Hamiltonian vector ΓH with H = F (U )G(V ) is integrable in the Jacobi sense if and only if λ −λ (17) F (U ) = (U + a) , G(V ) = (V + b) where a, b and λ are are arbitrary constants. Consequently ( )λ U +a H(U, V ) = , V +b Proof. For statement (i), we insert H = F (U ) + G(V ) into the Levi-Civita conditions (3) after some computations we get that (( )2 2 ( )2 2 ) ∂F ∂ G ∂G ∂ F {αj , βj }{αk , βk } + = 0, ∂u ∂v∂v ∂v ∂u∂u for j, k = 1, . . . , M and j ̸= k. Hence, from the previous equation we obtain ∂2G ∂2F − (∂V ∂V ∂U ∂U = 1 , )2 = ( )2 ∂G ∂F λ ∂V ∂U or equivalently ∂2G ( )−1 ∂ ∂G 1 − (∂v∂V)2 = = , ∂G ∂V ∂V λ ∂V ∂2F ( )−1 ∂u∂U = − ∂ ∂F 1 = . ( )2 ∂F ∂V ∂U λ ∂U After integration of these partial differential equations we obtain (16). So statement (i) is proved. For statement (ii), we insert H = F (U )G(V ) into the Levi-Civita condi- tions (3) after some computations we get that (( )2 ( ( )2 ( )2 ( )2 )) ∂F ∂2G ∂G ) ∂G ( ∂ 2 F ∂F {αj , βj }{αk , βk } G − + F − = 0. ∂U ∂V ∂V ∂V ∂V ∂u∂U ∂U Thus ( )2 ( )2 ∂2G ∂G 1 ∂G G − = , ∂V ∂V ∂V λ ∂V ( )2 ( )2 ∂2F ∂F 1 ∂F F − = − . ∂U ∂U ∂U λ ∂U Consequently ∂2G ( ) ∂G ∂2F ( ) ∂F ∂V ∂V = 1 + 1 ∂V , ∂U ∂U = 1 − 1 ∂U . ∂G λ G ∂F λ F ∂V ∂U 8 J. LLIBRE AND RAFAEL RAMÍREZ Hence after integration we get (17). Thus the proof of the proposition is done.  U +a By considering that the Hamiltonian system with Hamiltonian is V +b integrable ( in the)Jacobi sense, then from Proposition 5 we get that any func- U +a tion F is integrable in the Jacobi sense, where F is an arbitrary V +b C 2 function. Problem 7. Assume that the Hamiltonian systems with Hamiltonian F = F (z1 , . . . , zM , P1 , . . . , PM ) and G = G(z1 , . . . , zM , P1 , . . . , PM ) are integrable in the Jacobi sense. Establish the conditions on F and G under which F + G and F G are integrable in the Jacobi sense. (V) The Lie algebra is a vector space g together with a bilinear map g×g −→ g; (X, Y ) −→ [X, Y ] = XY −Y X, called the Lie bracket, satisfying the Jacobi identity [Z, [X, Y ]] + [X, [Y, Z]] + [Y, [Z, X]] = 0, for arbitrary X, Y, Z ∈ g. In particular the set of vector fields on a manifold M is a Lie algebra. M ∑ Proposition 8. Assume that the Hamiltonian vector field ΓH = Γj j=1 given in (5), is integrable in the Jacobi sense, then [Γj , Γn ](H) = 0. In other words if the Hamiltonian vector field ΓH is integrable in the Jacobi sense then Γj , Γk and [Γj , Γk ] are tangents to the hypersurface H = h. Proof. Consider ( )( ) ∂H ∂ ∂H ∂ ∂H ∂ ∂H ∂ [Γj , Γk ](f ) = − − (f ) ∂Pj ∂zj ∂zj ∂Pj ∂Pk ∂zk ∂zk ∂Pk ( )( ) ∂H ∂ ∂H ∂ ∂H ∂ ∂H ∂ − − − (f ) ∂Pk ∂zk ∂zk ∂Pk ∂Pj ∂zj ∂zj ∂Pj ( ) ( ) ∂H ∂ 2 H ∂H ∂ 2 H ∂f ∂H ∂ 2 H ∂H ∂ 2 H ∂f = − − − ∂Pj ∂Pk ∂zj ∂zj ∂Pk ∂Pj ∂zk ∂Pj ∂zk ∂zj ∂zj ∂Pj ∂zk ∂Pk ( 2 2 ) ( 2 ) ∂H ∂ H ∂H ∂ H ∂f ∂H ∂ H ∂H ∂ 2 H ∂f − − − − , ∂Pk ∂zk ∂Pj ∂zk ∂Pj ∂Pk ∂zj ∂Pk ∂zk ∂zj ∂zk ∂Pk ∂zj ∂Pj where f = f (z1 , . . . , zm , P1 , . . . , PM ) is an arbitrary C 2 function, or equiv- alently (18) ( ( ) ( ) ) ( ( ) ( ) ) ∂H ∂f ∂H ∂f ∂H ∂f ∂H ∂f [Γj , Γk ](f ) = Γj − Γk − Γk − Γk . ∂Pk ∂zk ∂zk ∂Pk ∂Pj ∂zj ∂zj ∂Pj Hence taking f = H from (13) we get that ( ( ) ( ) ) ( ( ) ( ) ) ∂H ∂H ∂H ∂H ∂H ∂H ∂H ∂H [Γj , Γk ](H) = Γk − Γk − Γj − Γk = 0. ∂Pj ∂zj ∂zj ∂Pj ∂Pk ∂zk ∂zk ∂Pk Since clearly Γj (H) = Γk (H) = 0, from the definitions of Γj and Γk the proposition is proved.  INTEGRABILITY BY SEPARATION OF VARIABLES 9 4. Integrability in the Jacobi and Frobenius sense Let D be a differential system in a manifold M and V (D) be the set of vector fields X such that X(y) ∈ D for all y ∈ M (see for instance [24]). Theorem 9 (Frobenius ’ Theorem). Differential system D is completely integrable if and only if V (D) is a Lie algebra, i.e. [X, Y ] ∈ V (D) for all vector fields X, Y ∈ V (D), or what is equivalent if X1 , . . . , XM generate V (D). Then there M ∑ k k k exist functions Cjn = −Cnj such that [Xj , Xn ] = Cjn Xk . k=1 Let Γj be the Hamiltonian vector fields given in (5). We say that the Hamiltonian ∑M vector field ΓH = Γj is integrable in the Frobenius sense if and only the vector j=1 M ∑ k fields Γj for j = 1, . . . , M are such that [Γj , Γn ] = Cjn Γk , The Hamiltonian k=1 M ∑ vector field ΓH = Γj is integrable in the Jacobi sense if and only the vector j=1 fields Γj for j = 1, . . . , M satisfy (15). The next result show that a hamiltonian is integrable in the Jacobi sense if and only if is integrable in the Frobenius sense. M ∑ Theorem 10. The Hamiltonian vector field ΓH = Γj is integrable in the Jacobi j=1 sense then if and only if it is integrable in the Frobenius sense. ∂H ∂ ∂H ∂ Proof. From the relation Γj = − it follows that ∂Pj ∂zj ∂zj ∂Pj ∂H ∂ Γj + ∂ ∂zj ∂Pj (19) = , ∂zj ∂H ∂Pj ∂ Inserting into (18) we obtain ∂zj ( ) ( ) ∂H ∂H [Γj , Γk ] = Γk log Γj − Γj log Γk ∂Pj ∂Pk (( ) ( ) ) ∂H ∂H ∂H ∂H ∂ (20) + Γk − Γk ∂Pj ∂zj ∂zj ∂Pj ∂Pj (( ) ( ) ) ∂H ∂H ∂H ∂H ∂ − Γj − Γj . ∂Pk ∂zk ∂zk ∂Pk ∂Pk 10 J. LLIBRE AND RAFAEL RAMÍREZ ∑M n Thus assuming that ΓH is integrable in the Frobenius sense, i.e. [Γj , Γk ] = Cjk Γn , ( ) ( ) n=1 ∂H ∂H ∂H ∂H then Γj −Γj = 0. Hence in view of Remark ?? (see formula ∂Pk ∂zk ∂zk ∂Pk (13)) we get that ΓH is integrable in the Jacobi sense. The reciprocity it is easy to obtain, indeed from (20) it follows that if ΓH is integrable in Jacobi sense then (13) holds, consequently we obtain that ( ) ( ) ∑M ∂H ∂H n (21) [Γj , Γk ] = Γk log Γj − Γj log Γk = Cjk Γn . ∂Pj ∂Pk n=1 n n Hence we get that the functions Cjk are such that Cjk = 0 if n ̸= j or n ̸= k and ( ) ( ) j ∂H k ∂H (22) Cjk = Γk log , Cjk := −Γj log . ∂Pj ∂Pk In short the proposition is proved.  Remark 11. The proof of Theorem 10 also can be done using ∂H ∂ −Γj + ∂ ∂Pj ∂zj = ∂Pj ∂H ∂zj ∂ instead of (19). Indeed inserting into (18) we obtain ∂Pj ( ) ( ) ∂H ∂H [Γj , Γk ] = Γk log | | Γj − Γj log | | Γk ∂zj ∂zk ( ( ) ( ) ) ∂H ∂H ∂H ∂H ∂ + Γk − Γk ∂Pj ∂zj ∂zj ∂Pj ∂zj ( ( ∂H ) ∂H ( ) ∂H ∂H ) ∂ − Γj − Γj . ∂Pk ∂zk ∂zk ∂Pk ∂zk n Thus if ΓH is integrable in the Frobenius sense, then [Γj , Γk ] = Cjk Γn and conse- ( ) ( ) ∂H ∂H ∂H ∂H quently Γj − Γj = 0, i.e. ΓH is integrable in the Jacobi ∂Pk ∂zk ∂zk ∂Pk sense. Hence we obtain that ( ) ( ) ∂H ∂H (23) [Γj , Γk ] = Γk log Γj − Γj log Γk , ∂zj ∂zk n n Consequently the functions Cjk are such that Cjk = 0 if n ̸= j or n ̸= k and ( ) ( ) j ∂H k ∂H Cjk = Γk log , Cjk = −Γj log . ∂zj ∂zk Proposition 12. System ( ) ( ) ∂H k ∂H ∂H k ∂H Γj = −Cjk , Γj = −Cjk , ∂Pk ∂Pk ∂zk ∂zk (24) ( ) ( ) ∂H j ∂H ∂H j ∂H Γk = Cjk , Γk = Cjk , ∂Pj ∂Pk ∂Pj ∂zj INTEGRABILITY BY SEPARATION OF VARIABLES 11 can be written in the matrix form (25) W ξ = 0, where   ∂2H ∂2H j  − Cjk 0   ∂Pk ∂zj ∂Pk ∂Pj   ∂2H ∂2H j   − 0 Cjk   ∂zk ∂zj ∂Pj ∂zk  W =     k ∂2H ∂2H   −Cjk 0 −   ∂Pj ∂zk ∂Pk ∂Pj   ∂2H ∂2H  k 0 −Cjk − ∂zk ∂zj ∂Pk ∂zj ( )T ∂H ∂H ∂H ∂H and ξ = , , , , . Moreover the differential system (25) has ∂Pj ∂zj ∂Pk ∂zk non-trivial solutions if and only if det(W ) = 0, or equivalently ( )2 k j ∂2H ∂2H ∂2H ∂2H −Cjk Cjk + − = 0. ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj Proof. From (25) by considering that the vector ξ is a non-zero vector, then in view of the relation ( )2 k j ∂2H ∂2H ∂2H ∂2H det W = −Cjk Cjk + − , ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj we obtain the proof of the proposition.  Theorem 13. Assume that ( 2 )2 ( 2 )2 ∂ H ∂ H (26) + ̸= 0, for j ̸= k, ∂Pj ∂Pk ∂zk ∂zj then the Levi-Civita second order partial differential equations (3) are equivalent to the equations k j ∂2H ∂2H ∂2H ∂2H (27) −Cjk Cjk + − = 0, ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj for j, k = 1, . . . , M and j ̸= k. Proof. The following identities hold ( ( ) ( ) ) ∂2H ∂H ∂H ∂H ∂H Γk − Γk ∂Pk ∂Pj ∂zj ∂Pj ∂Pj ∂zj ( 2 ) ∂H ∂H k j ∂ H ∂2H ∂2H ∂2H = −Cjk Cjk + − , ∂Pk ∂Pj ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj ( ( ) ( ) ) ∂2H ∂H ∂H ∂H ∂H Γk − Γk ∂zk ∂zj ∂zj ∂Pj ∂Pj ∂zj ( 2 ) ∂H ∂H k j ∂ H ∂2H ∂2H ∂2H =− Cjk Cjk + − , ∂zk ∂zj ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj 12 J. LLIBRE AND RAFAEL RAMÍREZ for j ̸= k. Indeed, from the relations ( ( ) ( ) ) ∂2H ∂H ∂H ∂H ∂H Γk − Γk ∂Pk ∂Pj ∂zj ∂Pj ∂Pj ∂zj ( 2 2 ) ( ) ( ) ∂H ∂H ∂ H ∂ H ∂2H ∂2H ∂H ∂H = − + Γj Γk . ∂Pk ∂Pj ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj ∂Pk ∂Pj and in view of (22) we get that ( )( ) ∂H ∂H k j ∂H ∂H Γj Γk = −Cjk Cjk . ∂Pk ∂Pj ∂Pk ∂Pj Hence we obtain the proof of the first identity. The proof of the second identity is obtained in analogous way. From these two identities under condition (26) we get that the Levi-Civita con- ditions or what is equivalent the conditions (13) are equivalent to (27). Thus the proof of theorem is done.  Proposition 14. The following equations hold j j j j j j k j m j (28) Γm Cjk + Γk Cmj + Cmj Ckj + Cjk Cmj + Ckm Cjk + Ckm Cjm =0 j for j, n = 1, . . . , M with n ̸= j. Moreover if Cmk =constants for j, k, m = 1, . . . , N, k j then Cjk Cjk = 0, for j ̸= k. Proof. From the Jacobi identity [Γm , [Γj , Γk ]] + [Γj , [Γk , Γm ]] + [Γk , [Γm , Γj ]] = 0, and after some computations we get that ( ) ( ) j k j k [Γm , [Γj , Γk ]] = Γm Cjk Γj + Cjk Γk − Cjk Γj + Cjk Γk Γm ( ) ( ) j j j k = Γm Cjk Γj + Γm (Ckj Γk + Cjk [Γm , Γj ] + Cjk [Γm , Γk ] ( ) ( ) j k j m j = Γm Cjk Γj + Γm (Cjk Γk + Cjk (Cmj Γm + Cmj Γj ) k m k +Cjk (Cmk Γm + Cmk Γk ) ( ) ( ) j j j k k k = Γm Cjk + Cjk Cmj Γj + Γm Cjk + Cjk Cmk Γk ( ) j m k m + Cjk Cmj + Cjk Cmk Γm . Consequently ( ) ( ) m m m m m j [Γk , [Γm , Γj ]] = Γk Cmj + Cmj Ckm Γm + Γk Cjm + Cjm Ckj Γj ( ) j j j m + Cjk Cjk + Ckj Cjm Γk , ( ) ( ) k k k k k m [Γj , [Γk , Γm ]] = Γj Ckm + Ckm Cjk Γk + Γj Cmk + Cmk Cjm Γm ( ) k j k j + Ckm Cjk + Cmk Cjm Γj , INTEGRABILITY BY SEPARATION OF VARIABLES 13 Hence, inserting these expression into the Jacobi identity we get 0= [Γm , [Γj , Γk ]] + [Γj , [Γk , Γm ]] + [Γk , [Γm , Γj ]] ( ) j j j j j j k j m j = Γm Cjk + Γk Cmj + Cmj Ckj + Cjk Cmj + Ckm Cjk + Ckm Cjm Γj + . . . Hence we get (28). The proof of the second statement we obtain by putting m = j in (28). In short the proposition is proved.  j Corollary 15. Under the assumptions of Proposition 14 if Cnk is a constant for j, n, k = 1, . . . , N then j k Cjk Cjk =0 for all j ̸= k. j Proof. It follows from (28) by considering that Cnk = constant for all j, n, k and by putting n = j.  Proposition 16. Relations (24) are equivalent to the equations ( )2 ( )2 ∂H ∂Rj k ∂H ∂H ∂Rj k ∂H = Cjk , = Cjk , ∂Pj ∂Pk ∂Pk ∂Pj ∂zk ∂zk (29) ( )2 ( )2 ∂H ∂Rk j ∂H ∂H ∂Rk j ∂H = Cjk , = Cjk , ∂Pk ∂Pj ∂Pj ∂Pj ∂zj ∂zj with j ̸= k, where the functions Rj for j = 1, . . . , M, are defined in (14) , Proof. From (24) we have   ∂H ( )2 ( )2 ( ) ∂H ∂ ∂H ∂   − ∂zj  = −Γj ∂H = Cjk k ∂H (Rj ) = . ∂Pj ∂Pk ∂Pj ∂Pk  ∂H  ∂Pk ∂Pk ∂Pj Hence after some similar computations we obtain the proof of the proposition.  5. integrability in the Jacobi sense in some particular cases Assume that the hamiltonian H satisfies the conditions ∂2H (30) ̸= 0, for all j ̸= k. ∂Pj ∂Pk If conditions (30) hold then the Levi-Civita conditions (3) can be written as (27). In this section we shall study conditions (27) when k j Cjk Cjk = 0, for for all j ̸= k. Therefore the Levi-Civita conditions (27) becomes ∂2H ∂2H ∂2H ∂2H (31) − = 0. ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj 14 J. LLIBRE AND RAFAEL RAMÍREZ Theorem 17. Assume that a Hamiltonian vector field ΓH (see formula (5)) is inte- grable in the Jacobi sense. Then ΓH admits a Lie algebra with the basis Γ1 , . . . , ΓM such that [Γj , Γk ] = 0, for j, k = 1, . . . , M, j ̸= k, (i.e admits an Abelian Lie algebra) if and only if (32) H = H(H1 (z1 , P1 ), H2 (z2 , P2 ), . . . , HM (zM , PM )). Proof. If H is given by the formula (32) then ∂H ∂H ∂Hj ∂H ∂H ∂Hj = , = , ∂Pj ∂Hj ∂Pj ∂zj ∂Hj ∂zj ∂2H ∂ 2 H ∂Hj ∂Hk ∂2H ∂ 2 H ∂Hj ∂Hk = , = , ∂Pk ∂Pj ∂Hj ∂Hk ∂Pj ∂Pk ∂Pk ∂zj ∂Hj ∂Hk ∂zj ∂Pk (33) ∂2H ∂ 2 H ∂Hj ∂Hk = , ∂zj ∂zk ∂Hj ∂Hk ∂zj ∂zk ∂2H ∂ 2 H ∂Hj ∂Hk = , ∂zj ∂Pk ∂Hj ∂Hk ∂zj ∂Pk Consequently ( ) ( ) ∂H j ∂H k Γk log = Cjk = 0, Γj log = −Cjk = 0. ∂Pj ∂Pk Hence by (23) we get that [Γj , Γk ] = 0. Thus the Lie algebra is Abelian. The reci- procity can be obtained as follows. By considering that the vector fields Γ1 , . . . , ΓM generates an Abelian Lie algebra, then [Γk , Γj ] = 0, consequently from (23) we get j k Cjk = Cjk = 0. Therefore from (29) we obtain that ∂Rj ∂Rk ∂Rj ∂Rk = = 0, = = 0, for all j ̸= k, ∂Pk ∂Pj ∂zk ∂zj Consequently Rj = R(zj , Pj ) for j = 1, . . . , M. Hence from (14) H must be such that ∂H ∂H + Rj (zj , Pj ) = 0, for j = 1, . . . , M. ∂zj ∂Pj The function H given by the formula (32) satisfies the previous equations. Indeed ∂H ∂H ∂Hj ∂Hj ∂zj ∂Hj ∂zj ∂zj Rj (zj , Pj ) = − =− =− , ∂H ∂H ∂Hj ∂Hj ∂Pj ∂Hj ∂Pj ∂Pj where Hj = Hj (zj , Pj ). Clearly that in view of (33) the Levi-Civita conditions (31) hold. In short the proposition is proved.  Theorem 18. Assume that a Hamiltonian vector field ΓH is integrable in the Jacobi sense. Then ΓH admits a Lie algebra satisfying j k [Γj , Γk ] = Cjk Γj , for j, k = 1, . . . , M, j > k, Cjk = 0, (34) j k [Γj , Γk ] = Cjk Γj , if Cjk = 0, INTEGRABILITY BY SEPARATION OF VARIABLES 15 for j, k = 1, . . . , M and j ̸= k, if and only if the Hamiltonian H is of the form (35) H = HM (zM , PM , HM −1 (zM −1 , PM −1 , HM −2 (zM −2 , PM −2 , HM −3 (. . . , H1 (z1 , P1 ) . . .))) . Proof. We shall study only the first case. The second case is proved in analogous form. k From (23) and (34) we get that Cjk = 0 for j > k. Consequently from (29) it follows that ∂R1 ∂R1 ∂R1 ∂R1 ∂R1 ∂R1 = = = = ... = = = 0, ∂P2 ∂z2 ∂P3 ∂z3 ∂PM ∂zM ∂R2 ∂R2 ∂R2 ∂R2 = = ... = = = 0, ∂P3 ∂z3 ∂PM ∂zM .. .. .. . . . ∂RM −1 ∂RM −1 = = 0, ∂PM ∂zM Thus R1 = R1 (z1 , P1 ), R2 = R2 (z2 , P2 , z1 , P1 ), . . . , RM −1 =, RM −1 (zM −1 , PM −1 , . . . , z1 , P1 ) Hence from (14) we obtain ∂H ∂H + R1 (z1 , P1 ) = 0, ∂z1 ∂P1 ∂H ∂H + R2 (z2 , P2 , z1 , P1 ) = 0, ∂z2 ∂P2 .. .. .. . . . ∂H ∂H + RM −1 (zM −1 , PM −1 , . . . , z1 , P1 ) = 0, ∂zM −1 ∂PM −1 ∂H ∂H + RM (zM , PM , zM −1 , PM −1 , . . . , z1 , P1 ) = 0, ∂zM ∂PM The solution of this system is the function (35). 16 J. LLIBRE AND RAFAEL RAMÍREZ The reciprocity is obtained as follows. Assume that H is given by the formula (35) then the following relations hold M M ∂H ∂H1 ∏ ∂Hj ∂H ∂H1 ∏ ∂Hj = , = , ∂z1 ∂z1 j=2 ∂Hj−1 ∂P1 ∂P1 j=2 ∂Hj−1 M M ∂H ∂H2 ∏ ∂Hj ∂H ∂H2 ∏ ∂Hj = , = , ∂z2 ∂z2 j=3 ∂Hj−1 ∂P2 ∂P2 j=3 ∂Hj−1 (36) .. .. .. . . . ∂H ∂HM −1 ∂HM ∂H ∂HM −1 ∂HM = , = , ∂zM −1 ∂zM −1 ∂HM −1 ∂PM −1 ∂zP −1 ∂HM −1 ∂H ∂HM ∂H ∂HM = , = . ∂zM ∂zM ∂PM ∂zPM Thus again from (14) we get that M ∏ ∂Hj ∂H1 (z1 , P1 ) ∂Hj−1 ∂z1 j=2 R1 (z1 , P1 ) = − ∂H1 (z1 , P1 ) ∏M ∂Hj ∂P1 ∂H j=2 j−1 ∂H1 (z1 , P1 ) ∂z1 = − , ∂H1 (z1 , P1 ) ∂P1 M ∏ ∂Hj ∂H2 (z2 , P2 , H1 (z1 , P1 )) ∂Hj−1 ∂z2 j=3 R2 (z2 , P2 , z1 , P1 ) = − ∂H2 (z2 , P2 , H1 (z1 , P1 ) ∏ ∂H M j ∂P2 ∂H j=3 j−1 ∂H2 (z2 , P2 , H1 (z1 , P1 )) ∂z2 = − , ∂H2 (z2 , P2 , H1 (z1 , P1 )) ∂P2 .. .. .. . . . INTEGRABILITY BY SEPARATION OF VARIABLES 17 ∂HM −1 ∂HM ∂z ∂HM −1 RM −1 (zM −1 , PM −1 , . . . , z1 , P1 ) = − M −1 ∂HM −1 ∂HM ∂PM −1 ∂HM −1 ∂ HM −1 (zM −1 , PM −1 , HM −2 (. . . , H1 (z1 , P1 ) . . .) ∂z = − M −1 , ∂ HM −1 (zM −1 , PM −1 , HM −2 (. . . , H1 (z1 , P1 ) . . .) ∂PM −1 ∂ HM (zM , PM , HM −1 (. . . , H1 (z1 , P1 ) . . .) ∂zM RM (zM , PM , . . . , z1 , P1 ) = − . ∂ HM (zM , PM , HM −1 (. . . , H1 (z1 , P1 ) . . .) ∂PM Hence ∂Rk ∂Rk = = 0, for j > k. ∂Pj ∂zj j From (29) we obtain that Cjk = 0 for j > k. Thus from (24) and (23) we get (34). In view of (36) the function (35) satisfies the Levi-Civita conditions (31). Indeed, from (36) for k > j it follows that M M ∂H ∂Hj ∏ ∂Hn ∂H ∂Hj ∏ ∂Hn = , = , ∂zj ∂zj n=j+1 ∂Hn−1 ∂Pj ∂Pj n=j+1 ∂Hn−1     M ∏ ∏M ∂2H ∂Hj ∂  ∂Hn  2 ∂ H ∂Hj ∂  ∂Hn  = , = ,, ∂Pj ∂Pk ∂Pj ∂Pk n=j+1 ∂Hn−1 ∂zj ∂zk ∂zj ∂zk n=j+1 ∂Hn−1 M M ∂H ∂Hj ∏ ∂Hn ∂H ∂Hj ∏ ∂Hn = , = , ∂zj ∂zj n=j+1 ∂Hn−1 ∂Pj ∂Pj n=j+1 ∂Hn−1     M M ∂2H ∂Hj ∂  ∏ ∂Hn  ∂2H ∂Hj ∂  ∏ ∂Hn  = , = ,, ∂Pj ∂zk ∂Pj ∂zk n=j+1 ∂Hn−1 ∂zj ∂Pk ∂zj ∂Pk n=j+1 ∂Hn−1 Consequently ∂2H ∂2H ∂2H ∂2H − ≡ 0, for k > j. ∂Pj ∂Pk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj In short the proposition is proved.  Remark 19. The functions (32) and (34) are well-known Hamiltonians (see for instance [10]). The originality of Theorems 17 and 18 consist in providing the relation with the Lie algebra of the Jacobi integrable vector field ΓH . 5.1. Example. The Stark problem in arbitrary dimensions. The main in- tegrable problems of Celestial Mechanics is the Kepler problem, the Euler problem (two center Newtonian gravitational motion), and the Stark problem. This small number of integrable problems explains why the Stark problem, corresponding to the motion in a Newtonian gravitational field subjected to an additional uniform 18 J. LLIBRE AND RAFAEL RAMÍREZ force of constant magnitude and direction, has received special attention over al- most two and one-half centuries (see for instance [17, 16, 3, 20]). The integrability in the Jacobi sense of two and three dimensional Stark problem was first established in [14, 17]. Here we extend this problem to arbitrary dimension and show that it continues being integrable in the Jacobi sense. The equations of motions of two-dimensional Stark problem are µx µy ẍ = − , ÿ = − + ε, r3 r3 √ where µ and ε are constants and r = x2 + y 2 , and for the three-dimensional case the equations of motions are µx µy µz (37) ẍ = − , ÿ = − z̈ = − + ε, r3 r3 r3 √ where µ and ε are constants and r = x2 + y 2 + z 2 . In [17] the integrals of motion are determined, and the resulting quadratures are analytically given. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. Now we shall illustrated the integrability in the Jacobi sense for the Stark prob- ∑N lem in dimension N > 3 whose kinetic energy for unit of mass is T = ẋ2j and its j=1 v uN µ u∑ 2 potential function is U = − − ε xN , where r = t xj , µ and ε are constants. r j=1 The equations of motion are µ xj µ xN ẍj = − , ẍN = − + ε, r3 r3 for j = 1, . . . , N − 1. Clearly the Hamiltonian vector field ΓH with H = T + U is not integrable in Jacobi sense in cartesian coordinates. However the authors of [3, 20] and others have found that it becomes separable in parabolic coordinates for N = 2 and N = 3. We developed these results and proved that ΓH is integrable in the Jacobi sense for N > 3. Proposition 20. The Hamiltonian vector field ΓH with N 1∑ 2 µ H= ẋ − − ε xN , 2 j=1 j r in coordinates ξ2 − η2 x = ξη, y= , 2 for N = 2 becomes Pξ2 + Pη2 − 4µ − ε(ξ 4 − η 4 ) H= 2(ξ 2 + η 2 ) INTEGRABILITY BY SEPARATION OF VARIABLES 19 and for N > 2 in coordinates N ∏ −2 x1 = ξη cos α1 sin αj , j=2 N ∏ −2 x2 = ξη sin α1 sin αj , j=2 N ∏ −2 x3 = ξη cos α2 sin αj , j=3 N ∏ −2 x4 = ξη sin α2 sin αj , j=3 .. .. .. .. . . . . xN −2 = ξη cos α1 cos αN −3 sin αN −2 , xN −1 = ξη cos αN −2 , ξ2 − η2 xN = , 2 becomes Pξ2 + Pη2 − 4µ − ε(ξ 4 − η 4 ) CN −2 (38) H= + 2 2 , 2(ξ 2 + η 2 ) ξ η where CN −2 = HN −2 (PN −2 , αN −2 , HN −3 (PN −3 , αN −3 , . . . , H1 (P1 , α1 ) . . .))) , Hj−1 H1 = P12 , Hj = Pj2 + , for j = 2, . . . , N − 2. sin2 αj N ∑ ( )2 ξ2 + η2 Proof. For N ≥ 2 we get that r = 2 x2j = . The kinetic energy T j=1 2 and potential energy U becomes 1 1 2µ ε( 2 ) T = (ẋ2 + ẏ 2 ) = (ξ 2 + η 2 )(ξ˙2 + η̇ 2 ), U =− − ξ − η2 , 2 2 ξ2 +η 2 2 respectively for N = 2 and N 1∑ 2 T = ẋ 2 j=1 j   N −2 N −2 ξ2 η2  2 ∏ ∏  (ξ 2 + η 2 ) ˙2 = α̇1 sin2 αj + α̇22 sin2 αj + . . . + α̇N 2 −2 + (ξ + η̇ 2 ), 2 j=2 j=3 2 2µ ε( 2 ) U= − − ξ − η2 , ξ2 +η 2 2 20 J. LLIBRE AND RAFAEL RAMÍREZ for N > 2, respectively. Consequently the Hamiltonian H in coordinates (ξ, η, α1 , . . . , αN −2 ) for N > 2 becomes     1  P12 P22 PN2 −3   2  H=  N −2 + N −2 + ... + + P N −2  2ξ 2 η 2 ∏ ∏ sin2 αN −2   sin2 αj sin2 αj  j=2 j=3 Pξ2 + Pη2 2µ ε( 2 ) + − − ξ − η2 , 2(ξ 2 + η 2 ) ξ 2 + η 2 2 or equivalently 1 ( 2 1 ( 2 1 ( H= P + P + ... 2ξ 2 η 2 N −2 sin2 αN −2 N −3 sin2 αN −3 (( 1 ( 2 P12 ))) ))) (39) P32 + P 2 + . . . sin2 α3 sin2 α2 Pξ2 + Pη2 2µ ε( 2 ) 2 + 2 − 2 2 − ξ − η2 . 2(ξ + η ) ξ + η 2 By introducing the Hamiltonians Hj−1 H1 = P12 , Hj = Pj2 + , for j = 2, . . . , N − 2, sin2 αj we get that (39) can be written as 1 H= (HN −2 (PN −2 , αN −2 , HN −3 (PN −3 , αN −3 , . . . , H1 (P1 , α1 ) . . .)))) 2ξ 2 η 2 Pξ2 + Pη2 2µ ε( 2 ) + + + ξ − η2 2(ξ 2 + η 2 ) ξ 2 + η 2 2 CN −2 Pξ2 + Pη2 2µ ε( 2 ) = 2 2 + 2 2 − 2 2 − ξ − η2 2ξ η 2(ξ + η ) ξ + η 2 where CN −2 is the level of the function HN −2 (PN −2 , αN −2 , HN −3 (PN −3 , αN −3 , . . . , H1 (P1 , α1 ) . . .)) . In short the proposition is proved.  Proposition 21. The Hamiltonian vector field ΓH with Hamiltonian (38) is inte- grable in the Jacobi sense. Moreover the N dimensional Stark problem has N − 1 independent first integrals Hj−1 H1 = P12 , Hj = Pj2 + , for j = 2, . . . , N − 2, sin2 φj (40) ∫ ∫ η dη ξ dξ HN −1 := √ − √ , R6 (η) S6 (ξ) where R6 (η) = −ε η 6 + 2hη 4 + (2µ + λ)η 2 + 2CN −2 , S6 (ξ) = εξ 6 + 2hξ 4 + (2µ − λ)ξ 2 + 2CN −2 , INTEGRABILITY BY SEPARATION OF VARIABLES 21 where CN −2 and λ are arbitrary constants. Proof. Now we prove the integrability in the Jacobi sense of ΓH . Indeed, from (38) it follows that Pξ2 + Pη2 − 4µ − ε(ξ 4 − η 4 ) CN −2 H= + 2 2 2(ξ 2 + η 2 ) ξ η 1 ( ) = 2 2 2 2 ξ 2 η 2 (Pξ2 + Pη2 − 4µ − ε(ξ 4 − η 4 )) + 2CN −2 (ξ 2 + η 2 ) . 2(ξ + η )ξ η Thus ξ 2 η 2 (Pξ2 + Pη2 − 4µ − ε(ξ 4 − η 4 )) + 2CN −2 (ξ 2 + η 2 )) − 2(ξ 2 + η 2 )ξ 2 η 2 h = 0. Consequently ( ) ( ) ξ 2 η 2 (Pη2 − 2µ + εη 4 − 2hη 2 ) + 2CN −2 +η 2 ξ 2 (Pξ2 − 2µ − εξ 4 − 2hξ 2 ) + 2CN −2 = 0. which is equivalently to η 2 (Pη2 − 2µ + εη 4 − 2hη 2 ) + 2CN −2 = λη 2 , ξ 2 (Pξ2 − 2µ − εξ 4 − 2hξ 2 ) + 2CN −2 = −λξ 2 , where λ is an arbitrary constant. Hence we get that η 2 (Pη2 −(2µ+λ)+εη 4 −2hη 2 )+2CN −2 = 0, ξ 2 (Pξ2 −(2µ−λ)−εξ 4 −2hξ 2 )+2CN −2 = 0. Thus the variable is separated. Hence √ √ ηPη = ± −ε η 6 + 2hη 4 + (2µ + λ)η 2 + 2CN −2 := R6 (η), √ √ ξPξ = ± εξ 6 + 2hξ 4 + (2µ − λ)ξ 2 + 2CN −2 := S6 (ξ), or equivalently √ √ η(ξ 2 + η 2 )η̇ = ± −ε η 6 + 2hη 4 + (2µ + λ)η 2 + 2CN −2 := R6 (η), √ √ ξ(ξ 2 + η 2 )ξ˙ = ± εξ 6 + 2hξ 4 + (2µ − λ)ξ 2 + 2CN −2 := S6 (ξ). By introducing the new time τ such that dt = (ξ 2 +η 2 )dτ we get after the integration ∫ ∫ η dη ξ dξ (41) √ = τ + τ0 , √ = τ + τ1 , R6 (η) S6 (ξ) where τ0 and τ1 are arbitrary constants. From these relations eliminating the new time variable τ we get the first integral ∫ ∫ η dη ξ dξ F := √ − √ =C R6 (η) S6 (ξ) where C = τ0 − τ1 . Hence the Hamiltonian vector field ΓH is integrable in the Jacobi sense, it has N − 1 independent first integral (40).  Proposition 22. The parametric expressions of the solutions of the N dimensional Stark problem is the following 2h √ 2h √ ξ2 = − + P( ε(τ + τ0 )), η2 = + P( −ε(τ + τ1 )). ε ε where P is the Weierstrass function. 22 J. LLIBRE AND RAFAEL RAMÍREZ 2h 2h Proof. After the change ξ 2 = X − and η 2 = Y + we get that the polynomial ε ε 4R6 (ξ)/ε and 4S6 (η)/(−ε) becomes 4 ( ) 8 ( ) P3 (X) = 4X 2 − 2 (3λ − 6µ)ε + 4h2 X + 3 27CN −2 ε2 + (9hλ − 18hµ)ε + 8h3 , 3ε 27ε 4 ( ) 8 ( ) Q3 (Y ) = 4Y 2 − 2 (3λ + 6µ)ε + 4h2 X − 27CN −2 ε2 + (9hλ + 18hµ)ε + 8h3 3ε 27ε3 Consequently (41) can be rewritten as ∫ ∫ dX √ dY √ (42) √ = ε(τ + τ0 ), √ = −ε(τ + τ1 ), P3 (X) Q3 (Y ) The problem is therefore reduced to quadratures, more specifically to elliptic inte- grals. The key is now to invert those integrals to find parametric expressions of the variables X and Y in functions of new time τ. By inverting integrals (42) we get that √ √ X = P( ε(τ + τ0 )), Y = P( −ε(τ + τ1 )). where P is the Weierstrass function. In short the proposition is proved.  Remark 23. For N = 3 the Stark problem does not conserve the generalized Laplace-Lenz vector F = (f1 , f2 , f3 ) defined by µx f1 = ẏ(xẏ − y ẋ) − ż(z ẋ − xż) − √ , x2 + y 2 + z 2 µy f2 = ż(y ż − z ẏ) − ẋ(xẏ − y ẋ) − √ , x + y2 + z2 2 µz f3 = ẋ(z ẋ − xż) − ẏ(y ż − z ẏ) − √ . x + y2 + z2 2 But since along the solutions of (37) we get that f˙1 = ε (2xż − z ẋ) , f˙2 = ε (2y ż − z ẏ) , f˙3 = −ε (xẋ + y ẏ) , we get the first integral µz ε (43) ẋ(z ẋ − xż) − ẏ(y ż − z ẏ) − √ + (x2 + y 2 ) = C. x2 + y2 + z2 2 In particular if we put in the previous equations y = ẏ = 0 then we obtain that f˙1 = ε (2xż − z ẋ) , f˙3 = −εxẋ. Hence we get that the first integral (43) becomes µz ε ẋ(z ẋ − xż) − √ + x2 = C 0 . x2 + z 2 2 This first integral was given in [17]. INTEGRABILITY BY SEPARATION OF VARIABLES 23 5.2. Hamiltonian vector field admitting a finite dimensional Lie algebra. We shall study the Levi-Civita integrable Hamiltonian systems which admit a finite dimensional Lie algebra formed by the vector fields Γ1 , . . . , ΓM such that j k (44) [Γj , Γk ] = Cjk Γj + Cjk Γk , j k where Cjk and Cjk are constants which in view of Corollary 15 satisfy the conditions j k Cjk Cjk = 0, for j, k = 1, . . . , M, and j ̸= k. Consequently (44) becomes k j j k [Γj , Γk ] = Cjk Γk when Cjk = 0, or [Γj , Γk ] = Cjk Γj when Cjk = 0, j k or [Γj , Γk ] = 0 when Cjk = 0, Cjk = 0, which is a particular case of (34). Proposition 24. A Hamiltonian vector field ΓH integrable in the Jacobi sense, admits a two dimensional Lie algebra with basis Γ1 and Γ2 satisfying 1 2 1 2 [Γ1 , Γ2 ] = C12 Γ1 + C12 Γ2 , C12 C12 = 0, if and only if 1 2 (i) H = H(H1 (z1 , P1 ), H2 (z2 , P2 )) when C12 = C12 = 0, 2 (ii) H = H(z2 , P2 , H1 (z1 , P1 )) when C12 = 0, 1 (iii) H = H(z1 , P1 , H2 (z2 , P2 )) when C12 = 0, Proof. Is a consequence of Theorem 17 and 18.  Proposition 25. The Jacobi integrable Hamiltonian vector field ΓH admits a three dimensional Lie algebra with basis Γ1 , Γ2 and Γ3 satisfying (45) [Γ1 , Γ2 ] = aΓ2 , [Γ2 , Γ3 ] = 0, [Γ3 , Γ1 ] = −aΓ3 , if and only if (46) H = H(z1 , P1 , H2 (z2 , P2 ), H3 (z3 , P3 )). Proof. For N = 3 we get that (see (21)) 1 2 1 2 [Γ1 , Γ2 ] = C12 Γ1 + C12 Γ2 , with C12 C12 = 0, 2 3 2 3 (47) [Γ2 , Γ3 ] = C23 Γ2 + C23 Γ3 , with C23 C23 = 0, 1 3 1 3 [Γ3 , Γ1 ] = C31 Γ1 + C31 Γ3 , with C31 C31 = 0, On the other hand from the Bianchi classification of three dimensional Lie-algebra we get that the Lie algebra with basis e1 , e2 and e3 has the following representation [e1 , e2 ] = ae2 + b3 e3 , [e2 , e3 ] = b1 e1 , [e3 , e1 ] = b2 e2 − ae3 . 1 3 2 1 By compare with (47) we obtain that C12 = C23 = C23 = C31 = 0. Hence in view of (29) we get that ∂R2 ∂R2 ∂R2 ∂R2 = = = = 0, ∂z1 ∂P1 ∂z3 ∂P3 ∂R3 ∂R3 ∂R3 ∂R3 = = = = 0, ∂z1 ∂P1 ∂z2 ∂P2 3 2 C31 = −C12 = −a. 24 J. LLIBRE AND RAFAEL RAMÍREZ Hence from (29) we obtain ∂H ∂H + R2 (z2 , P2 ) = 0, ∂z2 ∂P2 ∂H ∂H + R3 (z3 , P3 ) = 0. ∂z3 ∂P3 Clearly that the function (46) is a solution of these equations. In short the propo- sition is proved.  We observe that Proposition 24 and 25 is a consequence of Theorem 17 and 18. 6. Integrability in the Jacobi sense for the Hamiltonians such that ∂2H ∂2H = 0 and ̸= 0 with j ̸= k ∂Pj ∂Pk ∂zj ∂zk Clearly under theses conditions the function H can be written as M ∑ H= Hj (z1 , z2 , . . . , zM , Pj ). j=1 For these Hamiltonians the partial differential equations (3) become ( ) ( ) ∂2H ∂H ∂ ∂H ∂H ∂ ∂H (48) − log − log = 0, ∂zj ∂zk ∂zk ∂zj ∂Pk ∂zj ∂zk ∂Pj for j ̸= k, or equivalently from Theorem 13 we have j k ∂2H ∂2H (49) Cjk Cjk − = 0, ∂Pk ∂zj ∂zk ∂Pj where   ∂H   ( )   ∂H ∂Pj ∂ 2 H   −Γj log =− ,   ∂Pk ∂H ∂Pk ∂zj    ∂P k k Cjk = .   ∂H   ( )    ∂H ∂P ∂2H   −Γj log =− k ,   ∂zk ∂H ∂Pj ∂zk  ∂Pj Let |∆| be the determinant of the matrix   φ11 (z1 ) . . . φ1M (z1 )  .. ..  ∆= . ... .  φM 1 (zM ... φM M (zM ) where φjk (zj ) is a set of M 2 arbitrary functions. Let ψj (zj ) be a set of M functions. Then the vector field ΓH with M 1∑ ( ) H= Aj (z1 , . . . , zM ) Pj2 − ψj (zj ) 2 j=1 INTEGRABILITY BY SEPARATION OF VARIABLES 25 ( ) 1 ∂|∆| where Aj = is called the Stäckel vector field (see for instance [23, 6]) |∆| ∂φj1 Proposition 26. The Hamiltonian vector field ΓH with M ∑ H= Aj (z1 , z2 , . . . , zM )Hj (zj , Pj ), j=1 where Hj (zj , Pj ) is an arbitrary function for j = 1, . . . , M, is integrable in the Jacobi sense if and only if the functions Aj for j = 1, 2, . . . , M satisfy the partial differential equations ∂ 2 An ∂Ak ∂An ∂Aj ∂An (50) Aj Ak − Aj − Ak =0 ∂zj ∂zk ∂zj ∂zk ∂zk ∂zj for j, k, n = 1, . . . , M and j ̸= k. Moreover if Hj (zj , Pj ) = Pj2 − ψj (zj ) where ψj (zj ) is an arbitrary function, for j = 1, . . . , M, then we obtain an Stäckel system (see for instance [23, 6]). Proof. By considering that M ∑ ∂H ∂Aj = Hj (zj , Pj ), for n ̸= j, ∂zn j=1 ∂zn M ( ∑ ) ∂H ∂Aj ∂Hj (zj , Pj ) = Hj (zj , Pj ) + Aj , ∂zj j=1 ∂zj ∂zj M ( ∑ ) ∂H ∂Hj (zj , Pj ) = Aj , ∂Pj j=1 ∂Pj ∑M ∂2H ∂ 2 Aj ∂Aj ∂Hj (zj , Pj ) = Hj (zj , Pj ) + , for n ̸= j, ∂zj ∂zn j=1 ∂z j ∂z n ∂ zn ∂zj ∂2H = 0 for n ̸= j, ∂Pj ∂Pn Inserting these relations into (48) after some computations we get M ( ) ∂Hk ∂Hj ∑ ∂ 2 An ∂Ak ∂An ∂Aj ∂An Aj Ak Hn (zn , Pn ) Aj Ak − Aj − Ak = 0, ∂zj ∂zk n=1 ∂zj ∂zk ∂zj ∂zk ∂zk ∂zj for j, k, = 1, . . . , M and j ̸= k. Hence we obtain (50). It is interesting to observe that if we apply (49) then we obtain that ∂2H ∂2H ∂Ak ∂Aj ∂Hk ∂Hj = . ∂Pk ∂zj ∂zk ∂Pj ∂zj ∂zk ∂Pk ∂Pj On the other hand by considering that ( ) k ∂2H Aj ∂Hj ∂Ak Cjk = −Γj log =− , ∂Pk Ak ∂Pj ∂zj 26 J. LLIBRE AND RAFAEL RAMÍREZ we obtain k j ∂2H ∂2H Aj ∂Hj ∂Ak Ak ∂Hk ∂Aj ∂Ak ∂Aj ∂Hk ∂Hj Cjk Cjk − = − ≡ 0. ∂Pk ∂zj ∂zk ∂Pj Ak ∂Pj ∂zj Aj ∂Pk ∂zk ∂zj ∂zk ∂Pk ∂Pj Thus the condition (49) holds identically. On the other hand,since ( ) k Aj ∂Hj ∂Ak ∂H Cjk = = Γj log for j ̸= k, , Ak ∂Pj ∂zj ∂zk therefore after some computations we obtain that the functions Aj (z1 , . . . , zm ) is a solution of (50). Finally we observe that the integrability in the Jacobi sense of the Hamiltonian M ∑ ( ) H = Aj (z1 , z2 , . . . , zM ) Pj2 − ψj (zj ) was established by Stäckel see (see for j=1 instance [23, 6]). In short the proposition is proved.  Finally we state the following problem. Problems 27. (i) Let Aj = Aj (z1 , . . . , zM ) and αj = αj (H1 (z1 , P1 ), . . . , HM (zM , PM ) where Hj = Hj (zj , Pj ) for j = 1, . . . , M. Determine the conditions on the functions Aj and αj in such away that the Hamiltonian vector field ΓH with M ∑ H= Aj (z1 , . . . , zM )αj (H1 , . . . , HM ), j=1 is integrable in the Jacobi sense. (ii) Let Aj = Aj (z1 , . . . , zM ) and αj = αj (zM , PM , HM −1 (zM −1 , PM −1 , HM −2 (zM −2 , PM −2 , . . . , H1 (z1 , P1 ) . . .)) . Determine the conditions on the functions Aj and αj in such away that the Hamiltonian vector field ΓH with H equal to M ∑ An (z1 , . . . , zM )αn (zM , PM , HM −1 (zM −1 , PM −1 , HM −2 (zM −2 , PM −2 , . . . , H1 (z1 , P1 ) . . .)) , j=1 is integrable in the Jacobi sense. Remark 28. The functions (32) and (34) are well-known Hamiltonians (see for instance [10]). The originality of Theorems 17 and 18 consists in providing the relation with the Lie algebra of the Jacobi integrable vector field ΓH . The integrability in the Jacobi sense for the Hamiltonian vector field ΓH with such that ∂2H ∂2H (51) ̸= 0 and = 0 with j ̸= k, ∂Pj ∂Pk ∂zj ∂zk can be obtained by applying Proposition 4 on the integrable Hamiltonian vector fields in the Jacobi sense. Clearly under the condition (51) the function H can be written as M ∑ (52) H= Hj (zj , P1 , . . . , PM ). j=1 INTEGRABILITY BY SEPARATION OF VARIABLES 27 For these Hamiltonians the partial differential equations (3) become ( ) ( ) ∂2H ∂H ∂ ∂H ∂H ∂ ∂H − log − log = 0 for j ̸= k, ∂Pj ∂Pk ∂Pk ∂Pj ∂zk ∂Pj ∂Pk ∂zj or equivalently from Theorem 13 we have k j ∂2H ∂2H Cjk Cjk − = 0, ∂Pk ∂zj ∂zk ∂Pj where  ( )   ∂H   −Γj log , k ∂Pk Cjk = ( ) .   ∂H   −Γj log , ∂zk Proposition 29. The Hamiltonian vector field ΓH with M ∑ H= Aj (P1 , P2 , . . . , PM )Hj (zj , Pj ) j=1 is integrable in Jacobi sense if and only if the functions Aj for j = 1, 2, . . . , M satisfy the partial differential equations ∂ 2 An ∂Ak ∂An ∂Aj ∂An Aj Ak − Aj − Ak =0 ∂Pj ∂Pk ∂Pj ∂Pk ∂Pk ∂Pj for j, k, n = 1, . . . , M and j ̸= k. Proof. This proof is analogous to the proof of Proposition 26 after the change P ←→ z, by using the properties of solutions of the Levi-Civita conditions (see Proposition 4). In short the proposition is proved.  In view of Proposition 4 the problems 27 can be stated for the Hamiltonians (52) after the change Pj ←→ zj . 6.1. Integrability in the Jacobi sense and Lax pair. One of the best meth- ods for determining the involution set of first integrals for an integrable Liouville Hamiltonian vector field ΓH associated to a Hamiltonian differential system (6) is the Lax pair method, for more details then the ones given in what follows see [18] A Laix pair for a Hamiltonian vector field is a pair of smooth quadratic matrices A = (ajk ) and B = (bjk ), satisfying the equation called the Lax equation dA (53) = [B, A] = BA − AB, dt dA where the derivative is calculated along the solutions of (5). dt If the Hamiltonian system admits a Lax pair then it has the involution set of first integrals Fj = trace(Aj ), for j = 1, . . . , M. 28 J. LLIBRE AND RAFAEL RAMÍREZ For the particular case of the Hamiltonian vector field ΓH with M M 1∑ 2 ∑ H= P + akj (zk − zj )ajk (zj − zk ) 2 j=1 j k,j=1, k 2 is even and k = 1, . . . , . 2 Assuming that the Hamiltonian (54) is 2N N 1∑ 2 ∑ 2 (55) H= P − a2k−1,2k = T + V, 2 j=1 j k=1 30 J. LLIBRE AND RAFAEL RAMÍREZ if M = 2N , and 2N +1 N 1 ∑ 2 ∑ 2 (56) H= Pj − a2k−1,2k = T + V, 2 j=1 k=1 if M = 2N + 1, where a2k−1,2k = a2k−1,2k (z2k−1 − z2k ) are arbitrary functions. ∂V Then the Lax equation (53) becomes Ṗj = for j = 1, . . . , M . ∂zj Proof. After some computations it is easy to show that independently on the parity ∂V of M the Lax equation (53) becomes Ṗj = for j = 1, . . . , M. We note that the ∂zj Hamiltonian vector field with Hamiltonian ΓH with H given in (55) and (56) is integrable in the Liouville sense.  Proposition 32. The Hamiltonian vector field ΓH with H given in (55) and (56) is integrable in the Jacobi sense. Proof. Indeed, after the change of variables z2k−1 − z2k z2k−1 + z2k u2k−1 = , v2k−1 = , 2 2 for k = 1, . . . , N , we get P2k−1 = ż2k−1 = u̇2k−1 + v̇2k−1 , P2k = ż2k = u̇2k−1 − v̇2k−1 . Consequently the Hamiltonians (55) and (56) become 2N ∑ N ( ) ∑ H= u̇22j−1 + v̇2j−1 2 − a22j−1, 2j (2u2j−1 ) j=2 j=1 and 2N ∑ +1 N ( ) ∑ H= u̇22j−1 + v̇2j−1 2 − a22j−1,2j (2u2j−1 ) j=2 j=1 respectively. Hence the Hamiltonian system is integrable by variable separation.  Acknowledgments. The first author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author was partly supported by the Span- ish Ministry of Education through projects TIN2014-57364-C2-1-R, TSI2007-65406- C03-01 “AEGIS”. References [1] C. 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Woodhouse, Killing tensors and the separation of the Hamilton-Jacobi equation, Commun. Math. Phys. 44 (1975), 9–38 . 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain. E-mail address: jllibre@mat.uab.cat 2 Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Vir- gili, Avinguda dels Paı̈sos Catalans 26, 43007 Tarragona, Catalonia, Spain. E-mail address: rafaelorlando.ramirez@urv.cat