Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2016), 010, 16 pages Quasi-Bi-Hamiltonian Structuresof the 2-Dimensional Kepler Problem (cid:63)
Jose F. CARI ˜NENA and Manuel F. RA ˜NADADepartamento de F´ısica Te´orica and IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail: [email protected], [email protected]
Received September 29, 2015, in final form January 25, 2016; Published online January 27, 2016http://dx.doi.org/10.3842/SIGMA.2016.010
Abstract.
The existence of quasi-bi-Hamiltonian structures for the Kepler problem isstudied. We first relate the superintegrability of the system with the existence of twocomplex functions endowed with very interesting Poisson bracket properties and then weprove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions.The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonianstructure is obtained by making use of polar coordinates and in the second part a newquasi-bi-Hamiltonian structure is obtained by making use of the separability of the systemin parabolic coordinates.
Key words:
Kepler problem; superintegrability; complex structures; bi-Hamiltonian struc-tures; quasi-bi-Hamiltonian structures
In differential geometric terms, the phase space M of a Hamiltonian system is the 2 n -dimensionalcotangent bundle M = T ∗ Q of the n -dimensional configuration space Q . Cotangent bundles aremanifolds endowed, in a natural or canonical way, with a symplectic structure ω ; if { ( q i ) | i =2 , . . . , n } are local coordinates in Q and { ( q j , p j ); j = 1 , , . . . , n } the induced coordinates in T ∗ Q , then ω is given by ω = dq j ∧ dp j , ω = − dθ , θ = p j dq j , (we write all the indices as subscripts and summation convention on the repeated index is used).Given a differentiable function F in T ∗ Q , F = F ( q, p ), the vector field X F uniquely defined asthe solution of the equation i ( X F ) ω = dF is called the Hamiltonian vector field of the function F . In particular, given a Hamiltonian H = H ( q, p ), the dynamics is given by the corresponding Hamiltonian vector field X H , that is, i ( X H ) ω = dH .A vector field Γ ∈ X ( T ∗ Q ) is Hamiltonian if there is a function H such that Γ = X H , i.e., i (Γ) ω = dH , and locally-Hamiltonian when i (Γ) ω is a closed 1-form. This is equivalent toΓ-invariance of ω , i.e., L Γ ω = 0.A system of differential equations is called bi-Hamiltonian if it can be written in two differentways in Hamiltonian form. Suppose a manifold M equipped with two different symplectic (cid:63) a r X i v : . [ m a t h - ph ] J a n J.F. Cari˜nena and M.F. Ra˜nadastructures ω and ω . A vector field Γ on T ∗ Q is said to be a bi-Hamiltonian vector field if itis Hamiltonian with respect to both symplectic structures, i.e., i (Γ) ω = dH and i (Γ) ω = dH . The two functions, H and H , are integrals of motion for Γ. A weaker form of bi-Hamiltoniansystem is when the only symplectic form is the first one ( ω is a closed but nonsymplectic2-form).We point out that an important example of bi-Hamiltonian system is the rational harmonicoscillator (non-central harmonic oscillator with rational ratio of frequencies) [8] H = 12 (cid:0) p x + p y (cid:1) + 12 α (cid:0) m x + n y (cid:1) . A symplectic form determines a Poisson bivector Λ that satisfies the vanishing of the Schoutenbracket [Λ , Λ] = 0 (this property is equivalent to the Jacobi identity); we note that there existnon-constant rank Poisson structure not related with symplectic structures. The compatibilitycondition between two different Poisson structures, Λ and Λ , means that the linear combina-tion Λ λ = Λ − λ Λ is a Poisson pencil, that is, it is a Poisson bivector for every value of λ ;therefore the corresponding bracket {· , ·} λ = {· , ·} − λ {· , ·} is a pencil of Poisson brackets.Bi-Hamiltonian systems are systems endowed with very interesting properties but, in general,it is quite difficult to find a bi-Hamiltonian formulation for a given Hamiltonian vector field,and for this reason it is useful to introduce the concept of quasi-bi-Hamiltonian system. Definition 1.
A vector field X on a symplectic manifold ( M, ω ) is called quasi-Hamiltonian ifthere exists a (nowhere-vanishing) function µ such that µX is a Hamiltonian vector field µX ∈ X H ( M ) . Thus i ( µX ) ω = dh for some function h .We call µ an integrating factor of the quasi-Hamiltonian system, because it is an integratingfactor for the 1-form i ( X ) ω , and we note that in this case the function h is a first integral of X .Note that this condition can alternatively be written as as i ( X )( µω ) = dh , but the point is thatthe 2-form µω is not closed in the general case.The scarcity of bi-Hamiltonian systems leads to relax the condition for the vector field beingHamiltonian to a simpler situation of quasi-Hamiltonian with respect to the second symplecticstructure. Definition 2.
A Hamiltonian vector field X on a symplectic manifold ( M, ω ) is called quasi-bi-Hamiltonian if there exist another symplectic structure ω , and a nowhere-vanishing function µ ,such that µX is a Hamiltonian vector field with respect to ω .This concept was first introduced in [4] in the particular case of systems with two degrees offreedom and it was quickly extended in [23, 24] for a higher-dimensional systems. Some recentpapers considering properties of this particular class of systems are [1, 2, 3, 4, 5, 6, 8, 14, 15, 23,24, 25, 36].The nondegeneracy of the canonical form ω provides a vector bundle isomorphism (cid:99) ω of T ( T ∗ Q ) on T ∗ ( T ∗ Q ), inducing an identification of vector fields and 1-forms on the phase space.A consequence is that the pair ( ω , ω ) determines a (1 ,
1) tensor field R defined as R = (cid:99) ω − ◦ (cid:99) ω ,that is, ω ( X, Y ) = ω ( RX, Y ) , ∀ X, Y ∈ X ( T ∗ Q ) . uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 3Note that in the definition of R the 2-form ω is necessarily closed but it can be nonsymplec-tic. If Γ is bi-Hamiltonian with respect to ( ω , ω ) then R is Γ-invariant, that is, L Γ R = 0,where L denotes the Lie derivative (this means that the characteristic polynomial of R is aninvariant for Γ, and consequently the coefficients of the polynomial are constants of motion).The Nijenhuis tensor N R of the tensor field R is defined by N R ( X, Y ) = R [ X, Y ] + [
RX, RY ] − R [ RX, Y ] − R [ X, RY ] . It has been proved that if Γ is a Hamiltonian dynamical system of the type described above andsuch that (i) The tensor N R of R vanishes, (ii) The tensor field R has n distinct eigenfunctions(that is, they are maximally distinct), then the eigenfunctions of R are in involution and thesystem is therefore completely integrable [12, 16, 17, 18, 21]. It is important to note that theeigenvalues of R are constants of motion for Γ even in the case that the two properties (i) and (ii)are not satisfied (but then the eigenfunctions are not in involution).It is known that the Liouville formalism characterize the Hamiltonians that are integrablebut it does not provide a method for obtaining the integrals of motion; therefore it has beennecessary to elaborate different methods for obtaining constants of motion (Hamilton–Jacobiseparability, Lax pairs formalism, Noether symmetries, Hidden symmetries, etc); the existence ofa bi-Hamiltonian structure with the above two mentioned properties (Nijenhuis torsion conditionand maximally distinct eigenvalues) can be considered as method to establish the Liouvilleintegrability of a system; because of this, these two properties are frequently included in thedefinition of a bi-Hamiltonian structure.Most of systems admitting a bi-Hamiltonian structure are separable systems; so these twoproperties (separability and double Hamiltonian structure) are properties very close related(see [18] for a detailed discussion of this question and [35] for the case of multiple separability).Quasi-bi-Hamiltonian systems are very less known than the bi-Hamiltonian ones but it seemsthat they are also related with separability. Nevertheless, in this case the tensor field R is notΓ-invariant and the eigenvalues of the tensor field R are not constants of motion. An interestingproperty is that it was shown in [4] that in the particular case of two degrees of freedom thefunction µ / det R is a constant of the motion.The potential of the Kepler problem is spherically symmetric and therefore it admits al-ternative Lagrangians (the existence of alternative Lagrangians for central potentials is stud-ied in [13, 20, 26]); recall that if there exist alternative Lagrangian descriptions then one canfind non-Noether constants of motion [7]. This system has been studied as a bi-Hamilltoniansystem by making use of different approaches; Rauch-Wojciechowski proved the existence ofa bi-Hamiltonian formulation but introducing an extra variable so that the phase space is odd-dimensional and the Poisson brackets are degenerate [34] and more recently [19] a bi-Hamiltonianformulation for the perturbed Kepler problem has also been studied by making use of Delaunay-type variables. Now in this paper we will analyze certain properties of the Kepler problem related with theexistence of quasi-bi-Hamiltonian structures. The following two points summarize the contentsof the paper. • First, we will study the existence of certain complex functions with interesting Poissonproperties and then we will prove that the superintegrability of the system is directlyrelated with the properties of these complex functions. As is well-known this system ismulti-separable (it separates in both polar and parabolic coordinates); so we first presentthe study making use of polar coordinates and then we undertake a similar study by J.F. Cari˜nena and M.F. Ra˜nadaemploying parabolic coordinates (the parabolic complex functions are different from thepolar ones). • Second, we prove that the above mentioned complex functions determine the existenceof several quasi-bi-Hamiltonian structures. This is done in two steps: first with complex2-forms and then with several real 2-forms. The properties of these geometric structuresand of the associated recursion operators are analyzed.It is important to note that this study is concerned with the existence of quasi-bi-Hamil-tonian structures (instead of bi-Hamiltonian). So we recall that if i (Γ) ω = λdH then X Γ ω = dλ ∧ dH (cid:54) = 0. Consequently the tensor field R is not Γ-invariant and theeigenvalues of R are not constants of motion.We must clearly say that the structures obtained by this method (wedge product of thedifferentials of complex functions) do not satisfy the above mentioned Nijenhuis torsion con-dition. So perhaps it is convenient to name them as weak quasi-bi-Hamiltonian structures (inopposition to strong structures satisfying the Nijenhuis condition). Nevertheless, the purposein this paper is not to prove the integrability of a system as consequence of a bi-Hamiltonianstructure, since it is perfectly known that the Kepler problem is not only integrable but alsosuperintegrable. The purpose is to study new and interesting properties of the Kepler problem.In fact, it has been proved that if a dynamical vector field satisfies certain properties (existenceof canonoid transformations [9, 10] or existence of non-symplectic symmetries [27, 28]) then it isHamiltonian with respect to two different structures without satisfying necessarily the Nijenhuiscondition.The structure of the paper is as follows: In Section 2 we study the Kepler problem by makinguse of polar coordinates ( r, φ ); we relate the superintegrability of the system with the existenceof two complex functions M r and N φ endowed with very interesting Poisson bracket propertiesand then we prove the existence of a quasi-bi-Hamiltonian structure making use of these twofunctions. Then in Section 3 we consider once more the same system but in terms of paraboliccoordinates ( a, b ) and we obtain a new quasi-bi-Hamiltonian structure (different to the previousone) making use of a similar technique but with new complex functions M a and M b . Finally inSection 4 we make some final comments. After these rather general comments we restrict our study to the Kepler problem in the Euclideanplane that, as it is well known, is superintegrable and multiseparable (polar and paraboliccoordinates).Let us first notice that in some cases the two-dimensional Euclidean systems possess certaininteresting properties. For example, if the potential V ( x, y ) takes the form V = A ( u ) + B ( v ), u = x + y , v = x − y , then it admits a new Hamiltonian structure (and also a Lax pair) [22];unfortunately the new structure is in most of cases constant (we mean that the new Poissonbracket is determined by a symplectic form as ω = dx ∧ dp y + dy ∧ dp x ).In the next paragraphs we will relate the existence of bi-Hamiltonian structures with theproperties of the two complex functions with interesting Poisson bracket properties. It is well known that the Hamiltonian of the two-dimensional Kepler problem H K = 12 (cid:32) p r + p φ r (cid:33) + V K , V K = − gr , < g ∈ R , uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 5is Hamilton–Jacobi (H-J) separable in polar coordinates ( r, φ ) and it is, therefore, Liouvilleintegrable with the angular momentum J = p φ as the associate constant of motion. Moreover,it is also known that it is a super-integrable system with the two components of the Laplace–Runge–Lenz vector as additional integrals of motion. Now we will prove that this property ofsuperintegrability can be related to the existence of certain complex functions with interestingPoisson bracket properties.Let us denote by M rj and N φj , j = 1 ,
2, the following real functions M r = p r p φ , M r = g − p φ r , and N φ = cos φ, N φ = sin φ. Then we have the following properties(i) ddt M r = { M r , H K } = − λM r , ddt M r = { M r , H K } = λM r , (ii) ddt N φ = { N φ , H K } = − λN φ , ddt N φ = { N φ , H K } = λN φ , where λ denote the following function λ = p φ r . The property (ii), representing the behaviour of the angular functions N φj , is true for all thecentral potentials V ( r ); but the property (i), behaviour of the functions M rj , is specific of thepotential of the Kepler problem.Consider next the complexification of the linear space of functions on the manifold and extendby bilinearity the Poisson bracket. If we denote M r and N φ the complex functions M r = M r + iM r , N φ = N φ + iN φ , then they have the following properties { M r , H K } = iλM r , { N φ , H K } = iλN φ , and consequently the Poisson bracket of M r N ∗ φ with the Kepler Hamiltonian vanishes { M r N ∗ φ , H K } = { M r , H K } N ∗ φ + M r { N ∗ φ , H K } = ( iλM r ) N ∗ φ + M r ( − iλN ∗ φ ) = 0 . We can summarize this result in the following proposition.
Proposition 1.
Let us consider the Hamiltonian of the Kepler problem H K = 12 (cid:32) p r + p φ r (cid:33) + V K , V K = − gr , Then, the complex function J defined as J = M r N ∗ φ is a ( complex ) constant of the motion. Of course J determines two real first-integrals J = J + iJ , { J , H K } = 0 , { J , H K } = 0 , J.F. Cari˜nena and M.F. Ra˜nadawhose coordinate expressions turn out to be J = Re( J ) = p r p φ cos φ − p φ r sin φ + g sin φ,J = Im( J ) = p r p φ sin φ + p φ r cos φ − g cos φ. That is, the two functions J and J are just the two components of the two-dimensional Laplace–Runge–Lenz vector.Summarizing, we have got two interesting properties. First, the superintegrability of theKepler problem is directly related with the existence of two complex functions whose Poissonbrackets with the Hamiltonian are proportional with a common complex factor to themselves,and second, the two components of the Laplace–Runge–Lenz vector appear as the real andimaginary parts of the complex first-integral of motion. Remark that N φ is a complex functionof constant modulus one, while the modulus of M r is a polynomial of degree four in the momentagiven by M r M ∗ r = ( p r p φ ) + (cid:32) g − p φ r (cid:33) = 2 p φ H K + g . Let us denote by Y the (complex) Hamiltonian vector field of J i ( Y ) ω = dJ , that obviously satisfies Y ( H K ) = { H K , J } = 0, and by Y r and Y φ the Hamiltonian vectorfields of M r and N φ : i ( Y r ) ω = dM r , i ( Y φ ) ω = dN φ . Their local coordinate expressions are, respectively, given by Y r = p φ ∂∂r + (cid:16) p r − i p φ r (cid:17) ∂∂φ − i (cid:32) p φ r (cid:33) ∂∂p r , and Y φ = (sin φ − i cos φ ) ∂∂p φ . Then, the vector field Y appears as a linear combination of Y r and Y ∗ φ ; more specifically wehave Y = N ∗ φ Y r + M r Y ∗ φ = Y + Y (cid:48) , Y = N ∗ φ Y r , Y (cid:48) = M r Y ∗ φ . The vector field Y is certainly a symmetry of the Hamiltonian system ( T ∗ Q, ω , H K ), but thetwo vector fields, Y and Y (cid:48) , are neither symmetries of the symplectic form ω (that is, X Y ω (cid:54) = 0and X Y (cid:48) ω (cid:54) = 0) nor symmetries of the Hamiltonian (that is, X Y H K (cid:54) = 0 and X Y (cid:48) H K (cid:54) = 0).Moreover, remark that they are not symmetries of the dynamics, because[ Y, Γ K ] (cid:54) = 0 , [ Y (cid:48) , Γ K ] (cid:54) = 0 , i (Γ K ) ω = dH K . More specifically:uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 7
Proposition 2.
The Lie bracket of the dynamical vector field Γ K with Y is given by [Γ K , Y ] = iJ X λ , where X λ is the Hamiltonian vector field of the function λ . Proof .
A direct computation leads to[Γ K , Y ] = Γ K ( N ∗ φ ) Y r + N ∗ φ [Γ K , Y r ] = − iλN ∗ φ Y r + N ∗ φ (cid:0) − X { H K ,M r } (cid:1) . where we have used that the Lie bracket of two Hamiltonian vector fields satisfies [ X f , X g ] = − X { f,g } . Note also that the Hamiltonian vector field of a product f g is given by X fg = f X g + gX f , and then the above Lie bracket becomes[Γ K , Y ] = − iλN ∗ φ Y r + iN ∗ φ ( X λM r ) = − iλN ∗ φ Y r + iN ∗ φ ( λY r + M r X λ ) = i ( M r N ∗ φ ) X λ . (cid:4) The vector field X λ on the right hand side represents an obstruction for Y to be a dynamicalsymmetry. Only when λ be a numerical constant the vector field Y (and also Y (cid:48) ) is a dynamicalsymmetry of Γ K .In the following Ω will denote the complex 2-form defined asΩ = dM r ∧ dN ∗ φ . The two complex 2-forms ω Y and ω (cid:48) Y obtained by Lie derivative of ω , i.e., L Y ω = ω Y , L Y (cid:48) ω = ω (cid:48) Y , are such L Y ω = i Y ( dω ) + d ( i Y ω ) = d ( i Y ω ) = d ( N ∗ φ dM r ) = − Ω , L Y (cid:48) ω = i Y (cid:48) ( dω ) + d ( i Y (cid:48) ω ) = d ( i Y (cid:48) ω ) = d ( M r dN ∗ φ ) = Ω . Using the preceding results we can prove:
Proposition 3.
The Hamiltonian vector field Γ K of the Kepler problem is a quasi-Hamiltoniansystem with respect to the complex -form Ω . Proof .
The contraction of the vector field Γ K with the complex 2-form Ω gives i (Γ K )Ω = Γ K ( M r ) dN ∗ φ − Γ K ( N ∗ φ ) dM r , and recalling thatΓ K ( M r ) = { M r , H K } = iλM r , Γ K ( N ∗ φ ) = { N ∗ φ , H K } = − iλN ∗ φ , we arrive to i (Γ K )Ω = ( iλM r ) dN ∗ φ + ( iλN ∗ φ ) dM r = iλd ( M r N ∗ φ ) . (cid:4) The complex 2-form Ω can be written asΩ = Ω + i Ω where the two real 2-forms, Ω = Re(Ω) and Ω = Im(Ω), take the formΩ = dM r ∧ dN φ + dM r ∧ dN φ = d ( p r p φ ) ∧ d (cos φ ) + d (cid:32) g − p φ r (cid:33) ∧ d (sin φ ) J.F. Cari˜nena and M.F. Ra˜nada= α dr ∧ dφ + α dφ ∧ dp r + α dφ ∧ dp φ , Ω = − dM r ∧ dN φ + dM r ∧ dN φ = − d ( p r p φ ) ∧ d (sin φ ) + d (cid:32) g − p φ r (cid:33) ∧ d (cos φ )= β dr ∧ dφ + β dφ ∧ dp r + β dφ ∧ dp φ with α ij and β ij being given by α = (cid:32) p φ r (cid:33) cos φ, α = p φ sin φ, α = p r sin φ + 2 (cid:16) p φ r (cid:17) cos φ, and β = − (cid:32) p φ r (cid:33) sin φ, β = p φ cos φ, β = p r cos φ − (cid:16) p φ r (cid:17) sin φ. Then we have i (Γ K )Ω = − λdJ , i (Γ K )Ω = λdJ , what means that Γ K is also quasi-bi-Hamiltonian with respect to the two real 2-forms ( ω , Ω )or ( ω , Ω ).Remark that the complex 2-form Ω is well defined but it is not symplectic. In fact, fromthe above expressions in coordinates we have Ω ∧ Ω = 0, Ω ∧ Ω = 0, and Ω ∧ Ω = 0, andtherefore we obtainΩ ∧ Ω = (Ω ∧ Ω − Ω ∧ Ω ) + 2 i Ω ∧ Ω = 0 . The distribution defined by the kernel of Ω, that is two-dimensional, is given byKer Ω = (cid:8) f Z + f Z | f , f : R × R → C (cid:9) , where the vector fields Z and Z are Z = ( α + iβ ) ∂∂r + ( α + iβ ) ∂∂p r , Z = ( α + iβ ) ∂∂r + ( α + iβ ) ∂∂p φ . Therefore it satisfies[Ker Ω , Γ K ] ⊂ Ker Ω . That is, Γ K preserves the distribution Ker Ω.If Y and Y are the Hamiltonian vector fields (with respect to the canonical symplecticform ω ) of the first integrals J and J , then the dynamical vector field Γ K is orthogonal to Y with respect to the structure Ω and it is also orthogonal to Y with respect to the structure Ω ,that is, i (Γ K ) i ( Y )Ω = 0 , i (Γ K ) i ( Y )Ω = 0 . Just to close the section we remark that had we applied this technique to the isotropic two-dimensional harmonic oscillator with frequency α we had obtained the function M r as M r = (cid:18) r p r p φ (cid:19) + i (cid:32) p r − p φ r + α r (cid:33) , (the angular function N φ would be the same) and the constants so obtained are but the compo-nents of the Fradkin tensor. This shows that the harmonic oscillator is an example of dynamicalsystem both bi-Hamiltonian and quasi-bi-Hamiltonian.uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 9 The bi-Hamiltonian structure ( ω , Ω) determines a complex recursion operator R defined asΩ( X, Y ) = ω ( RX, Y ) , ∀ X, Y ∈ X ( T ∗ Q ) . But as Ω and R are complex, we can introduce two real recursion operator R and R definedas Ω ( X, Y ) = ω ( R X, Y ) , Ω ( X, Y ) = ω ( R X, Y ) . We recall that (cid:99) ω is the map (cid:99) ω : X ( T ∗ Q ) → ∧ ( T ∗ Q ) given by contraction, that is (cid:99) ω ( X ) = i ( X ) ω , and then the nondegenerate character of ω means that the map (cid:99) ω is a bijection. Usingthis notation we can write the two operators R and R as follows R = (cid:99) ω − ◦ (cid:99) Ω , R = (cid:99) ω − ◦ (cid:99) Ω . Then we have the following properties(i) The coordinates expressions of R and R are R = − α ∂∂p φ ⊗ dr + (cid:20) α ∂∂r + α ∂∂φ + α ∂∂p r (cid:21) ⊗ dφ + α ∂∂p φ ⊗ dp r + α ∂∂p φ ⊗ dp φ and R = − β ∂∂p φ ⊗ dr + (cid:20) β ∂∂r + β ∂∂φ + β ∂∂p r (cid:21) ⊗ dφ + β ∂∂p φ ⊗ dp r + β ∂∂p φ ⊗ dp φ . (ii) R and R have two different eigenvalues doubly degenerate and one of them is null (thatis, λ = λ = 0, λ = λ (cid:54) = 0). Therefore we havedet[ R ] = det[ R ] = 0 , what is a consequence of the singular character of Ω and Ω .We close this section summarizing the situation we have arrived. We have first introducedtwo complex functions, M r and N φ , mainly because of the behaviour of their Poisson brackets.Then we have proved that they are interesting for two reasons: first because they determinethe existence of superintegrability (existence of additional constants of motion) and secondbecause they determine quasi-bi-Hamiltonian structures (first complex ( ω , Ω) and then real( ω , Ω , Ω )).Concerning the first point, in this case the additional constants of motion are just the com-ponents of the Runge–Lenz vector (that have been highly studied making use of different ap-proaches). Now we have arrived to a new property: they can also be obtained as a consequenceof this complex formalism.Concerning the second point, the two complex functions M r and N φ determine the abovementioned geometric structures (first complex and then real) but unfortunately they are dege-nerated (we recall that Ω ∧ Ω = 0, Ω ∧ Ω = 0). This can be considered as a limitation of0 J.F. Cari˜nena and M.F. Ra˜nadathese geometric structures. If a bi-Hamiltonians structure satisfies all the appropriate proper-ties (that is, symplectic forms, vanishing of the Nijenhuis torsion of the recursion operator R ,diagonalizable recursion operator R with functionally independent real eigenvalues) then it de-termines the Liouville integrability of the system. In fact, the aim of the approach presentedin this paper is not to prove the integrability of a system as consequence of a bi-Hamiltonianstructure as we start with a system known to be not only integrable but also superintegrable.The existence of (Ω , Ω ) must be considered, not as a method for arriving to the integrabilityof the system, but as a new and interesting property of the Kepler problem (for the momentonly of the two-dimensional system, the generalization to the three-dimensional case must beconsidered as an open question). In this section we will study the existence of new bi-Hamiltonian structures for the Keplerdynamics by making use of parabolic coordinates ( a, b ) defined as x = (cid:0) a − b (cid:1) , y = ab. Of course all previous results can be translated to this new language in such a way that thefunctions M r and M r are now given by M r = ( ap b − bp a )( ap a + bp b ) √ a + b , M r = ( ap b − bp a ) √ a + b − g, while functions N φ and N φ become N φ = a − b √ a + b , N φ = 2 ab √ a + b . But the important point is that the behaviour of the Kepler Hamiltonian in these coordinateswill permit us to obtain new results different from the previous ones.
The general form of a natural Euclidean Hamiltonian is H = 12 m (cid:18) p a + p b a + b (cid:19) + V ( a, b ) . in such a way that if the potential V is of the form V ( a, b ) = A ( a ) + B ( b ) a + b , then the Hamiltonian is Hamilton–Jacobi separable and it is, therefore, Liouville integrable withthe following quadratic function J = 1( a + b ) ( ap b − bp a )( ap b + bp a ) + 2 (cid:18) a B − b Aa + b (cid:19) as the second constant of motion (the first one is the Hamiltonian itself).uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 11For simplifying the following expressions we introduce the following notation: J = ap b − bp a , P x = ap a − bp b a + b , P y = ap b + bp a a + b . The Hamiltonian of the Kepler problem when written in parabolic coordinates is H K = 12 (cid:18) p a + p b a + b (cid:19) + V K , V K = − ga + b , (1)and the Kepler dynamics is given by the following vector fieldΓ K = (cid:18) p a a + b (cid:19) ∂∂a + (cid:18) p b a + b (cid:19) ∂∂b + (cid:18) p a + p b − g ( a + b ) (cid:19) a ∂∂p a + (cid:18) p a + p b − g ( a + b ) (cid:19) b ∂∂p b , in such a way that, as ω = da ∧ dp a + db ∧ dp b , we have i (Γ K )( da ∧ dp a + db ∧ dp b ) = dH K . The Hamiltonian H K is Hamilton–Jacobi separable in coordinates ( a, b ) and the associatedquadratic constant of motion is the component R x of the Laplace–Runge–Lenz vector R x = J P y − g (cid:18) a − b a + b (cid:19) . Let us now introduce the functions M aj and M bj , j = 1 ,
2, defined by M a = J p a √ a + b , M a = 2 ga − J p b √ a + b , and M b = J p b √ a + b , M b = 2 gb + J p a √ a + b . Then, the important property is that the Poisson bracket of the function M a with H K isproportional to M a while the Poisson bracket of M a with H K is proportional to M a , but withthe opposite sign: { M a , H K } = − λM a , { M a , H K } = λM a , and the same is true for the functions M b and M b , { M b , H K } = − λM b , { M b , H K } = λM b , where now λ denotes the following function λ = ap b − bp a ( a + b ) . Therefore the two complex functions M a and M b defined as M a = M a + iM a , M b = M b + iM b , satisfy { M a , H K } = iλM a , { M b , H K } = iλM b . Proposition 4.
The complex function K defined as K = M a M ∗ b is a ( complex ) constant of the motion for the dynamics of the Kepler problem described by theHamiltonian (1) . We omit the proof because it is quite similar to the proof of the previous Proposition 1.Note that the modulus of the complex functions M a and M b are given by M a M ∗ a = 2 (cid:0) J H K − gR x + g (cid:1) , M b M ∗ b = 2 (cid:0) J H K + gR x + g (cid:1) , and then M a M ∗ a + M b M ∗ b = 4 (cid:0) J H K + g (cid:1) . Of course the complex function K determines two real functions that are first integrals forthe Kepler problem K = K + iK , { K , H K } = 0 , { K , H K } = 0 , with K and K given by K = Re( K ) = M a M b + M a M b = J P x + g (cid:18) aba + b (cid:19) ,K = Im( K ) = M a M b − M a M b = − J H K . Remark that the function K is the other Laplace–Runge–Lenz constant, while K , that isa fourth order polynomial in the momenta, determines the angular momentum J as a factor. First we recall that the complex functions M a y M b are given by M a = (cid:18) J p a √ a + b (cid:19) + i (cid:18) ga − J p b √ a + b (cid:19) , M b = (cid:18) J p b √ a + b (cid:19) + i (cid:18) gb + J p a √ a + b (cid:19) . Let us now denote by Z the Hamiltonian vector field of the function K , i.e., i ( Z ) ω = dZ , such that Z ( H K ) = 0, and by Z a and Z b the Hamiltonian vector fields of the complexfunctions M a and M b , that is, i ( Z a ) ω = dM a , i ( Z b ) ω = dM b . Their coordinate expressions are given by Z a = (cid:18) ∂M a ∂p a (cid:19) ∂∂a + (cid:18) ∂M a ∂p b (cid:19) ∂∂b − (cid:18) ∂M a ∂a (cid:19) ∂∂p a − (cid:18) ∂M a ∂b (cid:19) ∂∂p b = Z a + iZ a , with Z a and Z a given by Z a = 1 √ a + b (cid:18) ( ap b − bp a ) ∂∂a + ap a ∂∂b − ( ap a + bp b ) a + b (cid:18) bp a ∂∂p a − ap a ∂∂p b (cid:19)(cid:19) ,Z a = 1 √ a + b (cid:18) bp b ∂∂a + ( bp a − ap b ) ∂∂b − ( ap a + bp b ) p b − gba + b (cid:18) − b ∂∂p a + a ∂∂p b (cid:19)(cid:19) , uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 13and Z b = (cid:18) ∂M b ∂p a (cid:19) ∂∂a + (cid:18) ∂M b ∂p b (cid:19) ∂∂b − (cid:18) ∂M b ∂a (cid:19) ∂∂p a − (cid:18) ∂M b ∂b (cid:19) ∂∂p b = Z b + iZ b , with Z b and Z b given by Z b = 1 √ a + b (cid:18) − bp b ∂∂a + (2 ap b − bp a ) ∂∂b − ( ap a + bp b ) a + b (cid:18) bp b ∂∂p a − ap b ∂∂p b (cid:19)(cid:19) ,Z b = 1 √ a + b (cid:18) ( ap b − bp a ) ∂∂a + ap a ∂∂b − ( ap a + bp b ) p a − gaa + b (cid:18) b ∂∂p a − a ∂∂p b (cid:19)(cid:19) . Now recalling that dZ = d ( M a M ∗ b ) = M ∗ b d ( M a ) + M a d ( M ∗ b ) , we obtain Z = M ∗ b Z a + M a Z ∗ b = Z + Z (cid:48) , where Z = M ∗ b Z a , Z (cid:48) = M a Z ∗ b . The following proposition is similar to that of Proposition 2 and we omit the proof.
Proposition 5.
The Lie bracket of the Kepler dynamical vector field Γ K with the vector field Z is given by [Γ K , Z ] = iK X λ , where X λ is the Hamiltonian vector field of λ solution of the equation i ( X λ ) ω = dλ . In the following we will denote by Ω ab the complex 2-form defined asΩ ab = dM a ∧ dM ∗ b = d (cid:20)(cid:18) J p a √ a + b (cid:19) + i (cid:18) ga − J p b √ a + b (cid:19)(cid:21) ∧ d (cid:20)(cid:18) J p b √ a + b (cid:19) − i (cid:18) gb + J p a √ a + b (cid:19)(cid:21) . Then the two 2-forms ω Z and ω (cid:48) Z obtained by Lie derivation of ω with respect to Z and Z (cid:48) aregiven by L Z ω = ω Z = − Ω ab , L Z (cid:48) ω = ω (cid:48) Z = Ω ab . Proposition 6.
The Hamiltonian vector field Γ K of the Kepler problem is quasi-Hamiltonianwith respect to the complex -form Ω ab . Proof .
This can be proved by a direct computation i (Γ K )Ω ab = Γ K ( M a ) dM ∗ b − Γ K ( M ∗ b ) dM a = ( iλM a ) dM ∗ b + ( iλM ∗ b ) dM a = iλd ( M a M ∗ b ) . (cid:4) The complex 2-form Ω ab can be decomposed asΩ ab = Ω ab + i Ω ab , where the two real 2-forms, Ω ab = Re(Ω ab ) and Ω ab = Im(Ω ab ), take the formΩ ab = dM a ∧ dM b + dM a ∧ dM b , Ω ab = − dM a ∧ dM b + dM a ∧ dM b , i (Γ K )Ω ab = − λdK , i (Γ K )Ω ab = λdK , what means that Γ K is also quasi-bi-Hamiltonian with respect to the two real 2-forms ( ω , Ω ab )and ( ω , Ω ab ).Once more we obtain that the factor λ determines that the system is quasi-bi-Hamiltonianinstead of just bi-Hamiltonian.The complex 2-form Ω ab is well defined but it is not symplectic. The kernel is two-dimensionaland it is invariant under the action of Γ K [Ker Ω ab , Γ K ] ⊂ Ker Ω ab . The coordinate expressions of Ω ab and Ω ab areΩ ab = 2 J ( a + b ) ( α da ∧ dp a + α da ∧ dp b + α db ∧ dp a + α db ∧ dp b + α dp a ∧ dp b ) , Ω ab = 2 g ( a + b ) ( β da ∧ dp a + β da ∧ dp b + β db ∧ dp a + β db ∧ dp b ) , with α ij and β ij being given by α = (cid:0) gb − ap a p b − bp b (cid:1) b, α = − (cid:0) ga − ap a − bp a p b (cid:1) b,α = (cid:0) − gb + ap a p b + bp b (cid:1) a, α = (cid:0) ga − ap a − bp a p b (cid:1) a,α = 2 J (cid:0) a + b (cid:1) , and β = (cid:0) abp a − a p b − b p b (cid:1) b, β = (cid:0) abp b − a p a − b p a (cid:1) b,β = (cid:0) − abp a + a p b + b p b (cid:1) a, β = (cid:0) − abp b − a p a − b p a (cid:1) a. We close this section with the following properties:(i) The two real 2-forms are not symplectic. In fact we have verified that Ω ∧ Ω = 0,Ω ∧ Ω = 0, and also Ω ∧ Ω = 0.(ii) These two 2-forms, Ω ab and Ω ab , determine two recursion operators ((1 ,
1) tensor fields) R ab and R ab defined asΩ ab ( X, Y ) = ω ( R ab X, Y ) , Ω ab ( X, Y ) = ω ( R ab X, Y ) , or in an equivalent way R ab = (cid:99) ω − ◦ (cid:100) Ω ab , R ab = (cid:99) ω − ◦ (cid:100) Ω ab . As in Section 2.3, a consequence of the singular character of Ω ab and Ω ab is thatdet[ R ab ] = det[ R ab ] = 0 . (iii) If we denote by Z and Z the Hamiltonian vector fields (with respect to the canonicalsymplectic form ω ) of the integrals K and K , then the dynamical vector field Γ K isorthogonal to Z with respect to the structure Ω and it is also orthogonal to Z withrespect to the structure Ω , that is, i (Γ K ) i ( Y )Ω ab = 0 , i (Γ K ) i ( Y )Ω ab = 0 . uasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem 15 The Kepler problem is separable in two different coordinate systems, and because of this, itis superintegrable with quadratic integrals of motion. Now we have proved that this super-integrability is directly related with the existence of certain complex functions possessing veryinteresting Poisson bracket properties and also that these functions are also related with theexistence of quasi-bi-Hamiltonian structures.We finalize with some questions for future work. First, as stated in the Introduction, quasi-bi-Hamiltonian structures is a matter that still remain as slightly studied (in contrast to thebi-Hamiltonian systems); so the particular case of the Kepler problem can be a good motivationto undertake a better study of these systems. Second, the existence of these complex functionsis not a specific characteristic of the Kepler problem; in fact, it has been proved that thesuperintegrability of certain systems recently studied (as the isotonic oscillator or the TTWor PW systems) [11, 29, 30, 31, 32, 33] is also related with such a class of complex functions.Therefore, an interesting open question is whether these other superintegrable systems are alsoendowed with bi-Hamiltonian structures or with quasi-bi-Hamiltonian structures; the startingpoint for this study must be a deeper analysis of the properties of these complex functionsmaking use of the geometric (symplectic) formalism as an approach.
Acknowledgments
This work has been supported by the research projects MTM–2012–33575 (MICINN, Madrid)and DGA-E24/1 (DGA, Zaragoza).
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