On bi-Hamiltonian formulation of the perturbed Kepler problem
aa r X i v : . [ n li n . S I] A p r On bi-Hamiltonian formulation of the perturbed Keplerproblem
Yu. A. Grigoryev, A. V. Tsiganov
St.Petersburg State University, St.Petersburg, Russiae–mail: [email protected], [email protected]
Abstract
The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of thefact that the graph of the Hamilton function is not a hypersurface of translation, which isagainst a necessary condition for the existence of the bi-Hamiltonian structure accordingto the Fernandes theorem. In fact, both the initial and perturbed Kepler systems areisochronous systems and, therefore, the Fernandes theorem cannot be applied to them.
Over the last few years Magri’s approach [9] to integrability through bi-Hamiltonian structureshad become one of the powerful methods of integrability of evolution equations applicable instudying both finite and infinite dimensional dynamical systems.The global, topological obstructions to the existence of a bi-Hamiltonian structure for ageneral completely integrable Hamiltonian system are discussed in [3, 8, 13, 16]. Some coun-terexamples were given to show that an existence of a bi-Hamiltonian structure is not alwayssatisfied around a Liouville torus for a given Arnold-Liouville system. For instance, Fernandes[8] and Olver [13] announced that the perturbed Kepler problem is a completely integrable sys-tem without a bi-Hamiltonian formulation with respect to non-degenerate compatible Poissonstructures in contrast with the initial Kepler problem.Below we explicitly present a few non-degenerate bi-Hamiltonian formulations of the per-turbed Kepler problem using the Bogoyavlenskij construction of a continuum of compatiblePoisson structures for the isochronous Hamiltonian systems [3].A bi-Hamiltonian vector field is one which allows two Hamiltonian formulations X = P dH = P ′ dK . (1.1)Here P and P ′ are the two compatible Poisson bivectors with vanishing Schouten brackets[ P, P ] = [
P, P ′ ] = [ P ′ , P ′ ] = 0 . (1.2)In generic cases bivectors P and P ′ could be degenerate and Hamiltonians H and K could befunctionally dependent. However, under an additional assumption one can construct a completesequence of functionally independent first integrals of X [3, 9, 10].The aim of this note is to present a bi-Hamiltonian formulation of the perturbed Keplervector field X defined by the Hamilton function H = p x + p y + p z − r + ǫ r , r = p x + y + z , (1.3)and canonical Poisson bivector P = (cid:18) − I 0 (cid:19) . (1.4)1he r − correction can be seen as due to an asymmetric mass distribution of the attractingbody (e.g., the gravitational attraction of the earth on a nearby orbiting artificial satellite) or tonon-Newtonian perturbation from the theory of general relativity (e.g., the motion of a particlein the Schwarzschild spherically symmetric solution of the Einstein equations), see [17]. Following to [3, 8] we start with a discussion of the bi-Hamiltonian vector fields in terms of theaction-angle variables.According to the classical Arnold-Liouville theory in the action-angle variables J k and ω k a given Hamiltonian vector field X = P dH has the simple form X : ˙ J k = 0 , ˙ ω k = ∂H∂J k , k = 1 , . . . , n. (2.5)Here H = H ( J , . . . , J n ) is a Hamilton function and the Poisson bivector P is the canonicalone P = n X k =1 ∂∂J k ∧ ∂∂ω k . (2.6)The problem of the existence of the action-angle variables in the neighborhood of an orbit, of alevel set or globally is discussed in [6], see also the recent review [12] and references within. Wewill look for a bi-Hamiltonian formulation only in the domain of definition of the action-anglevariables.The vector field X is called non-degenerate or anisochronous if the Kolmogorov conditionfor the Hessian matrix det (cid:12)(cid:12)(cid:12)(cid:12) ∂ H ( J , . . . , J n ) ∂J i ∂J k (cid:12)(cid:12)(cid:12)(cid:12) = 0 (2.7)is met almost everywhere in the given action-angle coordinates. This condition implies that thedense subsets of the invariant n -dimensional tori of X are closures of trajectories.In [3] Bogoyavlenskij proposed a complete classification of the invariant Poisson struc-tures for non-degenerate and degenerate Hamiltonian systems, see Theorem 1 and Theorem 8,respectively.Let us consider one trivial example of the generic Bogoyavlenskij construction for thedegenerate or isochronous Hamiltonian system. If in the domain of definition of the action-angle variables we have some nonzero derivative a = ∂H∂J m = 0 , we can make the following canonical transformation˜ J k = J k , ˜ ω k = ω k − ∂H∂J k a − ω m , k = m ˜ J m = H , ˜ ω m = a − ω m . (2.8)This canonical transformation does not add new singularities to the initial action-angle variablesand reduces the Hamiltonian to the simplest form H = ˜ J m . It allows us to construct bi-Hamiltonian formulation of the initial vector field X with twofunctionally dependent Hamiltonians H = ˜ J m and K = g ( ˜ J m ) , (2.9)2ut with the non-degenerate second Poisson bivector P ′ = n X k = m β k ( ˜ J k ) ∂∂ ˜ J k ∧ ∂∂ω k + (cid:18) dgd ˜ J m (cid:19) − ∂∂ ˜ J m ∧ ∂∂ ˜ ω m , (2.10)where β k ( ˜ J k ) are arbitrary nonzero functions and g ( ˜ J m ) is such that g ′ = 0. In this case theeigenvalues of the corresponding recursion operator N = P ′ P − are integrals of motion only,see examples of this type bi-Hamiltonian formulations of the Kepler problem in [3, 11, 16].In fact the Bogoyavlenskij theorems are much more powerful, however, in the followingdiscussion, we need only this particular case. Let us introduce the spherical coordinates r, θ and φr = p x + y + z , θ = arctan (cid:16) yx (cid:17) , φ = arccos z p x + y + z ! which are the radius, longitude and azimuth, respectively. In order to describe the correspondingmomenta we can use the so-called Mathieu generating function F = p x r sin φ cos θ + p y r sin φ sin θ + p z r cos φ , so that p r = ∂F∂r , p θ = ∂F∂θ , p φ = ∂F∂φ . In the spherical variables Hamiltonian H (1.3) takes the form H = 12 p r + p θ r + p φ r sin θ ! − r + ǫ r , such that the Hamilton-Jacobi equation H = h has an additive separable solution S = S r ( r ) + S θ ( θ ) + S φ ( φ )that allows us to introduce the action-angle variables. Let us begin with the definition of twoadditional commuting integrals of motion ℓ = p θ + p φ sin θ , m = p φ which are the total angular momentum and the component of the angular momentum alongthe polar axis. Then for H = h < J φ = 12 π I p φ dφ = m ,J θ = 12 π I p θ dθ = 12 π I r ℓ − m sin θ dθ = ℓ − m ,J r = 12 π I p r dr = 12 π I r h + 2 r − ℓ + ǫr dr = 1 √− h − √ ℓ + ǫ (2.11)3sing the standard integration method [2]. Then the corresponding angle variables can beobtained from the Jacobi equations ω r = ∂S∂J r = − rp r √− h + arccos 1 + 2 rh p h ( ℓ + ǫ ) ,ω θ = ∂S∂J θ = ℓ √ ℓ + ǫ arccos 1 − ℓ + ǫr p h ( ℓ + ǫ ) − ω r ! + arcsin ℓ cos θ √ ℓ − m ,ω φ = ∂S∂J φ = ω θ + φ + arcsin m cot θ √ ℓ − m . (2.12)The Hamiltonian H (1.3) in these action-angle variables takes the form H = − (cid:16) J r + p ( J θ + J φ ) + ǫ (cid:17) . (2.13)It is easy to prove that the graph of H (2.13) is not a hypersurface of translation in the actionvariables (2.11) [8] in contrast with the initial Kepler problem at ǫ = 0.The perturbed Kepler problem at ǫ = 0 and the unperturbed Kepler problem at ǫ = 0 aredegenerate or isochronous systemsdet (cid:12)(cid:12)(cid:12)(cid:12) ∂ H ( J , . . . , J n ) ∂J i ∂J k (cid:12)(cid:12)(cid:12)(cid:12) = 0with well-defined derivative for H = h < a = ∂H∂J r = − ( − h ) / . According to the Bogoyavlenskij theorem it allows us to get bi-Hamiltonian formulations ofthese systems in the domain of definition of the action-angle variables (2.11,2.12).
The action coordinates play an important role in classical dynamics because of their adiabaticinvariance, i.e. invariance under infinitesimally slow perturbations. In the Kepler problem,there are few well-known types of orbits and, therefore, there are few types of the action-anglevariables associated with different orbits. For instance, the Delaunay elements are valid onlyin the domain in phase space where there are the elliptic orbits [5]. On the other hand the twofamilies of the Poincar´e variables, which are the action-angle coordinates in the phase space ofthe Kepler problem, in the neighborhood of horizontal circular motions when eccentricities andinclinations are small [7]. There are also Delaunay-similar elements, Poincar´e-similar elementsand some other action-angle variables, which are well-defined in the neighborhood of differentorbits.For the perturbed Kepler problem we can also introduce the Delaunay type variables J = J φ , J = J φ + J θ , J = J r + √ ℓ + ǫ ,ω = ω φ − ω θ , ω = ω θ − ℓ √ ℓ + ǫ ω r , ω = ω r . (2.14)Recall that the Delauney variables have a geometrical meaning directly related to the descriptionof the orbits and their variations are much more significant for the astronomers than those of4artesian or spherical variables [7, 5, 17]. For ǫ = 0 variables J k , ω k (2.14) coincide with theclassical Delaunay elements (l , g , h , L , G , H): J ≡ L = √ a , ω ≡ l = n ( t − τ ) ,J ≡ G = p a (1 − e ) , ω ≡ g = ω ,J ≡ H = p a (1 − e ) cos i , ω ≡ h = Ω , where n is the mean motion, a is the semimajor axis of the orbit, e is the eccentricity, i is theinclination, ω is the argument of the perigee, Ω is the longitude of the ascending node, τ is thetime when the satellite passes through the perigee.In the Delaunay type variables the Hamilton function H (2.13) takes the form H = − J (2.15)and, therefore, we can construct the bi-Hamiltonian formulation of the perturbed Kepler modelwith the second bivector P ′ given by (2.10). For instance, if β ( J ) = J , β ( J ) = J , and K = − J , second bivector is equal to P ′ = X k =1 J k ∂∂J k ∧ ∂∂ω k = J J
00 0 0 0 0 J − J − J − J . The corresponding recursion operator has three functionally independent eigenvalues which arethe first integrals.In the initial action-angle variables ( ω r , ω θ , ω φ , J r , J θ , J φ ) this bivector has a more compli-cated form P ′ = J r + √ ( J φ + J θ ) + ǫ − η J θ + J φ
00 0 0 − η J θ J φ − J r − √ ( J φ + J θ ) + ǫ η η − J θ − J φ − J θ − J φ where η = ℓ ( ℓ − J r − √ ℓ + ǫ ) √ ℓ + ǫ , ℓ = J θ + J φ . In much the same way we can obtain other bi-Hamiltonian formulations associated with the twofamilies of the Poincar´e type action-angles variables or with other known types of the action-angle variables for the perturbed Kepler problem. Recall, for instance, that action variables atthe first Poicar´e type coordinate system are equal to Z = J θ , Γ = J r + p ℓ + ǫ − J φ − J θ , Λ = J r + p ℓ + ǫ , see [7] and references within.Using similar action-angle variables for the relativistic Kepler problem we can also obtainthe non degenerate bi-Hamiltonian formulation in contradiction with the Fernandes statementin [8]. 5 Conclusion
Let us duplicate textually the Fernandes theorem from [8]:
Theorem : A completely integrable Hamiltonian system is bi-Hamiltonian (satisfying (BH))if and only if the graph of the Hamiltonian function is a hypersurface of translation,relative tothe affine structure determined by the action variables.
An additional assumption (BH) is that the corresponding recursion operator N = P ′ P − has n functionally independent real eigenvalues λ , . . . , λ n .This formulation of the theorem is often mentioned in modern literature, see for instance[1, 3, 4, 13, 14, 15, 16]. Nevertheless, there is some trivial misprint because the author omitsone more necessary condition of the non degeneracy (2.7) of the Hamiltonian function, whichcould be found in the assumption ”ND” on the page 5 in [8] and in the proof of the theorem.The author forgets about this condition simultaneously in the formulation of the theoremand by considering physical examples of the applicability of this theorem. Thus, Fernandes[8] proclaims that perturbed Kepler problem does not have a bi-Hamiltonian formulation incontrast with the Kepler problem, see page 13 in [8]: ”Also we note that for the unperturbed Kepler problem ( ǫ = 0 ) the graph of the Hamiltonian is asurface of translation, and so it has a bi-Hamiltonian formulation (on the other hand, one canshow that the relativistic Kepler problem also does not have a bi-Hamiltonian formulation).” Let us repeat that initial and perturbed Kepler systems are degenerate systems and, therefore,we can not use the Fernandes theorem for both these systems simultaneously.This work was partially supported by RFBR grant 13-01-00061 and SPbU grant 11.38.664.2013.
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