Anisotropic Kepler and anisotropic two fixed centres problems
 860 Downloads
Abstract
In this paper we show that the anisotropic Kepler problem is dynamically equivalent to a system of two point masses which move in perpendicular lines (or planes) and interact according to Newton’s law of universal gravitation. Moreover, we prove that generalised version of anisotropic Kepler problem as well as anisotropic two centres problem are nonintegrable. This was achieved thanks to investigation of differential Galois groups of variational equations along certain particular solutions. Properties of these groups yield very strong necessary integrability conditions.
Keywords
Anisotropic Kepler problem Anisotropic two fixed centres problem Morales–Ramis theory Differential Galois theory Nonintegrability1 Introduction
Remark 1.1
As it was pointed out by Gutzwiller (1989), an anisotropic mass tensor together with gravitational interactions arises fairly often. For example the planar isosceles three body problem that reduces to a natural Hamiltonian system with two degrees of freedom, standard kinetic energy and the anisotropic Kepler potential with an additional radial term (Devaney 1980, 1982). Anisotropic Kepler problem appears also as a subproblem of the rhomboidal charged four body problem (Alfaro and PerezChavela 2000). In this paper we propose another realisations of the anisotropic Kepler systems as systems of two point masses whose motion is restricted by two holonomic constraints.
If \(\mu _1=\mu _2=\mu _3\) the above systems are just the spatial or planar classical Kepler problems, respectively. Hence, for these values of parameters, the systems are maximally superintegrable and their dynamics is regular. However, if \(\mu _{1}\ne \mu _2\) or \(\mu _2\ne \mu _3\), the dynamics of the respective systems is very irregular. It was investigated intensively in numerous papers, see e.g. Casasayas and Llibre (1984); Devaney (1982); Gutzwiller (1990). In particular in Gutzwiller (1977) it was shown that there is onetoone correspondence between a certain set of trajectories of twodimensional anisotropic Kepler problem and binary Bernoulli sequence. Thus strong chaotic behaviour of the system was proved. Arribas et al. (2003) proved the nonintegrability of the planar as well as spatial anisotropic Kepler problems. These proofs were obtained by analysis of properties of differential Galois groups of variational equations around some particular solutions.
Our motivation to revisit the anisotropic Kepler problem aroused from an analysis of constrained motions of material points. We noticed that the anisotropic Kepler problem is equivalent to a system of two points whose motion is restricted to the coordinates axes, see Sect. 3. This rather unexpected observation justifies other anisotropic models with an arbitrary power law of interactions between material points. Additionally, it also gave us a motivation to revisit the generalised two fixed centres problem.
In this paper we answer the above posed question. Our results formulated in Sect. 2 have the form of three theorems. Two of them concern the generalised anisotropic Kepler problem and the third gives the necessary and sufficient conditions for the integrability of the generalised anisotropic two fixed centres problem. Proofs of all theorems given in Sects. 4 and 5 are based on the Morales–Ramis theory, see MoralesRuiz (1999). Basic notions and certain facts from this theory used in this paper are given in Appendix. In Sect. 6 we summarise the obtained results and give final remarks.
2 Results
As it is usually accepted in the context of Hamiltonian mechanics, here by the integrability we always understand the integrability in the Liouville sense. The system is considered as a complex Hamiltonian system. The first integrals required for the integrability are assumed to be complex rational functions. Although later we explain that our results extend to the wider class of meromorphic function, we keep this restriction on the class of first integrals in order to avoid technical difficulties in formulation of theorems. For the Liouville integrability it is required that the first integrals are functionally independent on a certain “large” set. In the case of rational functions the functional dependence is equivalent to the algebraic dependence. So we require that first integrals, necessary for the Liouville integrability, are algebraically independent.
If \(n\in \mathbb {Z}\), then the considered Hamiltonians are rational functions of canonical variables \((\varvec{q},\varvec{p})\) and in this case it is natural to ask about integrability in this class of first integrals. A problem appears when \(n=\tfrac{1}{2} +l\) for a certain \(l\in \mathbb {Z}\). In this case Hamiltonian (1.7) as well as Hamiltonian (1.9) are neither rational nor meromorphic functions of variables \((\varvec{q},\varvec{p})\). However they are algebraic over \(\mathbb {C}(\varvec{q},\varvec{p})\). For a study of the integrability of systems with algebraic Hamiltonians with the methods used in this paper, certain mathematical constructions must be used, see paper of Combot (2013). Generally, one can extend the system introducing new variables in such a way that the new system is still Hamiltonian with a rational Hamilton function. We explain this construction in Sect. 3.1.
This is why considering system given by Hamiltonian (1.7) with \(n=\tfrac{1}{2} +l\) for a certain \(l\in \mathbb {Z}\), we assume that first integrals required for the integrability are rational functions of variables \((\varvec{q},\varvec{p},r)\) where \(r^2=\mu _1q_1^2 + \mu _2 q_2^2 +\mu _3 q_3^2\).
Below we formulate two theorems where “integrability” means integrability in the above described sense with obvious modifications for planar version of the system.
Theorem 2.1
Hamiltonian system generated by (2.1) with \(2n\in \mathbb {Z}\) is integrable if and only if either \(\mu _1=\mu _2\), or \(n=\pm 1\).
Remark 2.2
Theorem 2.1 has a negative character—we did not find a new integrable case. However, the most interesting part of our considerations is hidden in its proof which is relatively simply except one case. Namely, we have to show the following.
Proposition 2.3
Let us recall that according to our definition, here nonintegrability means that the system does not admit an additional rational first integral \(F(\varvec{q},\varvec{p},r)\), where \(r^2=q_1^2  q_2^2\). To prove the above proposition quite involved mathematical tools must be used.
For the the spatial system given by (1.7) result similar to this obtained for the planar is given in the following theorem.
Theorem 2.4
Hamiltonian system generated by (1.7) is integrable if and only if either \(\mu _1=\mu _2=\mu _3\), or \(n=\pm 1\).
Theorem 2.5

\(n=1/2\); this is the classical two centres problem which is separable in elliptic coordinates;

\(n=2\); this a natural Hamiltonian system with a nonhomogeneous potential of degree four separable in elliptic coordinates, see Lakshmanan and Sahadevan (1993).
3 Anisotropic Kepler problems as constrained mechanical systems
3.1 Planar anisotropic Kepler problem
Poincaré crosssection presented in Fig. 2 shows chaotic behaviour of the system.
3.2 Spatial anisotropic Kepler problem
4 Proofs of integrability theorems for anisotropic Kepler problems
4.1 Preliminary remarks
If n is an integer, then the considered Hamiltonians are rational functions of \((\varvec{q},\varvec{p})\) and we have to prove that the systems are not integrable in the Liouville sense with rational first integrals.
Just to simplify the exposition we prove Theorem 2.1 assuming that Proposition 2.3 is valid. The proof of Proposition 2.3 is presented at the end of this section.
4.2 Proof of Theorem 2.1
We have shown already that if \(n=\pm 1\), or \(\mu _1=\mu _2\), then the system is integrable. Hence we have to prove that if \(n\ne \pm 1\) and \(\mu _1\ne \mu _2\), then the system is not integrable.
We prove our theorem by a contradiction. Thus, let us assume that the system is integrable with \(n\ne \pm 1\) and \(\mu \ne 1\). Then, by Theorem 6.4, \(\lambda =\mu \) and \(\lambda '=1/\mu \) belong to certain items in the Morales–Ramis table (6.16).
Notice that items 3 and 4 of table (6.16) are excluded by assumptions because they correspond to \(n=\pm 1\).
We have two possibilities: either \(\mu =1/\mu \), or \(\mu \ne 1/\mu \). If \(\mu =1/\mu \), then either \(\mu =1\) or \(\mu =1\). Case \(\mu =1\) is excluded by assumptions but case \(\mu =1\) appears only in item 1 for \(p=2\) and \(n=3/2\). However, for these values of parameters the system is not integrable by Proposition 2.3.
Now we can assume that \(\mu \ne 1/\mu \). If \(\mu \ne 0\) and \(\mu  <1\), then we find only a finite number of possible choices for \(\mu \) and \(k=2n\) in the Morales–Ramis table. They are listed in Table 1. By a direct check we can verify that if a pair \((k,\mu )\) belongs to the above table, then \((k,1/\mu )\) does not belong to an item in the Morales–Ramis table.
Admissible values of \((k,\mu )\) with \(\mu <1\)
n  k  \(\mu \) 

n  k  \(\frac{2n+1}{4n}\) 
\(3/2\)  3  1 / 8, 5 / 8, 5 / 96, 77 / 96 
11 / 600, 551 / 600, 119 / 600, 299 / 600  
\(2\)  4  7 / 72, 55 / 72 
\(5/2\)  5  19 / 360, 319 / 360, 7 / 40, 27 / 40 
3 / 2  \(3\)  3 / 8, 7 / 8, 19 / 96, 91 / 96 
49 / 600, 589 / 600, 301 / 600, 481 / 600  
2  \(4\)  17 / 72, 65 / 72 
5 / 2  \(5\)  41 / 360, 341 / 360, 13 / 40, 33 / 40 
4.3 Proof of Theorem 2.4
We have to show that if \(n\ne 1\) and \(\mu _i\ne \mu _j\) for certain \(i,j\in \{1,2,3\}\), then Hamiltonian system given by (1.7) is not integrable. Without loss of generality, we assume that \(\mu _1\ne \mu _2\).
Now, we restrict our attention to eigenvalues \(\lambda =\mu \) and \(\lambda '=1/\mu \). Repeating the same reasoning as in proof of Theorem 2.1 we show that if \(\mu _1\ne \mu _2\) the system is not integrable and this finishes the proof.
4.4 Proof of Proposition 2.3
5 Proof of Theorem 2.5
For \(\mu =1\) integrability of generalised two fixed centres problem was investigated in Maciejewski and Przybylska (2004). One would like to obtain a similar result for \(\mu \ne 1\), however the additional parameter in the system makes the problem hard. More precisely, one can prove the nonintegrability for fixed values of parameter n. However, we do not know how to proceed in general case, even restricting vales of n to half integers. This is why, at first we show the following.
Proposition 5.1
If the anisotropic two fixed centres system given by Hamiltonian (1.9) is integrable, then the corresponding anisotropic Kepler problem is integrable.
Proof
Now, we can pass to the proof of Theorem 2.5. The case with \(\mu =1\) was considered in Maciejewski and Przybylska (2004). This is why, taking into account Theorem 2.1, we have to analyse only case \(n=1\). The results of our analysis can be summarized shortly in the following lemma.
Lemma 5.2
Proof
6 Discussion and comments
In a case of integer n Theorems 2.1 and 2.4 remain valid after the change of a class of first integrals required for the integrability, from rational to meromorphic functions of canonical variables \((\varvec{q},\varvec{p})\). This follows from the main Theorem 6.3 of the Morales–Ramis theory. If \(n=\tfrac{1}{2} +m\) with a certain \(m\in \mathbb {Z}\), then these theorems remain true when we extend the class of first integrals to meromorphic functions of variables \((\varvec{q},\varvec{p},r)\). In fact, if the system given by Hamiltonian (1.9) admits a meromorphic first integral \(I(\varvec{q},\varvec{p},r)\), then it is also a first integral of extended system (4.3). To show that the extended system does not admit additional first integral we can invoke the Ayoul–Zung theorem (Ayoul and Zung 2010), and use the same particular solution to show that the differential Galois group of respective variational equations has a nonAbelian identity component. The mentioned paper contains the extension of Theorem 6.3 about differential Galois obstructions for the meromorphic integrability of Hamiltonian systems to the nonHamiltonian case.
We are not sure if Theorem 2.5 remains valid if we extend the admissible integrals to meromorphic functions of variables \((\varvec{q},\varvec{p},r_1,r_2)\).
Let us remark that for a long time the Ziglin and Morales–Ramis theories were used for study integrability of systems having algebraic, not single valued Hamiltonians. The first who pointed out that this lack of the respect for the basic assumption of the theory can lead to erroneous conclusions was Combot (2013).
In all cases with an algebraic but nonrational potential we can take it as a new variable and proceed as described in Maciejewski and Przybylska (2016) to obtain an extended system. Here we decided to introduce additional variables which are just algebraic expressions occuring in formulae for algebraic potentials. This simply shows other possibility for obtaining the desired result which is a transformation of Hamilton’s equations with algebraic righthand sides into a system with rational righthand sides.
Let us notice that the mechanical model given by Hamiltonian (3.5) is not integrable for any positive masses. In fact, according to our Theorem 2.4, the system is integrable only when \(\mu _1=\mu _2=\mu _3\), but for positive \(m_1\) and \(m_2\), it is impossible, see (3.7). We thank to the anonymous referee for this observation.
Notes
Acknowledgments
The authors are grateful to the anonymous referee for providing helpful corrections and suggestions improving the text, in particular for recalling the Yoshida scaling method. This work was partially supported by Grant No. DEC2013/09/B/ST1/04130 of National Science Centre of Poland.
References
 Alfaro, F., PérezChavela, E.: The rhomboidal charged four body problem. In: Delgado, J., Lacomba, E.A., PérezChavela, E., Llibre, J. (eds.) Hamiltonian Systems and Celestial Mechanics (HAMSYS98), vol. 6 of World Sci. Monogr. Ser. Math., pp. 1–19. World Scientific, Singapore (2000)Google Scholar
 Arribas, M., Elipe, A., Riaguas, A.: Nonintegrability of anisotropic quasihomogeneous Hamiltonian systems. Mech. Res. Commun. 30, 209–216 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 Audin, M.: Les systèmes hamiltoniens et leur intégrabilité. Cours Spécialisés [Specialized Courses], vol. 8. Société Mathématique de France, Paris (2001)Google Scholar
 Ayoul, M., Zung, N.T.: Galoisian obstructions to nonHamiltonian integrability. C. R. Math. Acad. Sci. Paris 348(23–24), 1323–1326 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 Baider, A., Churchill, R.C., Rod, D.L., Singer, M.F.: On the infinitesimal geometry of integrable systems. In: Mechanics Say (Waterloo, ON, 1992), vol. 7 of Fields Inst. Commun., pp. 5–56. American Mathematical Society, Providence, RI (1996)Google Scholar
 Braden, H.W.: A completely integrable mechanical system. Lett. Math. Phys. 6(6), 449–452 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Casale, G.: Morales–Ramis theorems via Malgrange pseudogroup. Ann. Inst. Fourier (Grenoble) 59(7), 2593–2610 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 Casasayas, J., Llibre, J.: Qualitative analysis of the anisotropic Kepler problem. Mem. Am. Math. Soc. 52, 312 (1984)MathSciNetzbMATHGoogle Scholar
 Combot, T.: A note on algebraic potentials and Morales–Ramis theory. Celest. Mech. Dyn. Astron. 115, 397–404 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Devaney, R.L.: Triple collision in the planar isosceles threebody problem. Invent. Math. 60(3), 249–267 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Devaney, R.L.: Blowing up singularities in classical mechanical systems. Am. Math. Mon. 89(8), 535–552 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
 Duval, G., Maciejewski, A.J.: Integrability of Hamiltonian systems with homogeneous potentials of degrees \(\pm 2\). An application of higher order variational equations. Discrete Contin. Dyn. Syst. 34(11), 4589–4615 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 Duval, G., Maciejewski, A.J.: Integrability of potentials of degree \(k\ne \pm 2\). Second order variational equations between Kolchin solvability and Abelianity. Discrete Contin. Dyn. Syst. 35(5), 1969–2009 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 Fischer, W., Lieb, I.: A Course in Complex Analysis. Vieweg+Teubner Verlag, Berlin (2012)CrossRefzbMATHGoogle Scholar
 Gutzwiller, M.C.: The anisotropic Kepler problem in two dimensions. J. Math. Phys. 14, 139–152 (1973)ADSMathSciNetCrossRefGoogle Scholar
 Gutzwiller, M.C.: Bernoulli sequences and trajectories in the anisotropic Kepler problem. J. Math. Phys. 18(4), 806–823 (1977)ADSMathSciNetCrossRefGoogle Scholar
 Gutzwiller, M.C.: Multifractal measures and stability islands in the anisotropic Kepler problem. Phys. D 38(1–3), 160–171 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
 Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Interdisciplinary Applied Mathematics, vol. 1. Springer, New York (1990)zbMATHGoogle Scholar
 Kimura, T.: On Riemann’s equations which are solvable by quadratures. Funkcial. Ekvac. 12:269–281 (1969/1970)Google Scholar
 Kovacic, J.J.: An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2(1), 3–43 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 Lakshmanan, M., Sahadevan, R.: Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type. Phys. Rep. 224(1–2), 93 (1993)Google Scholar
 Maciejewski, A.J., Przybylska, M.: Nonintegrability of the generalised twocenters problem. Celest. Mech. Dyn. Astron. 89(2), 145–164 (2004)ADSCrossRefzbMATHGoogle Scholar
 Maciejewski, A.J., Przybylska, M.: Integrability of Hamiltonian systems with algebraic potentials. Phys. Lett. A 380(1–2), 76–82 (2016)ADSMathSciNetCrossRefGoogle Scholar
 MoralesRuiz, J.J.: In: Differential Galois Theory and Nonintegrability of Hamiltonian Systems, vol. 179 of Progress in Mathematics. Birkhäuser Verlag, Basel (1999)Google Scholar
 MoralesRuiz, J.J., Ramis, J.P.: Galoisian obstructions to integrability of Hamiltonian systems: statements and examples. In: Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaró, 1995), vol. 533 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 509–513. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
 MoralesRuiz, J.J., Ramis, J.P.: A note on the nonintegrability of some Hamiltonian systems with a homogeneous potential. Methods Appl. Anal. 8(1), 113–120 (2001a)MathSciNetzbMATHGoogle Scholar
 MoralesRuiz, J.J., Ramis, J.P.: Galoisian obstructions to integrability of Hamiltonian systems. I. Methods Appl. Anal. 8(1), 33–95 (2001b)MathSciNetzbMATHGoogle Scholar
 MoralesRuiz, J.J., Ramis, J.P.: Galoisian obstructions to integrability of Hamiltonian systems. II. Methods Appl. Anal. 8(1), 97–111 (2001c)MathSciNetzbMATHGoogle Scholar
 MoralesRuiz, J.J., Ramis, J.P.: Integrability of dynamical systems through differential Galois theory: a practical guide. In: Differential Algebra, Complex Analysis and Orthogonal Polynomials, vol. 509 of Contemp. Math., pp. 143–220. American Mathematical Society, Providence, RI (2010)Google Scholar
 MoralesRuiz, J.J., Ramis, J.P., Simó, C.: Integrability of Hamiltonian systems and differential Galois groups of higher variational equations. Ann. Sci. Éc. Norm. Supér 40(6), 845–884 (2007)MathSciNetzbMATHGoogle Scholar
 Przybylska, M., Szumiński, W.: Nonintegrability of flail triple pendulum. Chaos Solitons Fractals 53, 60–74 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Tosel, Juillard, E.: Meromorphic parametric nonintegrability; the inverse square potential. Arch. Ration. Mech. Anal. 152, 187–205 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
 Yoshida, H.: A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential. Phys. D 29(1–2), 128–142 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
 Ziglin, S.L.: Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I. Funct. Anal. Appl. 16, 181–189 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.