Non-integrability of the dumbbell and point mass problem

This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. The non-integrability of this system is proven using differential Galois theory.

where r 1 , r 2 and r 3 are the position vectors of the respective masses, and U(r 1 , r 2 , is the potential. r d is the radius vector of the centre of mass of the dumbbell, and e := r 3 − r 2 is a unit vector along the dumbbell. It follows that (1.6) The system has five degrees of freedom. The configuration space of the point m 1 is R 2 , and the configuration space of the dumbbell is R 2 × S 1 . A configuration of the system is fully specified by r 1 , r d , and e. In these coordinates, the Lagrangian is as follows.
where I := l 2 m 2 m 3 m d , U(r 1 , r d , e) = U(r 1 , r d + µ 3 le, r d − µ 2 le). (1.8) The system possesses natural symmetries that are exploited to reduce the dimension of the system. Amongst these is translational symmetry. This is manifest by defining r := r d − r 1 as the vector between the mass m 1 and the dumbbell. A configuration of the system can be thus described by r, e, and r s defined by (1.9) As the Lagrangian (1.7) is written in the following form. is the reduced mass, and The components of r s are cyclic coordinates, and the motion of the centre of mass separates completely. Therefore, the term m s ṙ s 2 /2 is removed from the Lagrangian (1.11).
It is convenient to introduce dimensionless variables, taking l as the unit of length, m r as the unit of mass, and as the unit of time. Setting l = m r = T = 1, the Lagrangian (1.11) reduces to (1.16) The reduced system still has a symmetry: it is invariant with respect to the natural action of the group SO(2, R). In fact, the dynamics are oblivious to the orientation of the inertial frame. This symmetry can be used to reduce the dimension of the configuration space by one. This is achieved by describing the dynamics in a rotating frame in which the dumbbell is at rest. The transformation from the inertial frame to the rotating frame is given by r = AR, A ∈ SO(2, R). (1.17) An additional assumption, that e is parallel to x-axis of the rotating frame, allows for a coordinate representation of the transformation as follows.
The Lagrangian expressed in the coordinates R = [X 1 , X 2 ] T , and ϕ has the form where The generalised momenta P := [P 1 , P 2 ] T and P ϕ , are given by Coordinate ϕ is cyclic, so P ϕ is a first integral of the system. Thus, the Hamiltonian of the system is written as where α = 1/I, and γ is a fixed value of the first integral P ϕ corresponding to the cyclic variable ϕ. Hence, the equations of motion have the form (1.25) P Numerical experiments suggest that the considered system is not integrable. The complex behaviour of the system is apparent from the Poincaré cross-section. For fixed values of the parameters and the energy the equations (1.25) are integrated numerically by the Burlish-Stoer method [5]. The cross-section plane is specified as X 1 = 0, and the cross direction is chosenẊ 1 > 0. The coordinates of the cross-section are (X 2 , P 2 ). The crosssections presented in the subsequent figures are obtained for parameters µ = 1 and α = 4 61 . Figure 2 is a cross-section corresponding to energy e = −0.1 and γ = −1. Chaotic behaviour is evident. Two cross-sections for γ = 0 are shown in Figure 3 and Figure 4. Both of them attest to the non-integrability of the system. The problem is proving the non-integrability of the system analytically. It is attractive to try using the differential Galois approach to the problem of integrability (see e.g. [4]). However, as is explained in [1], application of this theory directly to the system (1.25) is invalid as the Hamiltonian function (1.24) is not single-valued.
Proposed in this paper is a solution addressing this deficiency, based on an extension of the system into a larger phase space. Coordinates of these additional dimensions are denoted by R 1 and R 2 . This approach further requires a definition of the time derivative consistent with system (1.25). Following this, coordinates R 1 and R 2 are considered as lengths of vectors (2.2) defines R 1 and R 2 as algebraic functions of (X 1 , X 2 ). The extended phase space is C 6 with coordinates Z := [X 1 , X 2 , P 1 , P 2 , R 1 , R 2 ]. The Hamiltonian (1.24) expressed in these coordinates reads system (1.25) can be extended to the following one.
It can be shown that matrix J(Z) defines a Poisson structure on C 6 . The corresponding Poisson bracket is given by the following formula where F and G are two smooth functions. The Poisson structure given by J(Z) is degenerated and has rank 4. It is demonstrable that polynomials G 1 and G 2 are Casimir functions of this structure, whose common levels are symplectic manifolds. In particular, is a symplectic manifold. System (2.5), when restricted to M 4 µ , is equivalent to (1.25). In the setting thus described the main result of this paper can be formulated in the following theorem.
On this manifold it can be taken that so the system (2.5), restricted to N, reduces tȯ where (X, P) := (X 1 , P 1 ) ∈ C 2 . This is a Hamiltonian system with one degree of freedom. Thus the manifold N is foliated by phase curves Γ e on energy levels K |N = e, that is For a generic value of e the level M e contains three phase curve. If e is real, two or three of these levels have a non-empty intersection with the real part of the phase space. In further consideration it is assumed that e ≥ 0, and the chosen phase curve Γ e contains the half-line (1/(1 + µ), ∞) × {0} ⊂ C 2 . In the phase space C 6 this curve is given by Application of the Morales-Ramis theory necessitates the linearisation of equations (2.5) along Γ e . It has the form This system has three first integrals k(z) := ∇K(Z) · z, g 1 (z) := ∇G 1 (Z) · z, g 2 (z) := ∇G 2 (Z) · z, Z ∈ Γ e . (3.7) On the level g 1 (z) = g 2 (z) = 0, the following equalities are satisfied where it is assumed that z = [x 1 , x 2 , p 1 , p 2 , r 1 , r 2 ] T . On Γ e , coordinates (R 1 , R 2 ) can be expressed by X 1 , Hence, on the level g 1 (z) = g 2 (z) = 0, variables (r 1 , r 2 ) can be eliminated from the variational equations (3.6). The reduced system of variational equations has the form d dt where and (X, P) ∈ Γ e . This system splits into two subsystems. Variables (u 1 , v 1 ) describe variations in the invariant plane N. In fact, (u 1 , v 1 ) is a vector tangent to N at point (X, P), hence the subsystem in variables (u 1 , v 1 ) is called tangential. The second subsystem, corresponding to variables (u 2 , v 2 ), yield the normal variational equations. The latter subsystem can be expressed by a single second-order equation, where u = u 2 , and a := − 2αXP 1 + αX 2 , b := 2αP 2 1 + αX 2 + where the energy integral has been used.
Making the next change of variables, (3.14) is transformed to the canonical form, with coefficients given as (3.17) Here β 2 = (1 + µ) 2 /α and coefficients p i of P are given in the Appendix in (A.1). This equation has seven singularities, and z 7 = ∞, providing energy is chosen such that Points z 1 , . . . , z 6 are regular poles of second order and the order of infinity is 1 provided that E(1 + µ) = 0. Here, the order of a singular point is defined as in [2].
The differences of exponents at the singularities in C are as follows.
It is evident that the differences of the exponents at singularities z 3 and z 4 are integer, and thus, in local solutions around these points, logarithmic terms may appear. Application of the method described in Appendix B verifies the presence of such terms. For singularity z 3 , solutions of the indicial equation are α 1 = 3/2 and α 2 = −1/2. The relevant one is α = α 1 = 3/2. The expansion of r(z − z 3 ) 2 , according to (B.4), gives coefficients r 0 , r 1 , and r 2 . Then f 1 , f 2 and g 1 , g 2 are determined from (B.6) and (B.12) respectively. Since s = α 1 − α 2 = 2, the coefficient g 2 , which multiplies the logarithm, must be found, and it is given as . (3.20) Examination of the real and imaginary part of g 2 yields the following conditions.
The condition precluding a logarithm in a local solution around z 4 is g * 2 = 0, where * denotes complex conjugation. The corresponding solutions are the same as those given previously.
In the first solution in (3.21), condition gives with the only non-negative solution m 1 = 0. The second solution in (3.21) also yields m 1 = 0 only. If conditions (3.21) are not satisfied, the two linearly-independent local solutions w 1 and w 2 of (3.16) in a neighbourhood of z * = z 3 or z * = z 4 have the following forms.
where f (z) and h(z) are holomorphic at z * and f (z * ) = 0. The local monodromy matrix, which corresponds to the continuation of the matrix of fundamental solutions along a small loop encircling z * counter-clockwise, gives rise to a triangular monodromy matrix, (For details, see [3].) A subgroup of SL(2, C) generated by a triangular non-diagonalizable matrix is not finite, and thus the differential Galois group is not finite either. Moreover, the differential Galois group G of this equation is not any subgroup of the dihedral group because such subgroups contain only diagonalizable matrices. Thus, G is either the full triangular group or SL(2, C). But since the order of infinity is one, the necessary condition that G is the full triangular group is not satisfied (see Lemma B.2). This means the that the differential Galois group of (3.16) is SL(2, C), with a non-Abelian identity component equal to the whole group SL(2, C).