Non-integrability Criterium for Normal Variational Equations around an integrable Subsystem and an example: The Wilbeforce spring-pendulum

In this paper we analyze the non-integrability of the Wilbeforce pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.

1. Introduction. The Wilberforce pendulum consists of a mass hanging on a flexible spiral spring that is free to oscillate in both the standard longitudinal mode and the torsional mode. When the mass is lifted above its equilibrium point and released from rest, it oscillates up and down along a vertical line, slowly transferring its energy into a rotational oscillation. If the nuts screwed onto the vanes protruding from the sides of the mass are adjusted to give the appropriate moment of inertia such that the frequencies of the longitudinal mode and the torsional mode are the same, the pendulum will transfer its energy back and forth completely between these two modes os oscillation, see [3]. We shall refer to the Wilberforce springpendulum when the spring is free to swing in a plane, thus adding an extra degree of freedom. This system displays evidence inherent to chaotic dynamical systems.
Hence, establishing its non-integrability is our goal. The main tool for studying the non-integrability of this kinds of Hamiltonian systems is Morales-Ramis theory. One important criterion to obtain non-integrability of autonomous Hamiltonian systems by means of differential Galois theory is due to J. J. Morales-Ruiz and J.P Ramis [10], and Morales-Ruiz [9]. The Morales-Ramis theorem connects two notions: integrability of Hamiltonian systems and integrability of linear systems of equations. In particular, Morales-Ramis theory has been used to prove the non-integrability of spring pendulum systems, see [5,7,9]. This paper is organized as follows. We start with a brief description of the basics of the Morales-Ramis theory of non-integrability of the Hamiltonian system, geometric objects associated and the theory of linear homogeneous system of differential equation with singular points, following [4] and [1]. In the section 4 we present the necessary theoretical background to apply Morales-Ramis theory to the Wilberforce spring-pendulum Hamiltonian system. Finally, the Section 5 is devoted to a detailed analysis of apply the Theorem 3.4 in order to prove that the Wilberforce spring-pendulum Hamiltonian system is non-integrable in terms of meromorphic first integrals.

Variational equations and connections.
Let M be complex analytic manifold and X a meromorphic vector field on M. Let us denote by K the field of meromorphic functions on M and ∂ : K → K the derivative with respect to X so that (K, ∂) is a differential field. From now on we will write f for ∂f with f an element of K.
Let π : E → M be a vector bundle, that we assume to be meromorphically trivial. Let us recall that this always holds in the algebraic case due to the vanishing of the first Galois cohomology group of the linear group, and also holds if M is an open Riemann surface, due to Brauer theorem.
Let E be the space of meromorphic sections of π. A linear connection in the direction of X is a K ∂ linear map ∇ : E → E satisfying Leibniz rule ∇(f e) = (∂f )e + f ∇(e). Let us consider two vector bundles (E, ∇) and (E , ∇ ) over M endowed with linear connections in the direction of X. A morphism of connections is a K-linear map φ : Let {e 1 , . . . , e n } be a basis of E. The n 2 coefficients a ij ∈ K in the following expression are uniquely determined.
A meromorphic section e ∈ E is called horizontal if it satisfies ∇(e) = 0. Let us write e = n i=1 x i e i , x = (x 1 , . . . , x n ) t and x = δx whenever c ∈ K δ . It follows from equation (1) that x satisfy the system of linear differential equations, Therefore, the choice of a linear basis of E over K gives a one-to-one correspondence between linear connections in E and n × n systems of linear differential equations with coefficients in K.

Remark 1.
Let us assume now that M is a Riemann surface Γ, the differential Galois theory in this case is developed in [15], and can be generalized to the case in which the field C of constants of K is the field of complex numbers C, id est, the vector field X does not admit meromorphic first integrals. To a connection ∇ in the direction of X it corresponds an algebraic group Gal(E, ∇) which is embedded into GL(n, C) up to a conjugacy class. In the development of our application we will just make use of the following geometric properties of the differential Galois group.
. Then the canonical projections induce an injective morphism of algebraic groups, Those geometric properties and many other were exhaustively studied in the more general case of the Lie-Vessiot systems in [4]. The necessary differential Galois theory background here is given in Appendix A.
2.1. Generic variational connection. Let us consider M, X, K and ∂ as in Section 1. There are two equivalent ways of introducing the variational equation for X. The vector field X can be naturally prolonged to the tangent bundle TM by application of the chain rule as done in [1]. Equivalently we can consider the Lie bracket with X as a linear connection. Let T M be the space of meromorphic vector fields in M, then is a linear connection in TM in the direction of X. This is called the first variational connection of X on the generic point of M 2.2. Normal variational connection and equations. Let N be an invariant manifold of M with respect to the flow of X. The following exact sequence of vector bundles over N, implicitly defines the normal bundle to N in M. It is clear that the connection ∇ X restrict to TM| N and TN and therefore we have also an exact sequence of connections.
Let us consider a local system of coordinates x 1 , . . . , x r , y 1 , . . . , y m in M such that the equations of N are written, y 1 = 0, . . . , y m = 0, and those of X, Then, the equations of horizontal section for ∇, the variational equation along X restricted to N are written, In this system of coordinates, we can easily project the equations onto the normal bundle obtaining the equations for horizontal sections of∇, Let Γ be an invariant curve (Riemann surface) of X contained in N. Its invariance by X means that the connections discussed above specialize to the bundles restricted to Γ, The connection ∇ X is called the variational connection of X along the integral curve Γ, and∇| Γ is called the variational connection of X normal to N in M along Γ. The equations of horizontal sections for this last one are written a square system of m linear differential equations with meromorphic coefficients in Γ, or equivalently a meromorphic linear connection in Norm(N, M)| Γ . This is the so called variational equations of the flow of X normal to the invariant manifold N along the integral curve Γ.

Remark 2.
It is important to remark that the invariant curve Γ may not corresponds to a unique trajectory of the flow of X but also contain equilibrium points. This is important because equilibrium points in Γ are poles of ( X| Γ τ ) −1 and therefore singularities of the system (4). In many applications we also replace the original manifold M by some suitable compactification such that the considered invariant curve Γ has some equilibrium points at the infinity (see [9] pp. 70-74). The curve Γ may then contain singular points like nodes or cusps. This is coherent with the computations above, providing that the field of meromorphic functions on a singular Riemann surface coincides with the one of its desingularization 3. Non-integrability criterium.
3.1. Morales-Ramis theory. From now on let M be a complex symplectic manifold, of dimension 2n with simplectic form Ω. Let us consider a symplectic system of coordinates x 1 , . . . , x n , y 1 , . . . , y n . If H is an holomorphic function in M then the Hamiltonian vector field X H is defined intrinsically by the formula, and its expression in local coordinates is, The symplectic structure induces a Poisson bracket for functions defined on M , namely {F, G} = X F G. The functions F, G in M are said to be in involution if {F, G} = 0. A Hamiltonian system is called completely integrable by meromorphic functions if it admits n independent meromorphic first integrals in involution. Let Γ be an invariant curve for X H . Let us define the variational connection ∇ X H in direction of X as in Section 1. Then , (TM| Γ , ∇ X H | Γ ) is the variational conection to X along the integral curve Γ. The equation of horizontal sections, so called the first variational equation along Γ is written in symplectic coordinates, The Morales-Ramis theory, relates the Liouville integrability of Hamiltonian system and the Galois groups of differential equations. The following result (Morales-Ramis [10]) and subsequent generalizations are some of the most effective and useful known for the theoretical proof of non-integrability of Hamiltonian systems. Let us recall that an algebraic group G is said to be virtually abelian it its connected component of the identity G 0 is abelian (see Appendix A).
Theorem 3.1. Let H be a Hamiltonian, and X H its associated Hamiltonian vector field in M. Let Γ be an invariant curve for H. Assume that H is completely integrable by meromorphic functions independent in a neighborhood of Γ but not necessarily on Γ itself. Then the group Gal(TM| Γ , ∇ X H | Γ ) is virtually abelian.

3.2.
Analysis of normal variational connection. From now on let N be a symplectic submanifold of M, i.e., the pair (N, Ω| N ) is a symplectic manifold. Let 2r be the dimension of N and n = r + m.
There is a natural decomposition of the bundle TM| N → N as the direct sum of the tangent bundle to N and the normal bundle to N in M, Proof. Let us consider TN ⊥ ⊂ TM the bundle of vectors orthogonal to TN with respect to the symplectic form Ω. By hypothesis Ω| N is non-degenerated and therefore TN ⊥ is a supplementary bundle for TN. The exact sequence (3) identifies TN ⊥ with Norm(N, M) and we get the result.
Proof. It suffices to proof that TN ⊥ is an invariant bundle for ∇ X H | N . Let us take Y meromorphic section of TN ⊥ and Z meromorphic vector field in M. Let us note that L X Ω = Ω for any Hamiltonian vector field, and that [ X, Z] is defined as vector field tangent to N. Therefore, Proof. First, let us assume that Gal(∇ X | Γ ) is virtually abelian. We consider the exact sequence (3). By Remark 1 there is exhaustive group morphism, Assume that Gal(∇| Γ ) is virtually abelian. From Theorem 3.1 we have that Gal(∇ X| N | Γ ) is virtually abelian. It follows that Gal(∇ X ) is virtually abelian.
The following corollary directly follows from Theorem 3.1. Therefore, we can ask what is the use of Theorem 3.4. There are examples, e.g. [12], of application of the Morales-Ramis theorem in which the non-integrability is not seen in the normal variational equation but in the total variational equation. Theorem 3.4 describes a theoretical situation in which the non-integrability, if captured by the first order variational equation then it is captured with the normal variational equation. Corollary 1. Let H be a Hamiltonian system in M. Assume that N is invariant by the flow of X H . Assume that X H | N is completely integrable, as Hamiltonian system in N, by meromorphic functions. Then if X H is completely integrable in M by meromorphic functions then the differential Galois group of the variational equation to the flow of X H normal to N along Γ, Gal(∇| Γ ), is virtually abelian.

4.
The Wilbeforce-spring pendulum. The classical Wilbeforce springpendulum consists of a solid cylinder attached to a spring. The cylinder is free to perform oscillations along the vertical. At the same time when the spring is elongated it produces a torque proportional to the angle of torsion of the cylinder φ (see Figure 1). We add an extra degree of freedom by allowing the spring to swing on a fixed plane in space containing the vertical motions. Let r, θ denote polar coordinates, where r is the distance from the fixed point of the spring to center o mass of the cylinder and θ the angular deviation from the vertical axis. The Lagrangian is where k is the constant of the spring, r 0 is the unstretched length of the spring, λ gives the constant of proportionality of the angular restoring and is the coupling among the vertical and torsional modes. The mass of the cylinder is m and J its moment of inertia. Figure 1. The Wilberforce spring-pendulum. Taken from reference [3].
Remark 4. The invariant subsystem θ = 0,θ = 0 corresponds to the classical Wilberforce pendulum which is a quadratic Lagrangian thus a linear system.

4.1.
Adimensionalization. It will be convenient to reduce the seven parameters m, J, k, r 0 , , g to four by the introduction to variables a follows: Let = r 0 + mg/k, where the parameter is the ratio of frequencies of the linear pendulum and the spring. Then if r = ρ, Also we introduce a new time variable τ via dτ dt = ω s , with this substitution the Lagrangian (6) becomes where prime ( ) indicates differentiation with respect τ.
One checks easily that the constant factor can be dismissed leading to the adimensional Lagrangian, Let us introduce the following (dimensionless) parameters then the Lagrangian finally becomes The Hamiltonian corresponding to this coordinates assume the form In fact by scaling the angle φ and its conjugate momenta we can suppose in what follows that a = 1 resulting with three parameters: b, c, f 4.2. The Wilberforce linear subsystem. The equations of motion are given by From these equations, one checks easily that θ = P θ = 0 is an invariant subsystem, as asserted previously. The systems reduces in this case to the linear system That can be recast as two coupled oscillators with Hamiltonian (set P θ = θ = 0 in (9)) Let us exhibit the reduced system (12) as an integrable Hamiltonian system in a convenient normal form. For this purpose let z = ρ − 1 + f , so (13) becomes where of course the constant term can be neglected. The following proposition makes explicit the integrals of motion in this case.
Then naturally P 2 x1 + P 2 x2 = P 2 z + P 2 φ for any α. Imposing the condition that the mixed terms x 1 x 2 vanish we get the choice

Then the Hamiltonian becomes
where ω 2 1 = cos 2 α + b sin 2 α + c cos α sin α = 1 4.3. The normal variational equations. We next calculate the full variational equations of (10) along of the solution of (12), which can be written as It follows immediately from equations (18) that the tangencial equations along the invariant plane θ = P θ = 0 are Then this can be rewritten as The 2 × 2 lower matrix has precisely the eigenvalues −ω 2 1,2 given in (16), (17). Thus the eigenvalues of the full matrix are ±iω 1 , ±iω 2 whenever ω 2 1,2 = 0. The case ω 2 1,2 > 0 is obtained whenever c 2 < 4b, and the general solution of this system is with arbitrary constants A, B, C and D. In the same way, the case ω 2 1,2 < 0 is obtained whenever c 2 > 4b, and the general solution of this system is ρ(t) = A 2 sinh ω 1 t + B 2 cosh ω 2 t, φ(t) = C 2 sinh ω 1 t + D 2 cosh ω 2 t with arbitrary constants A 2 , B 2 , C 2 and D 2 .
In general, for ω 2 1,2 = 0, provided whenever c 2 = 4b, the general solution of such system is with arbitrary constants A 3 , B 3 , C 3 and D 3 .
On the invariant plane the normal variational equation becomes Then we may rewrite this systems as a second-order equation of the form 5. Galoisian analysis of the variational equation. Considering the solution ρ(t) = A sin ω 1 t + B cos ω 2 t, the variational equation (19) becomes to where ∂ t =˙= d dt (see appendices). To avoid triviality we assume f = 0, ω 1 = 0 and ω 2 = 0. By means of the change of dependent variable η = (A sin ω 1 t + B cos ω 2 t)θ this equation is transformed in the reduced form: The equations (20) and (21) cannot be algebrized using Hamiltonian Algebrization (see Appendix C) due to in general there is not exists r ∈ Q such that ω 2 = rω 1 , see [1,2] and references therein. To apply Morales-Ramis theory, and in particular our result (Theorem 2.) we need one particular solution for the variational equation in where its Galois group is not virtually abelian. Thus, we can assume A = 0 and the equation (21) is reduced to On the other hand, if we assume B = 0, we obtain Now, we proceed to apply the Hamiltonian algebrization for (22) considering B = 0. We can see that x = x(t) = cos(ω 2 t) is a Hamiltonian change of variable, where √ α = −ω 2 √ 1 − x 2 . Therefore, the algebrization of (22) is Thus, the equation (24) is equivalent to The equation (25) has four singularities: 0, 1, −1, ∞ which are of regular type. This means that this equation can be transformed into a Heun equation: With the translation x → x − 1 we obtain the Heun equation with parameters Through the change of variable ξ = 4 x(x − 2) η we obtain the reduced form of the equation (26) to apply the Kovacic algorithm: For our purposes, in order to apply the Kovacic algorithm (see Appendix B and see also [6]), we can write r as follows: Now we start the analysis of the equation (27) applying Kovacic's algorithm. We can see that Γ = {0, 1, 2, ∞} and •(r 0 ) = •(r 2 ) = •(r ∞ ) = 2 and •(r 1 ) = 1. By case 1, step 1, we fall into the conditions (c 1 ) for c = 1, (c 2 ) for c = 0, 2 and (∞ 2 ) for ∞. In this way we obtain By step 2 we obtain D = {0} and by step 3 we obtain that P 0 = 1 does not satisfy the relation (30). This means, for differential Galois theory (see Appendix A), that the Galois group of the variational equation is not a subgroup of the Borel group (except for λ = 0 or ω 2 = 0, avoided from the start). We follow with case 2 of Kovacic's algorithm in where we fall into the conditions (c 1 ) for c = 1, (c 2 ) for c = 0, 2 and (∞ 2 ) for ∞. In this way, by step 1, we obtain Now, by step 2, we obtain D = {0} and by step 3 we obtain that P 0 = 1 does not satisfy the relation (31). This means, again for differential Galois theory, that the Galois group of the variational equation is not conjugated to a subgroup of the infinite Dihedral group. Finally we look into the case 3 of Kovacic's algorithm. We first consider m = 4, thus, by step 1, we fall into the conditions (c 1 ) for c = 1, (c 2 ) for c = 0, 2 and (∞) for ∞. In this way we obtain By step 2 we obtain D = {0} and for instance the rational function θ and the polynomial S are given by By step 3, we have the monic polynomial of degree 0, that is P = 1, and the sequence of polynomials is obtained by means of the relation (32): , P 1 = 24x 6 + · · · + 3, P 0 = −216x 8 + · − 51 2 , P −1 = 3024x 10 + · + 144 non-identically zero! and for instance the Galois Group is not the tetrahedral group. Now we consider m = 6, thus, by step 1, we fall into the conditions (c 1 ) for c = 1, (c 2 ) for c = 0, 2 and (∞) for ∞. In this way we obtain By step 2 we obtain D = {0} and for instance the rational function θ and the polynomial S are given by By step 3, we have the monic polynomial of degree 0, that is P = 1, and the sequence of polynomials is obtained by means of the relation (32): . . . P 0 = −38016x 12 + · · · − 3123 4 , P −1 = 798336x 14 + · · · + 6156 non-identically zero! and for instance the Galois Group is not the octahedral group. Finally we consider m = 12, thus, by step 1, we fall into the conditions (c 1 ) for c = 1, (c 2 ) for c = 0, 2 and (∞) for ∞. In this way we obtain By step 2 we obtain D = {0} and for instance the rational function θ and the polynomial S are given by By step 3, we have the monic polynomial of degree 0, that is P = 1, and the sequence of polynomials is obtained by means of the relation (32): , P −1 = 473775654223872x 26 + · · · + 85760073216 non-identically zero! and for instance the Galois Group is not the icosahedral group. In conclusion, the Galois group of the equation (27) is the connected and unsolvable group SL(2, C).
In the same way we can consider the case when c 2 > 4b, that is, ρ(t) = A 2 sinh ω 1 t+B 2 cosh ω 2 t. Owing to the change of variable t → it transforms trigonometric functions into hyperbolic ones and after similar change of variables as considered before, we arrive again to a non-integrable Heun equation. In general, for c 2 = 4b which corresponds to ρ(t) = A 3 e ω1t + B 3 e ω2t we arrive to a non-integrable confluent Heun equation in where not all singularity is of regular type. We can use the fact that the identity connected component of the Galois group is preserved under algebrization process (see [1,2]), thus, in our case, the Galois group of the normal variational equation will be the same no matter the way in where we algebrize the differential equation.
In this paper we shown that all the singularities of the normal variational equation (27) are of regular type (Heun equation), as well we proven that such differential equation has not Liouvillian solutions and also we obtained that and Thus, assuming f ω 1 ω 2 = 0 for any B ∈ C * , we have proven the following result.
Theorem 5.1. Let the parameters of the Wilberforce-spring pendulum satisfy c 2 − 4b = 0, then the system is not integrable through meromorphic first integrals.
Proof. Observe that ω 2 1 is always positive. On the other hand ω 2 2 = 0 if and only if c 2 = 4b.
Appendix A. Differential Galois theory. The Galois theory of differential equations, also called Differential Galois Theory has been developed by Picard, Vessiot, Kolchin and currently by a lot of researchers. In particular, we focus in the Galois theory of linear differential equations, also known as Picard-Vessiot theory. Following [2] and also [1], we present here an algebraic model for functions and the corresponding Galois theory. Differential Fields. Let K be a commutative field of characteristic zero. A derivation of K is a map ∂ x : for all a, b ∈ K. We then say that (K, ∂ x ) (or just K, when there is no ambiguity) is a differential field with the derivation ∂ x . We assume that K contains an element x such that ∂ x (x) = 1. Let C denote the field of constants of K: It is also of characteristic zero and will be assumed to be algebraically closed. The coefficient field for a differential equation is defined as the smallest differential field containing all the coefficients of the equation.
Due to we will mostly analyze second order linear homogeneous differential equations, i.e equations of the form L := ∂ 2 x y + a∂ x y + by = 0, a, b ∈ K. so the rest of the theory will be explained in this context.
Picard-Vessiot Extension. Let L be a differential field containing K (a differential extension of K). We say that L is a Picard-Vessiot extension of K for L if there exist two linearly independent y 1 , y 2 ∈ L solutions of L such that L = K y 1 , y 2 (i.e L = K(y 1 , y 2 , ∂ x y 1 , ∂ x y 2 )) and L and K has the same field of constants C.
In what follows, we choose a Picard-Vessiot extension and the term "solution of L" will mean "solution of L in L". So any solution of L is a linear combination (over C) of y 1 and y 2 .
Differential Galois Groups A K-automorphism σ of the Picard-Vessiot extension L is called a differential automorphism if it leaves K fixed and commutes with the derivation. This means that σ(∂ x a) = ∂ x (σ(a)) for all a ∈ L and ∀a ∈ K, σ(a) = a.
The group of all differential automorphisms of L over K is called the differential Galois group of L over K and is denoted by DGal(L/K).
As σ commutes with the derivation, this extends naturally to an action on a fundamental solution matrix of the companion first order system associated with L.
This defines a faithful representation DGal(L/K) → GL(2, C) and it is possible to consider DGal(L/K) as a subgroup of GL(2, C). It depends on the choice of the fundamental system {y 1 , y 2 }, but only up to conjugacy.
Recall that an algebraic group G is an algebraic manifold endowed with a group structure. Let GL(n, C) denote, as usual, the set of invertible n × n matrices with entries in C (and SL(n, C) be the set of matrices with determinant equal to 1). A linear algebraic group will be a subgroup of GL(n, C) equipped with a structure of algebraic group. One of the fundamental results of the Picard-Vessiot theory is the following theorem.
The differential Galois group DGal(L/K) is an algebraic subgroup of GL(2, C). In fact, the differential Galois group measures the algebraic relations between the solutions (and their derivatives). It is sometimes viewed as the object which should tell "what algebra sees of the dynamics of the solutions".
In an algebraic group G, the largest connected algebraic subgroup of G containing the identity, noted G • , is a normal subgroup of finite index. It is often called the connected component of the identity. If G = G 0 then G is a connected group. When G 0 satisfies some property, we say that G virtually satisfies this property. For example, virtually solvability of G means solvability of G 0 and virtual abelianity of G means abelianity of G 0 .. Lie-Kolchin Theorem. Let G ⊆ GL(2, C) be a virtually solvable group. Then G 0 is triangularizable, i.e it is conjugate to a subgroup of upper triangular matrices.
Algebraic Subgroups of SL(2, C). We present below some examples of subgroups of SL(2, C). Irreducible subgroups The infinite dihedral group (also called meta-abelian group):

Reducible subgroups
, d ∈ C * and its finite subgroups D 2n (where c and d spans the n-th roots of unity). There are also three other finite irreducible (primitive) groups: the tetrahedral group A SL2 4 of order 24, the octahedral group S SL2 4 of order 48, and the icosahedral group A SL2 5 of order 120.
Integrability. We say that the linear differential equation L is (Liouville) integrable if the Picard-Vessiot extension L ⊃ K is obtained as a tower of differential fields K = L 0 ⊂ L 1 ⊂ · · · ⊂ L m = L such that L i = L i−1 (η) for i = 1, . . . , m, where either 1. η is algebraic over L i−1 , that is η satisfies a polynomial equation with coefficients in L i−1 . 2. η is primitive over L i−1 , that is ∂ x η ∈ L i−1 . 3. η is exponential over L i−1 , that is ∂ x η/η ∈ L i−1 .
We remark that the usual terminology in differential algebra for integrable equations is that the corresponding Picard-Vessiot extensions are called Liouvillian. The following theorem is due to Kolchin.
The equation L is integrable if and only if DGal(L/K) is virtually solvable.