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Published under licence by IOP Publishing Ltd V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 1 Some tastings in Morales-Ramis theory P Acosta-Humánez1,2, and G Jiménez3 1 Universidad Simón Bolívar, Barranquilla, Colombia 2 Instituto Superior de Formación Docente Salomon Urea, Santiago de los Caballeros, Republica Dominicana 3 Universidad del Norte, Barranquilla, Colombia E-mail: primitivo.acosta-humanez@isfodosu.edu.do, gjimenez@uninorte.edu.co Abstract. In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system. 1. Introduction The Morales-Ramis theory is a powerful tool for showing the nonintegrability of Hamiltonian systems. To understand the Morales-Ramis theory, we need to introduce two different notions of integrability: the integrability of Hamiltonian systems in Liouville sense and the integrability of linear differential equations in Picard-Vessiot sense. Further developments of Morales-Ramis theory, with contributions of the first author, can be found in [1-5]. This work summarizes some results of these papers among others. Our main contribution corresponds to the obtaining the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system. 1.1. Integrability of Hamiltonian systems Let us consider a n degrees of freedom Hamiltonian 𝐻. The equations of the flow of the Hamiltonian system, in a system of canonical coordinates, 𝑥#,⋯ , 𝑥&, 𝑦#,⋯𝑦& are written Equation (1). ẋ = ∂H ∂y , ẏ = ∂H ∂y (1) And they are known as Hamilton equations. We recall that the Poison brackets between f(x#, x0, x1, x2) and g(x#, x0, x1, x2) is given by Equation (2). {f, g} = 78 ∂f ∂y9 ∂g ∂x9 − ∂f ∂x9 ∂g ∂y9 ; < 9=# (2) We say that f and g are in involution when {f, g} = 0 also we say in this case that f and g commute under the Poisson bracket. In this way, we can write the Hamilton equations as follows: V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 2 ẋ = {H, x}, ẏ = {H, y} (3) A Hamiltonian H in C0< is called integrable in the sense of Liouville if there exist n independent first integrals of the Hamiltonian system in involution [6,7]. We will say that H is integrable by terms of rational functions if we can find a complete set of integrals within the family of rational functions. Respectively, we can say that H is integrable by terms of meromorphic functions if we can find a complete set of integrals within the family of meromorphic functions [8,9]. We denote by XA the Hamiltonian vector field, that is, the right-hand side of the Hamilton equations. In a general way, we deal with non-linear Hamiltonian systems. For suitability, without lost of generality, we can consider Hamiltonian systems with two degrees of freedom that is a Hamiltonian H in C2. Let Γ be an integral curve of XA, being parametrized by γ: t → (q#(t), q0(t), p#(t), p0(t)) the first variational equation (VE) along Γ is given by the Equation (4). ξ̇ = Hes(H(γ(t))) ∗ M 0 I −I 0O ξ , ξ̇ = Qξ#̇, ξ0̇, ξ1̇, ξ2̇R S , ξ = (ξ#, ξ0, ξ1, ξ2)S (4) 1.2. Picard-Vessiot theory The Picard-Vessiot theory is the Galois theory of linear differential equations. In the classical Galois theory, the main object is a group of permutations of the roots, while in the Picard-Vessiot theory is a linear algebraic group. For other applications of the Picard Vessiot theory due to the first author can be found in [10] and [11]. In the remainder of this paper we only work, as particular case, with linear differential equations of second order (see Equation (5)). yTT + ayT + by = 0, a, b ∈ ℂ(x) (5) Suppose that y#, y0 is a fundamental system of solutions of the differential equation. This means that 𝑦#, 𝑦0 are linearly independent over ℂ and every solution is a linear combination of these two. Let L = ℂ(x)[y#, y0] = ℂ(x)[y#, y0, y′#, y′0], that is the smallest differential field containing to ℂ(x) and {y#, y0}. The group of all differential automorphisms of L over ℂ(x) is called the Galois group of L over ℂ(x) and denoted by Gal(L/ℂ(x)). This means that for σ: L → L, σ(aT) = σT(a) and σ(a) = a, ∀a ∈ ℂ(x). If σ ∈ Gal(L/ℂ(x)) then σy#, σy0 is another fundamental system of solutions of the linear differential equation. Hence there exists a matrix A ∈ GL(2, ℂ) such that, Equation (6). σ 8 y# y0 ; = 8 σy# σy0 ; = A 8 y# y0 ; (6) • Theorem 1. The Galois group G = Gal(L/ℂ(x)) is an algebraic subgroup of GL(2, C). Moreover, the Galois group of a reduced linear differential equation ξTT = rξ, r ∈ ℂ(x) is an algebraic subgroup of SL(2, C). • Theorem 2. A linear differential equation is solvable integrable by terms of, Liouvillian functions, if and only if the connected component of the identity element of its Galois group is a solvable group. 2. Morales-Ramis theory We want to relate integrability of Hamiltonian systems to Picard-Vessiot theory. The following theorems treat this problem. • Theorem 3. Morales-Ramis [12]. Let H be a Hamiltonian in C0<, and γ a particular solution such that the normal variational equation (NVE) has regular (resp. irregular) singularities at the points of γ at infinity. Then, if H is completely integrable by terms of meromorphic (resp. rational) functions, then the Identity component of Galois Group of the NVE is abelian. V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 3 To understand completely this technical result, it is required a formal study of concerning to differential Galois theory and Morales-Ramis theory. We can illustrate this theorem through the following examples, but only for a basic level. Example, consider the Hamiltonian presented in Equation (7). H = 1 2 P#0 + 1 2 P00 − 2q#0 − 6q#q00 (7) The Hamiltonian equations, Equation (8), are: q̇# = p#, q̇0 = p0, ṗ# = 6q#0 + 6q00, ṗ0 = 12q#q0 (8) Taking the invariant plane q0 = p0 = 0 we have q̈# = 6q#0 a solution for this equation is q#(t) = # kl and the variational equation is 12ξ# = t0ξ̈#, which corresponds to a Cauchy-Euler equation, thus, the Galois group is abelian due to the Hamiltonian system is integrable. The following examples were taken from [12]. Consider the Hamiltonian of the Equation (9): H = # 0 P#0 + # 0 P00 − Q(q#) nll 0 + β(q#, q0)q01, (9) where Q(q#) is a polynomial and β(q#, q0) is a function of two variables with continuous partial derivative and lim nl→r stu(nv,nl) snl < ∞, 0 ≤ j ≤ 2. The Hamilton equations, Equation (10) and Equation (11), are: q̇# = p#, ṗ# = Q′(q#) q00 2 − ∂β(q#, q0) ∂q# q01 (10) q̇0 = p0 , ṗ0 = Q(q#)q0 − ∂β(q#, q0) ∂q0 q01 − 3 β(q#, q0)q00 (11) Taking the invariant plane q0 = p0 = 0 we have q#(t) = at + b and the variational equation, Equation (12), is: } 0 0 1 0 0 0 0 0 0 1 0 0 0 θ 0 0 � � ξ# ξ0 ξ1 ξ2 � = ⎣ ⎢ ⎢ ⎢ ⎡ξ̇# ξ̇0 ξ̇1 ξ̇2⎦ ⎥ ⎥ ⎥ ⎤ , (12) where θ = Q(q#), then Q(q#)ξ0 = ξ0̈. If Q(q#) is a polynomial then the Galois group is not abelian, although in some case is solvable [1,4], hence the Hamiltonian system is not integrable by Morales- Ramis Theorem. Example considerer the Hamiltonian of the Equation (13). H = # 0 P#0 + # 0 P00 − �� (�l�0��nv)l + λ� − λ#q00 − λ0q#q00 − λ1q#0q00 + β(q#, q0)q01, (13) where λ� ∈ ℂ, with λ1 ≠ 0, β(q#, q0) is a function of two variable with continuous partial derivative and lim nl→r stu(nv,nl) snl < ∞, 0 ≤ j ≤ 2. The Hamilton equations, Equation (14) and Equation (15), are: V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 4 q̇# = p#, ṗ# = −4λ1λ2 (λ0 + 2λ1q#)1 + λ0q00 + 2λ1q#q00 − ∂β(q#, q0) ∂q# q01 (14) q̇0 = p0 , ṗ0 = 2λ#q0 + 2λ0q#q0 + 2λ1q0q#0 − ∂β(q#, q0) ∂q0 q01 − 3 β(q#, q0)q00 (15) Taking q0 = p0 = 0 and setting H(q#, 0, p#, 0) = h we see that Equation (16) and Equation (17). h = 1 2 P#0 − λ2 (λ0 + 2λ1q#)0 + λ� (16) q̇# = P# = 82h + λ2 (λ0 + 2λ1q#)0 − 2λ�; # 0 (17) Now, we pick h = λ�, thus we have Equation (18) and Equation (19). dq# dt = 8 λ2 (λ0 + 2λ1q#)0 ; # 0 (18) λ0q# + λ1q#0 = ±(λ2) # 0t + c (19) For instance, the variational equation is, Equation (20). � 0 0 1 0 0 α 0 0 0 1 0 0 0 β 0 0 � � ξ# ξ0 ξ1 ξ2 � = ⎣ ⎢ ⎢ ⎢ ⎡ξ̇# ξ̇0 ξ̇1 ξ̇2⎦ ⎥ ⎥ ⎥ ⎤ , α = 24λ10λ2 (λ0 + 2λ1q#)2 , β = 2λ# + 2λ0q# + 2λ1 q#0 (20) Then, Equation (21). (2λ# + 2λ0q# + 2λ1 q#0)ξ0 = ξ0̈, (21) replacing in Equation (21), we have Equation (22). p(t)ξ0 = ξ0̈, (22) where p(t) = 2λ# + 2(±(λ2) v lt + c) and consequently, the Galois group is not abelian hence the Hamiltonian system is not integrable. Example. Consider the following Hamiltonian, Equation (23). H = 1 2 p#0 + 1 2 p00 + 1 aq#1 + bq#0q0 + cq01 (23) Where a, b, c ∈ ℂ The Hamilton equations are, Equation (24) and Equation (25): V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 5 q̇# = p#, p#̇ = 3aq#0 + 2bq#q0 (aq#1 + bq#0q0 + cq01)0 (24) q0̇ = p0, p0̇ = �nvl�1�nll Q�nv���nv�nl��nl�R l, (25) taking the invariant plane, 𝑞# = 𝑝# = 0 we have Equation (26) and Equation (27). p0̇ = 3 cq2 , q0̈ = 3 cq02 (26) q0(t) = 8− 25 2c; # � t 0 � (27) 𝑍(𝑡) = Q0, 𝑞0(𝑡), 0, 𝑝0(𝑡)R, the variational equation is Equation (28). ⎣ ⎢ ⎢ ⎢ ⎡ξ̇# ξ̇0 ξ̇1 ξ̇2⎦ ⎥ ⎥ ⎥ ⎤ = � 0 0 1 0 0 γ 0 0 0 1 0 0 0 β 0 0 � � ξ# ξ0 ξ1 ξ2 � , γ = − 4b 25t0 , β = 24 −25t0 (28) q̇# = p#, ṗ# = − 2� 0�kl q# then q̈# = − 2� 0�kl q# is a Cauchy-Euler equation, the Galois group is the identity, this group is abelian but we cannot state that the Hamiltonian system is integrable. The previous theorem of Morales-Ramis was extended by Morales-Ramis-Simó. • Theorem 4. Morales-Ramis-Simo [13]. Let 𝐻 be a Hamiltonian in 𝐶0&, and 𝛾 a particular solution such that the NVE has regular (resp. irregular) singularities at the points of 𝛾 at infinity. Then, if 𝐻 is completely integrable by terms of meromorphic (resp. rational) functions, then the identity component of Galois Group of any linearized high order variational equation is abelian. The following examples illustrate the way to compute the second order variational equation considerer the Hamiltonian, Equation (29). H = # 0 p#0 + # 0 p00 + # 0 a#q#0 + # 0 a0q00 + # 2 a�q#2 + # 2 a1q02 + # 0 a2q#0q00, (29) where the Hamilton equations are given by Equation (30) and Equation (31). q#̇ = p#, p#̇ = −a#q# − a�q#1 − a2q#q00 (30) q0̇ = p0, p0̇ = −a0q0 − a1q01 − a2q#0q0 (31) Taking as invariant plane Γ = {(q#, q0, p#, p0): q0 = p0 = 0} we obtain first variational, Equation (32). ξ̇(#) = A(t)ξ(#), A(t) = 8 00¡0 I0¡0 B0¡0 00¡0 ; , B0¡0 = Mc 0 0 ẟO , c = −a# − 3a�q#0, (32) V International Conference Days of Applied Mathematics Journal of Physics: Conference Series 1414 (2019) 012011 IOP Publishing doi:10.1088/1742-6596/1414/1/012011 6 where ξ(#) = Mξ# (# ) , ξ0 (# ), ξ1 (# ), ξ2 (# )O S , ẟ = −a0 − a2q#0 and q# = q#(t), being (q#(t), 0, q#̇(t),0) a particular solution of the Hamiltonian system over the invariant plane. The second variational equation is given by Equation (33) ξ̇(0) = A(t)ξ(0) + f(t), f(t) = (0,0, p, µ)S, ρ = −3a� Mξ# (# )O 0 − a2q# Mξ# (# )O 0 , (33) where ξ(0) = Mξ# (0 ) , ξ0 (0 ), ξ1 (0 ), ξ2 (0 )O S and µ = −2a2q#ξ# (#)ξ0 (#). 3. Contribution The following proposition is our original contribution to this paper. Assume the Hamiltonian system given by Equation (34). H = P#0 + P00 2 − 1 aq#¥ + bq0¥ , a ≠ 0,m > 2 (34) The differential Galois group of the variational equation corresponding to the invariant plane q0 = p0 = 0 and energy level ℎ = 0, is virtually abelian. Furthermore, the Galois group is independent of the choice of a and b. Proof. The subsystem in invariant plane is Equation (35). h = ¨vl 0 − # �nv© , a ≠ 0,m > 2, (35) then we obtain a particular solution for 𝑞# given by Equation (36). q#(t) = M¥�0 uk O l ©ªl , β = √2a, (36) for instance, the variational equation is given by Equation (37). ξ̇ = A(t)ξ , A(t) = 8 00¡0 Ι0¡0 B0¡0 00¡0 ; , B0¡0 = Mc 0 0 0O , c = ¥(¥�#) �nv©ªl , (37) where ξ = (ξ#, ξ0, ξ1, ξ2)S thus, we arrive to the Cauchy-Euler equation, Equation (38). ­l®v ­kl = 0¥(¥�#) (¥�0)lkl ξ#, (38) and for instance, the Galois group is always abelian. References [1] Acosta-Humánez P and Blazquez-Sanz D 2008 Non integrability of some hamiltonians with rational potentials Discrete and Continuous Dynamical Systems-B 10(2) 265 [2] Acosta-Humánez P 2009 Nonautonomous hamiltonian systems and Morales-Ramis Theory 1. 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