We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.
We show that the main theorem of Morales--Ramis--Simo about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. Namely, if a dynamical system is meromorphically integrable in the non-Hamiltonian sense, then the differential Galois groups of the variational equations (of any order) along its solutions must be virtually Abelian
In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.
We study the planar three-body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the Lagrangian solution.
We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.
We study integrability of the Euler-Poisson equations describing the motion of a rigid body with one fixed point in a constant gravity field. Using the Morales-Ramis theory and tools of differential algebra we prove that a symmetric heavy top is integrable only in the classical cases of Euler, Lagrange, and Kovalevskaya and is partially integrable only in the Goryachev-Chaplygin case. Our proof is alternative to that given by Ziglin ({\em Funktsional. Anal. i Prilozhen.}, 17(1):8--23, 1983; {\em Funktsional. Anal. i Prilozhen.}, 31(1):3--11, 95, 1997).
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.
In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schroedinger type with polynomial or trigonometrical potential. We analyze the integrability of Normal Variational Equation in Liouvillian sense using the Kovacic's algorithm. We compute all Galois groups of Schroedinger type equations with polynomial potential. We also introduce a method of algebrization that transforms equations with transcendental coefficients in equations with rational coefficients without changing essentially …
Let us consider a vector field $X$ meromorphic on a neighbourhood of an algebraic curve $\barΓ\subset \mathbb{P}^n$ such that $Γ$ is a particular solution of $X$. The vector field $X$ is $(l,n-l)$ integrable if it there exists $Y_1,\dots,Y_{l-1},X$ vector fields commuting pairwise, and $F_1,\dots,F_{n-l}$ common first integrals. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of $X$ in a neighbourhood of $Γ$. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation $NVE_1$ has a virtually dia…
We derive necessary conditions for integrability in the Liouville sense of natural Hamiltonian systems with homogeneous potential of degree zero. We derive these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution $\vd\in\C^n$ of nonlinear equations $\grad V(\vd)=\vd$. We proved that if the system integrable then the Hessian matrix $V''(\vd)$ has only integer eigenvalues and is semi-simple.
In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constru…
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,
n this paper we formulate necessary conditions for the integrability in the Jacobi sense of Newton equations $\ddot \vq=-\vF(\vq)$, where $\vq\in\C^n$ and all components of $\vF$ are polynomial and homogeneous of the same degree $l$. These conditions are derived from an analysis of the differential Galois group of the variational equations along special particular solutions of the Newton equations. We show that, taking all admissible particular solutions, we restrict considerably the set of Newton's equations satisfying the necessary conditions for the integrability. Moreover, we apply the obt…
We prove the non integrability of the colinear $3$ and $4$ body problem, for any masses positive masses. To deal with resistant cases, we present strong integrability criterions for $3$ dimensional homogeneous potentials of degree $-1$, and prove that such cases cannot appear in the $4$ body problem. Following the same strategy, we present a simple proof of non integrability for the planar $n$ body problem. Eventually, we present some integrable cases of the $n$ body problem restricted to some invariant vector spaces.
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability [40]. Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a b…
The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.
We study the planar three-body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the Lagrangian solution. We use the Ziglin's method and study the monodromy group of the corresponding normal variational equations.
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.
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.
In our previous paper: Integrability of Homogeneous potentials of degree $k = \pm 2$. An application of higher order variational equations, we tried to extract some particular structures of the higher variational equations (the ${VE}_p$ for $p >1 $), along particular solutions of some Hamiltonian systems. Then, we use them to get new Galois obstructions to the integrability of natural Hamiltonian with potential of degree $k = \pm 2$. In the present work, we apply the results of the previous paper, to the complementary cases, when the degrees of the potentials are relative integers $k$, with $|…
The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the higher variational equations $VE_{p}$ for $p\geq 2$, although complicated they are, have very particular algebraic structure. Preceisely they are solvable if $VE_{1}$ is virtually Abelian since they are solvable inductively by what we call the \emph{second level integrals}. We then give necessary and sufficient conditions in terms of these second level integral…
We consider a special type of triple pendulum with two pendula attached to end mass of another one. Although we consider this system in the absence of the gravity, a quick analysis of of Poincaré cross sections shows that it is not integrable. We give an analytic proof of this fact analysing properties the of differential Galois group of variational equation along certain particular solutions of the system.
We study the integrability of the general two-dimensional Zakharov-Shabat systems, which appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems, in the meaning of differential Galois theory, i.e., their solvability by quadrature. It becomes a key for obtaining analytical solutions to the PDEs by using the IST. For a wide class of potentials, we prove that they are integrable in that meaning if and only if the potentials are reflectionless. It is well known that for such potentials partic…
In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for…
This is an example of application of Ziglin-Morales-Ramis algebraic studies in Hamiltonian integrability, more specifically the result by Morales, Ramis and Simó on higher-order variational equations, to the well-known Friedmann-Robertson-Walker cosmological model. A previous paper by the author formalises said variational systems in such a way allowing the simple expression of notable elements of the differential Galois group needed to study integrability. Using this formalisation and an alternative method already used by other authors, we find sufficient conditions whose fulfillment would en…
In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in\C^n$, of degree $k\in\Z^\star$. The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V"(c)$ calculated at a non-zero point $c\in\C^n$, such that $V'(c)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V"(c)$ is not diagonalizable. We prove, among other things, that if $V"(…
We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theo…
This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=α(q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $α\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even are polynomial integrable. We present an elementary proof of this fact in the context of the polynomial bi-homogeneous potentials, as was introduced by Combot et al. \cite{Combot2020}. In addition, we…
We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the…
We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $ \sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.
This paper is a sequel of the reference \cite[§4.2, p.p. 1782--1783]{almp}, in where some families of quadratic polynomial vector fields related with orthogonal polynomials were studied. We extend such results that contain some details related with differential Galois Theory as well the inclusion of Darboux theory of integrability and qualitative theory of dynamical systems.
We study the integrability in the Liouville sense of natural Hamiltonian systems with a homogeneous rational potential $V(\vq)$. Strong necessary conditions for the integrability of such systems were obtained by an analysis of differential Galois group of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions putted on eigenvalues of Hessian $V"(\vd)$ calculated at a non-zero solution $\vd$ of equation $\grad V(\vd)=\vd$. Such solutions are called proper Darboux points. It was recently proved that for generic polynomial homogeneous p…
