Kovacic's Algorithm Research Articles

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

Integrability by quadratures of the problem of rolling motion of a heavy homogeneous ball on a surface of revolution of the second order

In this paper, we consider the problem of rolling of a heavy homogeneous ball on a perfectly rough surface of revolution. Usually, when considering this problem, it is convenient to specify explicitly the surface along which the center of the ball moves during rolling, instead of the surface along which the ball rolls. The surface on which the center of the ball moves is equidistant to the surface on which the ball is rolling. It is well known, that the considered problem is reduced to the integration the second order linear homogeneous differential equation. In this paper we assume, that the surface along which the center of the ball moves is a non - degenerate surface of revolution of the second order. Using the Kovacic algorithm we prove that the general solution of the corresponding linear differential equation can be found explicitly. This means, that in this case the problem of rolling of a ball on a surface of revolution can be integrated by quadratures.

Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case

AbstractIn 1890, Hess found new special case of integrability of Euler–Poisson equations of motion of a heavy rigid body with a fixed point. In 1892, Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second‐order linear differential equation. In this paper, the corresponding linear differential equation is derived and its coefficients are presented in the rational form. Using the Kovacic algorithm, we proved that the Liouvillian solutions of the corresponding second‐order linear differential equation exist only in the case, when the moving rigid body is the Lagrange top, or in the case, when the constant of the area integral is zero.

Liouvillian solutions of Whittaker-Ince equation

We apply Kovacic's algorithm, a tool that is developed from differential Galois theory, to discuss the existence of Liouvillian solutions of Whittaker-Ince equation, ellipsoidal wave equation and the Picard-Fuchs equation of a K3 surface. We determine the necessary and sufficient conditions of having Liouvillian solutions for Whittaker-Ince equation when a parameter vanishes; as well as a sufficient condition of having Liouvillian solution when this parameter is non-zero. On the other hand, we prove that ellipsoidal wave equation has no Liouvillian solution. We generalize a Picard-Fuchs equation for certain K3 surface and show that a particular case of this Picard-Fuchs equation cannot have any Liouvillian solutions.

Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case

Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case

Liouvillian solutions in the problem of rolling of a heavy homogeneous ball on a surface of revolution

The problem of rolling without sliding of a homogeneous ball on a fixed surface under the action of gravity is a classical problem of nonholonomic system dynamics. Usually, when considering this problem, following the E. J. Routh approach it is convenient to define explicitly the equation of the surface, on which the ball’s centre is moving. This surface is equidistant to the surface, over which the contact point is moving. From the classical works of E. J. Routh and F. Noether it was known that if the ball rolls on a surface such that its centre moves along a surface of revolution, then the problem is reduced to solving the second order linear differential equation. Therefore it is interesting to study for which surface of revolution the corresponding second order linear differential equation admits Liouvillian solutions. To solve this problem it is possible to apply the Kovacic algorithm to the corresponding second order linear differential equation. In this paper we present our own method to derive the corresponding second order linear differential equation. In the case when the centre of the ball moves along the ellipsoid of revolution we prove that the corresponding second order linear differential equation admits a liouvillian solution.

Investigation of the Motion of a Heavy Body of Revolution on a Perfectly Rough Plane by the Kovacic Algorithm

Investigation of various problems of mechanics and mathematical physics is reduced to the solution of second-order linear differential equations with variable coefficients. In 1986, the American mathematician J. Kovacic proposed an algorithm for solution of a second-order linear differential equation in the case where the solution can be expressed in terms of so-called Liouville functions. If a linear second-order differential equation has no Liouville solutions, the Kovacic algorithm also allows one to ascertain this fact. In this paper, we discuss the application of the Kovacic algorithm to the problem of the motion of a heavy body of revolution on a perfectly rough horizontal plane. The existence of Liouville solutions of the problem is examined for the cases where the rolling body is an infinitely thin disk, a disk of finite thickness, a dynamically symmetric torus, a paraboloid of revolution, and a spindle-shaped body.

Galoisian approach to complex oscillation theory of some Hill equations

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponent of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. These are solutions obtainable by suitable differential field extension constructions. In particular, we establish a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of a biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality, and satisfy Fredholm integral equations over suitable integration regions in $\mathbb{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

Solutions in quadratures to the CEV model

Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β, the elasticity of volatility. We use Kovacic's algorithm to derive, for all half-integer values of β, all solutions 'in quadratures' of the CEV ordinary differential equation (CEV ODE). These solutions give rise, by separation of variables, to simple solutions to the CEV partial differential equation. In particular, when β = …, …5⁄2, …2, …3⁄2, …1, 1, 3⁄2, 2, 5⁄2, …, we obtain four classes of denumerably infinite elementary function solutions, when β = …½ and β = ½ we obtain two classes of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions.

Existence of Liouvillian Solutions in the Problem of Rolling Motion of a Dynamically Symmetric Ball on a Perfectly Rough Sphere

The problem of rolling without sliding of a rotationally symmetric rigid body on a motionless sphere is considered. The rolling body is assumed to be subjected to forces whose resultant force is applied to the center of mass G of the body, directed to center O of the sphere, and depends only on the distance between G and O. In this case, the process of solving this problem is reduced to integrating the second-order linear differential equation with respect to the projection of the angular velocity of the body onto its axis of dynamic symmetry. Using the Kovacic algorithm, we search for Liouvillian solutions of the corresponding second-order linear differential equation. We prove that all solutions of this equation are Liouvillian in the case when the rolling rigid body is a nonhomogeneous dynamically symmetric ball. The paper is organized as follows. In the first paragraph, we briefly discuss the statement of the general problem of motion of a rotationally symmetric rigid body on a perfectly rough sphere. We prove that this problem is reduced to solving the second-order linear differential equation. In the second paragraph, we find Liouvillian solutions to this equation for the case when the rolling rigid body is a dynamically symmetric ball.

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.

Nonexistence of liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough plane

Using the Kovacic algorithm, the nonexistence of liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough horizontal plane for almost all values of parameters of the problem is proven.

Solvable Cases in the Problem of Motion of a Heavy Rotationally Symmetric Ellipsoid on a Perfectly Rough Plane

We consider a classical problem of nonholonomic system dynamics – the problem of motion of a rotationally symmetric body on a fixed perfectly rough plane in a case when the moving body is a rotationally symmetric ellipsoid. Using the Kovacic algorithm we found several conditions under which equations of motion of the ellipsoid can be completely solved in quadratures.

Liouvillian propagators, Riccati equation and differential Galois theory

In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.

Non-integrability of a class of Painlevé IV equations as Hamiltonian systems

In this paper, we will prove the rational non-integrability of a class of Hamiltonian systems associated with Painlevé IV equation by using Morales-Ramis theory and Kovacic's algorithm, which, to some extent, also implies the non-integrability of the fourth Painlevé equation itself.

Non-integrability of geodesic flow on certain algebraic surfaces

This Letter addresses an open problem recently posed by V. Kozlov: a rigorous proof of the non-integrability of the geodesic flow on the cubic surface xyz=1. We prove this is the case using the Morales–Ramis theorem and Kovacic algorithm. We also consider some consequences and extensions of this result.

Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales–Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.

Galoisian approach to integrability of Schrödinger equation

In this paper, we examine the nonrelativistic stationary Schrodinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second-order ordinary linear differential operators, so as to achieve rational function coefficients (“algebrization”), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrodinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.

On the integrability of Friedmann–Robertson–Walker models with conformally coupled massive scalar fields

In this paper, we use a nonintegrability theorem by Morales and Ramis to analyse the integrability of Friedmann–Robertson–Walker cosmological models with a conformally coupled massive scalar field. We answer the long-standing question of whether these models with a vanishing cosmological constant and non-self-interacting scalar field are integrable: by applying Kovacic's algorithm to the normal variational equations, we prove analytically and rigorously that these equations and, consequently, the Hamiltonians are nonintegrable. We then address the models with a self-interacting massive scalar field and cosmological constant and show that, with the exception of a set of measure zero, the models are nonintegrable. For the spatially curved cases, we prove that there are no additional integrable cases other than those identified in the previous work based on the non-rigorous Painlevé analysis. In our study of the spatially flat model, we explicitly obtain a new possibly integrable case.

Liouvillian perturbations of black holes

We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests.