In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations.

In this short communication, we deal with an integrability analysis of nonlinear three-dimensional differential systems. Right-hand sides of these systems are linear in one variable, which enables one to find explicitly a particular solution and to calculate variational equations along this solution. The conditions for the complete integrability with two functionally independent rational first integrals for B-integrability and the partial integrability are obtained from an analysis of properties of the differential Galois group of variational equations. They have a very simple form of numbers, which is necessary to check whether they are appropriate integers. An application of the obtained conditions to some exemplary nonlinear three-dimensional differential systems is shown.

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. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems.Keywords Three-body problem · Morales-Ramis theory · Differential Galois theory · Non-integrability Equations of motion, symmetries and reductionConsidered is the gravitational three-body problem with a single constraint. Three point masses, m 1 , m 2 and m 3 move in a plane under mutual gravitational interaction. Masses m 2 and m 3 are connected by a massless inflexible rod of length l > 0 to form a dumbbell. A pictorial description of the problem is given in Fig. 1. Various celestial objects are considered to possess such bimodal mass distribution because do not have enough gravitational force to form their shape into a spherical object, e.g. some asteroids such as (51) Nemausa and (216) Kleopatra; meteorites, especially large irons or nucleus of some comets e.g. Comet Borrelly (for detailed references see Povenmire 2002).The dumbbell satellite has attracted the attention of scientists since the middle of 20th century because it is suitable for an investigation of the general properties of the rigid body motion in a gravity field and provides important

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 Poincar\'e 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.Comment: 22 pages, 13 figure

In this paper, we consider two chaotic finance models recently studied in the literature. The first one, introduced by Huang and Li, has a form of three first-order nonlinear differential equationṡ x = z + (y − a)x,ẏ = 1 − by − x 2 ,ż = − x − cz. The second system, called a hyperchaotic finance model, is defined bẏ x = z + (y − a)x + u, z = − x − cz,ẏ = 1 − by − x 2 , u = − dx y − ku. In both models, (a, b, c, d, k) are real positive parameters. In order to present the complexity of these systems Poincaré cross sections, bifurcation diagrams, Lyapunov exponents spectrum and the Kaplan-Yorke dimension have been calculated. Moreover, we show that the Huang-Li system is not integrable in a class of functions meromorphic in variables (x, y, z), for all real values of parameters (a, b, c), while the hyperchaotic system is not integrable in the case when k = c and := 1 + d(a + d − c) > 0. We give analytic proofs of these facts analyzing properties of the differential Galois groups of variational equations along cer-The research has been supported by the National Science Centre of Poland under Grants DEC