Andrzej Maciejewski
Licealna 9, PL65–417 Zielona Gora, Poland
University of Zielona Góra
Publications:
Maciejewski A. J., Przybylska M.
Global Properties of Kovalevskaya Exponents
2017, vol. 22, no. 7, pp. 840850
Abstract
This paper contains a collection of properties of Kovalevskaya exponents which are eigenvalues of a linearization matrix of weighted homogeneous nonlinear systems along certain straightline particular solutions. Relations in the form of linear combinations of Kovalevskaya
exponents with nonnegative integers related to the presence of first integrals of the weighted homogeneous nonlinear systems have been known for a long time. As a new result other nonlinear relations between Kovalevskaya exponents calculated on all straightline particular
solutions are presented. They were obtained by an application of the Euler–Jacobi–Kronecker formula specified to an appropriate $n$form in a certain weighted homogeneous projective space

Maciejewski A. J., Przybylska M.
Integrable Variational Equations of Nonintegrable Systems
2012, vol. 17, no. 34, pp. 337358
Abstract
Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler–Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler–Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler–Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.

Maciejewski A. J., Przybylska M.
Partial integrability of Hamiltonian systems with homogeneous potential
2010, vol. 15, no. 45, pp. 551563
Abstract
In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form
$H = \frac{1}{2} {\bf p}^T {\bf Mp} + V({\bf q})$, where ${\bf q} = (q_1, \ldots, q_n) \in \mathbb{C}^n$, ${\bf p}= (p_1, \ldots, p_n) \in \mathbb{C}^n$, are the canonical coordinates and momenta, $\bf M$ is a symmetric nonsingular matrix, and $V({\bf q})$ is a homogeneous function of degree $k \in Z^*$. We assume that the system admits $1 \leqslant m < n$ independent and commuting first integrals $F_1, \ldots F_m$. Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral $F_{m+1}$ such that all integrals $F_1, \ldots F_{m+1}$ are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain $n$ body problem on a line and the planar three body problem. 
Bardin B. S., Maciejewski A. J., Przybylska M.
Integrability of generalized Jacobi problem
2005, vol. 10, no. 4, pp. 437461
Abstract
We consider a point moving in an ellipsoid $a_1x_1^2+a_2x_2^2+a_3x_3^2=1$ under the influence of a force with quadratic potential $V=\frac{1}{2}(b_1x_1^2+b_2x_2^2+b_3x_3^2)$. We prove that the equations of motion of the point are meromorphically integrable if and only if the condition $b_1(a_2a_3)+b_2(a_3a_1)+b_3(a_1a_2)=0$ is fulfilled.

Maciejewski A. J., Przybylska M.
Nonintegrability of restricted two body problems in constant curvature spaces
2003, vol. 8, no. 4, pp. 413430
Abstract
We consider a restricted problem of two bodies in constant curvature spaces. The Newton and Hooke interactions between bodies are considered. For both types of interactions, we prove the nonintegrability of this problem in spaces with constant nonzero curvature. Our proof is based on the Morales–Ramis theory.

Maciejewski A. J., Przybylska M.
NonIntegrability of the Suslov Problem
2002, vol. 7, no. 1, pp. 7380
Abstract
In this paper we study integrability of the classical Suslov problem. We prove that in a version of this problem introduced by V.V. Kozlov the problem is integrable only in one known case. We consider also a generalisation of Kozlov version and prove that the system is not integrable. Our proofs are based on the Morales–Ramis theory.

Bardin B. S., Maciejewski A. J.
Nonlinear oscillations of a Hamiltonian system with one and half degrees of freedom
2000, vol. 5, no. 3, pp. 345360
Abstract
We study nonlinear oscillations of a nearly integrable Hamiltonian system with one and half degrees of freedom in a neighborhood of an equilibrium. We analyse the resonance case of order one. We perform careful analysis of a small finite neighborhood of the equilibrium. We show that in the case considered the equilibrium is not stable, however, this instability is soft, i.e. trajectories of the system starting near the equilibrium remain close to it for an infinite period of time. We discuss also the effect of separatrices splitting occurring in the system. We apply our theory to study the motion of a particle in a field of waves packet.

Maciejewski A. J., Strelcyn J.
Nonintegrability of the Generalized Halphen System
1996, vol. 1, no. 2, pp. 312
Abstract
A generalized of the well known Halphen system is considered. It is proved that this generalized system in odd dimension does not admit a nonconstant rational first integral.
