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Andrzej Maciejewski

Licealna 9, PL-65–417 Zielona Gora, Poland
University of Zielona Góra


Maciejewski A. J., Przybylska M.
Integrable Variational Equations of Non-integrable Systems
2012, vol. 17, no. 3-4, pp.  337-358
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.
Keywords: rigid body, Euler–Poisson equations, solvability in special functions, differential Galois group
Citation: Maciejewski A. J., Przybylska M.,  Integrable Variational Equations of Non-integrable Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 337-358
Maciejewski A. J., Przybylska M.
Partial integrability of Hamiltonian systems with homogeneous potential
2010, vol. 15, no. 4-5, pp.  551-563
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 non-singular 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.
Keywords: integrability, non-integrability criteria, monodromy group, differential Galois group, hypergeometric equation, Hamiltonian equations
Citation: Maciejewski A. J., Przybylska M.,  Partial integrability of Hamiltonian systems with homogeneous potential, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 551-563
Bardin B. S., Maciejewski A. J., Przybylska M.
Integrability of generalized Jacobi problem
2005, vol. 10, no. 4, pp.  437-461
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 $b_1(a_2-a_3)+b_2(a_3-a_1)+b_3(a_1-a_2)=0$ is fulfilled.
Keywords: Jacobi problem, integrability, differential Galois group, monodromy group
Citation: Bardin B. S., Maciejewski A. J., Przybylska M.,  Integrability of generalized Jacobi problem , Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 437-461
DOI: 10.1070/RD2005v010n04ABEH000325
Maciejewski A. J., Przybylska M.
Non-integrability of restricted two body problems in constant curvature spaces
2003, vol. 8, no. 4, pp.  413-430
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 non-integrability of this problem in spaces with constant non-zero curvature. Our proof is based on the Morales–Ramis theory.
Citation: Maciejewski A. J., Przybylska M.,  Non-integrability of restricted two body problems in constant curvature spaces, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 413-430
Maciejewski A. J., Przybylska M.
Non-Integrability of the Suslov Problem
2002, vol. 7, no. 1, pp.  73-80
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.
Citation: Maciejewski A. J., Przybylska M.,  Non-Integrability of the Suslov Problem, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 73-80
Bardin B. S., Maciejewski A. J.
Non-linear oscillations of a Hamiltonian system with one and half degrees of freedom
2000, vol. 5, no. 3, pp.  345-360
We study non-linear 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.
Citation: Bardin B. S., Maciejewski A. J.,  Non-linear oscillations of a Hamiltonian system with one and half degrees of freedom, Regular and Chaotic Dynamics, 2000, vol. 5, no. 3, pp. 345-360
Maciejewski A. J., Strelcyn J.
Non-integrability of the Generalized Halphen System
1996, vol. 1, no. 2, pp.  3-12
A generalized of the well known Halphen system is considered. It is proved that this generalized system in odd dimension does not admit a non-constant rational first integral.
Citation: Maciejewski A. J., Strelcyn J.,  Non-integrability of the Generalized Halphen System, Regular and Chaotic Dynamics, 1996, vol. 1, no. 2, pp. 3-12

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