Maria Przybylska
Publications:
RauchWojciechowski S., Przybylska M.
On Dynamics of Jellet's Egg. Asymptotic Solutions Revisited
2020, vol. 25, no. 1, pp. 4058
Abstract
We study here the asymptotic condition $\dot E=\mu g_n {\boldsymbol v}_A^2=0$ for an eccentric rolling and sliding ellipsoid with axes of
principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett's egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions.
There are 4 types of stationary solutions: tumbling, spinning, inclined rolling and
rotating on the side solutions.
In the generic situation of tumbling solutions concise explicit formulas for stationary angular velocities
$\dot\varphi_{\mathrm{JE}}(\cos\theta)$, $\omega_{3\mathrm{JE}}(\cos\theta)$ as functions of JE parameters
$\widetilde{\alpha},\alpha,\gamma$ are given. We distinguish the case $1\widetilde{\alpha}<\alpha^2<1+\widetilde{\alpha}$, $1\widetilde{\alpha}<\alpha^2\gamma<1+\widetilde{\alpha}$
when velocities $\dot\varphi_{\mathrm{JE}}$, $\omega_{3\mathrm{JE}}$ are defined for the whole range of inclination angles $\theta\in(0,\pi)$. Numerical simulations illustrate
how, for a JE launched almost vertically with $\theta(0)=\tfrac{1}{100},\tfrac{1}{10}$, the inversion of the JE depends
on relations between parameters.

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

RauchWojciechowski S., Przybylska M.
Understanding Reversals of a Rattleback
2017, vol. 22, no. 4, pp. 368385
Abstract
A counterintuitive unidirectional (say counterclockwise) motion of a toy rattleback takes place when it is started by tapping it at a long side or by spinning it slowly in the clockwise sense of rotation. We study the motion of a toy rattleback having an ellipsoidalshaped bottom by using frictionless Newton equations of motion of a rigid body rolling without sliding in a plane. We simulate these equations for tapping and spinning initial conditions to see the contact trajectory, the force arm and the reaction force responsible for torque turning the rattleback in the counterclockwise sense of rotation. Long time behavior of such a rattleback is, however, quasiperiodic and a rattleback starting with small transversal oscillations turns in the clockwise direction.

Przybylska M., RauchWojciechowski S.
Dynamics of a Rolling and Sliding Disk in a Plane. Asymptotic Solutions, Stability and Numerical Simulations
2016, vol. 21, no. 2, pp. 204231
Abstract
We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straightline rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.

Maciejewski A. J., Przybylska M.
Integrable Variational Equations of Nonintegrable Systems
2012, vol. 17, nos. 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, nos. 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. 
Przybylska M.
Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom. Nongeneric Cases
2009, vol. 14, no. 3, pp. 349388
Abstract
In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function, which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k > 2$, is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic if it admits a nonzero solution of equation $V'(\bf d)=0$. The existence of such a solution gives very strong integrability obstructions obtained in the frame of the Morales–Ramis theory. This theory also gives additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1, \ldots, \lambda_n)$ of the Hessian matrix $V''(\bf d)$ calculated at a nonzero $\bf d \in \mathbb{C}^n$ satisfying $V'(\bf d)=\bf d$. In our previous work we showed that for generic potentials some universal relations between $(\lambda_1, \ldots, \lambda_n)$ calculated at various solutions of $V'(\bf d)={\bf d}$ exist. These relations allow one to prove that the number of potentials satisfying the necessary conditions for the integrability is finite. The main aim of this paper was to show that relations of such forms also exist for nongeneric potentials. We show their existence and derive them for the case $n = k = 3$ applying the multivariable residue calculus. We demonstrate the strength of the results analyzing in details the nongeneric cases for $n = k = 3$. Our analysis covers all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n = k = 3$, thanks to this analysis, a threeparameter family of potentials integrable or superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.

Przybylska M.
Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom
2009, vol. 14, no. 2, pp. 263311
Abstract
We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k > 2$. The well known Morales–Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set $\mathcal{M}_k
\subset \mathbb{Q}$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\bf d)$ calculated at a nonzero $\bf d \in \mathbb{C}^n$ satisfying $V'(\bf d) = \bf d$, belong to $\mathcal{M}_k$.
The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set $\mathcal{I}_{n, k} \subset \mathcal{M}_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\bf d)$ belong to $\mathcal{I}_{n, k}$. We give an algorithm which allows to find sets $\mathcal{I}_{n, k}$. We applied this results for the case $n = k = 3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly nontrivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta. 
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.
