arXiv subjects

Maria Przybylska

Publications and source records attributed to Maria Przybylska.

At least 19 recordsLinked to original sources

arXiv2026

Translational dynamics of diatomic molecule in magnetic quadrupole trap

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3\Sigma$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincar\'e section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

math-ph↗

arXiv2025

Non-integrability of charged three-body problem

We consider the problem of $n$ points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational $n$-body problem. Under this general form of interaction, we investigate the integrability problem of three bodies. We show that the system is not integrable except in one case when two among three interaction constants vanish. In our investigation, we used the Morales-Ramis theorem concerning the integrability of a natural Hamiltonian system with a homogeneous potential and its generalization.

nlin.CD↗

arXiv2025

Top on a smooth plane

We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration $g\neq0$ and without external field $g=0$ are considered. The non-integrability proof for $g\neq0$ based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of constant gravitational field $g=0$ the integrability is much more difficult. At first, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied one of the Ziglin theorem concerning the splitting of separatrices phenomenon. Then we prove the non-integrability of the symmetric sliding top using differential Galois group of variational equations except two the same as for $g\neq0$ cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.

nlin.CD↗

arXiv2025

Non-integrability of the $n$-body problem

We prove that the classical planar $n$-body problem when restricted to a common level of the energy and the angular momentum is not integrable except in the case when both values of these integrals are zero. In the proof of our theorem, we use methods of differential Galois theory.

math-ph↗

arXiv2024

Integrability of quantum dots

We determine the frequency ratios $τ\equiv ω_z/ω_ρ$ for which the Hamiltonian system with a potential \[ V=\frac{1}{r}+\frac{1}{2}\Big({ω_ρ}^2(x^2+y^2)+{ω_z}^2 z^2\Big) \] is completely integrable. We relate this result to the existence of conformal Killing tensors of the associated Eisenhart metric on $\mathbb{R}^{1, 4}$. Finally we show that trajectories of a particle moving under the influence of the potential $V$ are not unparametrised geodesics of any Riemannian metric on $\mathbb{R}^3$.

hep-th↗

arXiv2023

Destructive relativity

Relativistic Hamiltonian equations describing a motion of a point mass in an arbitrary homogeneous potential are considered. For the first time, the necessary integrability conditions for integrability in the Liouville sense for this class of systems are formulated. These conditions are obtained by means of an analysis of the differential Galois groups of variational equations. They are simple and effective in applications. For instance, an application of the necessary integrability conditions for systems with two degrees of freedom shows that relativity almost completely destroys integrability, that is, in almost all cases relativistic versions of integrable systems are not integrable. The paper has been already published in ,,Chaos: An Interdisciplinary Journal of Nonlinear Science'', and the final journal version is available under the link: https://doi.org/10.1063/5.0140633

math-ph↗

arXiv2023

The geodesic flow of the BGPP metric is Liouville integrable

We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The $\mathrm{SO}(3,\mathbb{R})$ symmetry of the model allows to reduce the system from four to two degrees of freedom. Moreover, solutions of the reduced system and its degenerations can be solved explicitly or reduced to a certain quadrature. In degenerated cases BGPP metric coincides with the Eguchi-Hanson metric and for this case the mentioned quadrature can be calculated explicitly in terms of elliptic integrals.

math-ph↗

arXiv2017

Probing the eigenstates thermalization hypothesis with many-particle quantum walks on lattices

We simulate dynamics of many-particle systems of bosons and fermions using discrete time quantum walks on lattices. We present a computational proof of a behavior of the simulated systems similar to the one observed in Hamiltonian dynamics during quantum thermalization. We record the time evolution of the entropy and the temperature of a specific particle configuration during the entire dynamics and observe how they relax to a state we call quantum walks thermal state. This observation is made on two types of lattices while simulating different numbers of particles walking on two grid graphs with 25 vertices. In each case, we observe that the vertices counting statistics, the temperature of the indexed configuration and the dimension of the effective configuration Hilbert space relax simultaneously and remain relaxed for the rest of the many-particle quantum walks.

quant-ph↗

arXiv2016

Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces

We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations.

nlin.SI↗

arXiv2016

Constrained N-body problems

We consider a problem of mass points interacting gravitationally whose motion is subjected to certain holonomic constraints. The motion of points is restricted to certain curves and surfaces. We illustrate the complicated behaviour of trajectories of these systems using Poincaré cross sections. For some models we prove the non-integrability analysing properties of the differential Galois group of variational equations along certain particular solutions of considered systems. Also some integrable cases are identified.

nlin.CD↗

arXiv2016

Note on integrability of certain homogeneous Hamiltonian systems

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.

nlin.SI↗

arXiv2016

Thermalization in many-particle quantum walks

Many-particles quantum walks of particles obeying Bose statistics moving on graphs of various topologies are introduced. A single coin tossing commands the conditional shift operation over the whole graph. Vertices particle densities, the mean values of the phase space variables, second order spatial correlation and counting statistics are evaluated and simulated. Evidence of an universal dynamics is presented.

quant-ph↗

arXiv2015

An exactly solvable system from quantum optics

We investigate a generalisation of the Rabi system in the Bargmann-Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in terms of differential Galois group of the system is also given.

math-ph↗

arXiv2015

Comment on "Solvability of the two-photon Rabi Hamiltonian"

An implicit formula for the spectrum of the two-photon Rabi model was presented by Travenec in Phys. Rev. A 85, 043805 (2012) in analogy to the method of Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The spectrum is given by common zeros of four functions $G_c(z, E)$ taken with a fixed value of $z\in\mathbb{C}$. We point out that all the $G_c(z,E)$ functions defined by Travenec are identically zero, thus the spectrum is not in fact given by their roots.

math-ph↗

arXiv2014

Analytical method of spectra calculations in the Bargmann representation

We formulate a universal method for solving an arbitrary quantum system which, in the Bargmann representation, is described by a system of linear equations with one independent variable, such as one- and multi-photon Rabi models, or $N$ level systems interacting with a single mode of the electromagnetic field and their various generalizations. We explain three types of conditions that determine the spectrum and show their usage for two deformations of the Rabi model. We prove that the spectra of both models are just zeros of transcendental functions, which in one case are given explicitly in terms of confluent Heun functions.

math-ph↗

arXiv2013

Full spectrum of the Rabi model

It is shown that in the Rabi model, for an integer value of the spectral parameter $x$, in addition to the finite number of the classical Judd states there exist infinitely many possible eigenstates. These eigenstates exist if the parameters of the problem are zeros of a certain transcendental function; in other words, there are infinitely many possible choices of parameters for which integer $x$ belongs to the spectrum. Morover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.

math-ph↗

arXiv2013

The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlevé property of the solutions.

nlin.SI↗

arXiv2013

Comment on "Nonexistence of the final first integral in the Zipoy-Voorhees space-time"

The accuracy of the numerical findings of Lukes-Gerakopoulos PRD 86, 044013 (2012), regarding the existence of additional first integrals in the Zipoy-Voorhees space-time, was recently questioned by Maciejewski et al. PRD 88, 064003 (2013). In this comment, it is shown that the discrepancy between the results of Lukes-Gerakopoulos (2012) and Maciejewski et al. (2013) is not due to issues related to numerical accuracy, as claimed in Maciejewski et al (2013), but due to different choice of coordinates used in Maciejewski et al.(2013).

gr-qc↗