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Andrzej J. Maciejewski

Publications and source records attributed to Andrzej J. Maciejewski.

At least 19 recordsLinked to original sources
arXiv2026

Integrability of non-homogeneous Hamiltonian systems with gyroscopic coupling

We study the integrability of a two-dimensional Hamiltonian system with a gyroscopic term and a non-homogeneous potential composed of two homogeneous components of different degrees. The model describes the motion of a particle in a plane under the combined influence of a central (Kepler-type) potential, a uniform magnetic field, and a superposition of homogeneous forces. By combining the Levi--Civita regularization with the so-called coupling constant metamorphosis transformation, and employing differential Galois theory, we derive analytical necessary conditions for integrability in the Liouville sense. They put restrictions on the degrees of homogeneity of the potential terms and their values in particular points. The obtained results encompass and generalize several classical galactic and astrophysical models, including the generalized Hill model, the Hénon--Heiles and Armbruster--Guckenheimer--Kim systems, providing a unified framework for studying non-homogeneous Hamiltonians. We demonstrate the effectiveness of the derived integrability obstructions by proving the non-integrability of these models in the presence of a uniform rotational field. The numerical analysis via the Poincaré cross-sections further confirms the analytical results, illustrating the transition from regular to chaotic dynamics as the rotational and non-homogeneous terms are introduced. Moreover, we show that, without the Kepler-type term, a generalized non-homogeneous extension of the exceptional potential remains integrable. The explicit forms of the first integrals are given.

nlin.SI↗
arXiv2026

Translational dynamics of diatomic molecule in magnetic quadrupole trap

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3Σ$ 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 Poincaré 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 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↗
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↗
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↗
arXiv2024

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system's dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system's dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits $S^1$ symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilisation of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilised a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

nlin.CD↗
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↗
arXiv2021

Non-integrability of the Kepler and the two-body problem on the Heisenberg group

The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense.

math-ph↗
arXiv2019

A novel approach to the spectral problem in the two photon Rabi model

We explore the spectral problem of the two photon Rabi model from the point of view of complex differential equations in the Bargmann representation. The wave-functions are automatically entire but to ensure finite norm one has to effectively construct asymptotic expansions. This is achieved by means of the Mellin transformation and convergent factorial series, which allow direct computation of the spectral determinant. By further analysing the differential equation satisfied by the Mellin transform, we obtain a new form of the spectral conditions -- in terms of holonomy matrices and contour integrals.

quant-ph↗
arXiv2019

Level crossings and new exact solutions of the two-photon Rabi model

An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.

quant-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

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↗