Maria Przybylska

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 both 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, quasiperiodic, 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 centimeters). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

This paper is a continuation of our investigations of the dynamics and integrability of a relativistic particle moving under the influence of potential forces. Here, we examine particle motion in a general two-dimensional Riemannian manifold subjected to an external potential. In the limit of small momentum this system transforms into the corresponding non-relativistic system in a curved space extensively investigated in recent years. Our integrability results are formulated for relativistic systems with the metric and potential which are homogeneous functions of degrees n and k, respectively. First, we show that the integrability of a non-relativistic Hamiltonian is a necessary condition for the integrability of its relativistic version. Next, we derive the necessary integrability conditions from the analysis of the differential Galois groups of variational equations along two different particular solutions. These conditions, together with those for non-relativistic counterparts, give the final integrability obstructions. Obtained obstructions are used to explain integrability results for relativistic versions of well-known classical systems: the Elmandouh system with variable Gaussian curvature, the Hénon–Heiles-type systems, the position-dependent mass systems and the swinging Atwood machine. We compare the dynamics of the classical and corresponding relativistic versions systems in curved space using Poincaré sections and demonstrate that relativistic corrections frequently induce chaos. Moreover, we prove the non-integrability of these systems using the obtained necessary integrability conditions. Unexpectedly, we have also identified a new family of relativistic integrable Hamiltonian systems with position-dependent mass, which extends both classical integrable potentials and conditionally integrable systems defined at specific energy levels. Our results provide a deeper understanding of integrability and chaos in relativistic mechanics, which is currently of great importance for mathematical physics and astrophysics.

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 first integrals are zero. In the proof of our theorem, we use methods of differential Galois theory.

We discuss the trapping of heteronuclear diatomic molecules prepared in their electronic and vibrational ground states. We tune and shape the trapping potential for bosonic polar molecules in superpositions of rotational states by dressing rotational excitations with a static sextupole electric field. The translational motion of a molecule is treated classically. We examine the Hamiltonian which governs the center of mass dynamics. The effective potential has a global minimum that provides the trapping ability of this trap. The first term of its Taylor series expansion, corresponding to the quadratic Stark shifts, results in the integrable potential. In terms of cylindrical coordinates the center of mass Hamiltonian splits into axial and radial parts. Corresponding trajectories are parameterized by elliptic functions. At low electric fields, the non-approximated Hamiltonian is treated as a small perturbation of the mentioned integrable system described by Kolmogorov–Arnold–Moser theory. The applicability of this approximation is discussed and illustrated using the Poincaré cross-section method. We present results of numerical simulations illustrating the trapping and confinement of a polar molecule in the trap.

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≠0 and without external field g=0 are considered. The non-integrability proof for g≠0 is 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 a constant gravitational field g=0, the integrability is much more difficult. First, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied the Ziglin theorem concerning the splitting of separatrices phenomenon. Then, we prove the non-integrability of the symmetric sliding top using the differential Galois group of variational equations except two the same as for g≠0 cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.

We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The S O ( 3 , 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 given 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.