Searcharxiv⌕ Search

arXiv subjects

Ondřej Kubů

Publications and source records attributed to Ondřej Kubů.

8 recordsLinked to original sources
arXiv2026

Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

We study lifts of integrable systems by means of generalized Stäckel geometry. To this end, we present the notion of Stäckel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The Stäckel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent Stäckel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure. We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the Stäckel framework by means of modified Stäckel bases. Finally, we show that explicitly momentum-dependent lifting matrices generate Platonic-wave geometries with potential applications in modified gravity theories, or momentum-dependent metrics of Hamilton and Finsler geometries.

math-ph↗
arXiv2026

Haantjes algebras, Zernike system and separation of variables

The generalized Zernike family $H_{(N)} = p_1^2 + p_2^2 + \sum_{n=1}^N γ_n\,(q_1 p_1 + q_2 p_2)^n$ is a parametric family of two-dimensional superintegrable Hamiltonians, admitting $N$ integrals of motion of degree $N$ in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux--Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for $N = 2$ -- the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature -- covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of $H_{(2)}$ have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For $N \geq 3$ we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.

math-ph↗
arXiv2024

Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically; besides, the underlying Stäckel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

math-ph↗
arXiv2023

New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases

We study quadratic integrability of systems with velocity dependent potentials in three-dimensional Euclidean space. Unlike in the case with only scalar potential, quadratic integrability with velocity dependent potentials does not imply separability in the configuration space. The leading order terms in the pairs of commuting integrals can either generalize or have no relation to the forms leading to separation in the absence of a vector potential. We call such pairs of integrals generalized, to distinguish them from the standard ones, which would correspond to separation. Here we focus on three cases of generalized non-subgroup type integrals, namely elliptic cylindrical, prolate / oblate spheroidal and circular parabolic integrals, together with one case not related to any coordinate system. We find two new integrable systems, non-separable in the configuration space, both with generalized elliptic cylindrical integrals. In the other cases, all systems found were already known and possess standard pairs of integrals. In the limit of vanishing vector potential, both systems reduce to free motion and therefore separate in every orthogonal coordinate system.

math-ph↗
arXiv2022

Cylindrical first order superintegrability with complex magnetic fields

This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space $\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [X. Peng et al., Phys. Rev. Lett. 114 (2015)], they were so far not addressed in the growing literature on superintegrability. Here we venture into this field by searching for additional first order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.

math-ph↗
arXiv2022

Integrable and superintegrable systems of cylindrical type in magnetic fields

The goal of this thesis is the search for integrable and superintegrable systems with magnetic field. We formulate the quantum mechanical determining equations for second order integrals of motion in the cylindrical coordinates and we find all quadratically integrable systems of the cylindrical type. Among them we search for systems admitting additional integrals of motion. We find all systems with an additional first order integral both in classical and quantum mechanics. It turns out that all these systems have already been known and no other exist. We also find all systems with an additional integral of type $L^2+\ldots$, respectively $L_y p_y-L_x p_y+\ldots$, of which the majority is new to the literature. All found superintegrable systems admit the first order integral $L_z$ and we solve their Hamilton-Jacobi and Schrödinger equations by separation of variables in the cylindrical coordinates, for the first order systems in the Cartesian coordinates as well.

nlin.SI↗
arXiv2020

On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

We extend the investigation of three-dimensional (3D) Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry, namely the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. More precisely, we focus on linear and some special cases of quadratic superintegrability. In the linear case, no new superintegrable system arises. In the quadratic case, we found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases and another one at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems. These two systems are linked respectively with the caged and harmonic oscillators without magnetic fields through a time-dependent canonical transformation.

math-ph↗