Pairs of commuting quadratic elements in the universal enveloping algebra of Euclidean algebra and integrals of motion
Authors
Marchesiello, A.; Šnobl, L.
Year
2022
Published
Journal of Physics A: Mathematical and Theoretical. 2022, 55(14), ISSN 1751-8113.
Type
Article
Departments
Annotation
Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant (p) over right arrow . (l) over right arrow vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms.
Superintegrability of separable systems with magnetic field: the cylindrical case
Authors
Kubů, O.; Marchesiello, A.; Šnobl, L.
Year
2021
Published
Journal of Physics A: Mathematical and Theoretical. 2021, 54(42), ISSN 1751-8113.
Type
Article
Departments
Annotation
We present a general method simplifying the search for additional integrals of motion of three dimensional systems with magnetic fields. The method is suitable for systems possessing at least one conserved canonical momentum in a suitable coordinates system. It reduces the problem either to consideration of lower dimensional systems or of particular constrained forms of the hypothetical integral. In particular, it is applicable to all separable systems in the Euclidean space since they are known to possess at least one cyclic coordinates when magnetic field is present. Next, we focus on systems which separate in the cylindrical coordinates. Using our method, we are able to classify all superintegrable systems of this kind under the assumption that all considered integrals are at most second order in the momenta. In addition to already known systems, several new minimally superintegrable systems are found and we show that no quadratically maximally superintegrable ones can exist. We also construct some examples of systems with higher order integrals.
Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
Authors
Marchesiello, A.; Šnobl, L.
Year
2020
Published
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020, 16 ISSN 1815-0659.
Type
Article
Departments
Annotation
We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.
On the Detuned 2:4 Resonance
Authors
Hanssmann, H.; Marchesiello, A.; Pucacco, G.
Year
2020
Published
Journal of nonlinear science. 2020, 30(6), 2513-2544. ISSN 0938-8974.
Type
Article
Departments
Annotation
We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1 : 2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials, this concerns the short axial orbits, and in galactic dynamics, the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the coordinate planes whence the potential-and the normal form-both have no cubic terms. This Z2xZ2 symmetry turns the 1 : 2 resonance into a higher-order resonance, and one therefore also speaks of the 2 : 4 resonance. In this paper, we study the 2 : 4 resonance in its own right, not restricted to natural Hamiltonian systems where H=T+V would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.
Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance
Authors
Efstathiou, K.; Hanßmann, H.; Marchesiello, A.
Year
2019
Published
Journal of Geometry and Physics. 2019, 146 ISSN 0393-0440.
Type
Article
Departments
Annotation
We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:−2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:−2 resonance
An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
Authors
Marchesiello, A.; Šnobl, L.
Year
2018
Published
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018, 14 ISSN 1815-0659.
Type
Article
Departments
Annotation
We construct an additional independent integral of motion for a class of three dimensional minimally superintegrable systems with constant magnetic field. This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages] and it is known to possess periodic closed orbits. In the present paper we demonstrate that it is maximally superintegrable. Depending on the values of the parameters of the system, the newly found integral can be of arbitrarily high polynomial order in momenta.
Spherical type integrable classical systems in a magnetic field
Authors
Marchesiello, A.; Šnobl, L.; Winternitz, P.
Year
2018
Published
Journal of Physics A: Mathematical and Theoretical. 2018, 51(13), ISSN 1751-8113.
Type
Article
Departments
Annotation
We show that four classes of second order spherical type integrable classical systems in a magnetic field exist in the Euclidean space E-3, and construct the Hamiltonian and two second order integrals of motion in involution for each of them. For one of the classes the Hamiltonian depends on four arbitrary functions of one variable. This class contains the magnetic monopole as a special case. Two further classes have Hamiltonians depending on one arbitrary function of one variable and four or six constants, respectively. The magnetic field in these cases is radial. The remaining system corresponds to a constant magnetic field and the Hamiltonian depends on two constants. Questions of superintegrability-i. e. the existence of further integrals-are discussed.
Superintegrable 3D systems in a magnetic field corresponding to Cartesian separation of variables
Authors
Marchesiello, A.; Šnobl, L.
Year
2017
Published
Journal of Physics A: Mathematical and Theoretical. 2017, 50(24), ISSN 1751-8113.
Type
Article
Departments
Annotation
We consider three dimensional superintegrable systems in a magnetic field. We study the class of such systems which separate in Cartesian coordinates in the limit when the magnetic field vanishes, i.e. possess two second order integrals of motion of the 'Cartesian type'. For such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories. We observe that the structure of the leading order terms of the Cartesian type integrals should be considered in a more general form than for the case without magnetic field.