A geometric approach to time evolution operators of Lie quantum systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrodinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.Comment: Accepted for publication in the International Journal of Theoretical Physic

Nonstandard Hamiltonian structures of the Liénard equation and contact geometry

The construction of nonstandard Lagrangians and Hamiltonian structures for Liénard equations satisfying Chiellini condition is presented and their connection to time-dependent Hamiltonian formalism is shown. We also show that such nonstandard Lagrangians are deformations of simpler standard Lagrangians. We also exhibit their connection with contact Hamiltonian mechanics

Jacobi multipliers and Hamel''s formalism

In this work we establish the relation between the Jacobi last multiplier, which is a geometrical tool in the solution of problems in mechanics and that provides Lagrangian descriptions and constants of motion for second-order ordinary differential equations, and nonholonomic Lagrangian mechanics where the dynamics is determined by Hamel''s equations. © 2021 IOP Publishing Ltd

Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator

Ladder operators for the simplest version of a rationally extended quantum harmonic oscillator (REQHO) are constructed by applying a Darboux transformation to the quantum harmonic oscillator system. It is shown that the physical spectrum of the REQHO carries a direct sum of a trivial and an infinite-dimensional irreducible representation of the polynomially deformed bosonized osp(1|2) superalgebra. In correspondence with this the ground state of the system is isolated from other physical states but can be reached by ladder operators via nonphysical energy eigenstates, which belong to either an infinite chain of similar eigenstates or to the chains with generalized Jordan states. We show that the discrete chains of the states generated by ladder operators and associated with physical energy levels include six basic generalized Jordan states, in comparison with the two basic Jordan states entering in analogous discrete chains for the quantum harmonic oscillator

Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere S3S^3 and on the Hyperbolic space H3H^3

The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the harmonic oscillator, the Smorodinsky-Winternitz system and the harmonic oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the three-dimensional sphere S-3 (kappa > 0) and on the hyperbolic space H-3 (kappa 0) and H-3 (kappa 0, kappa = 0, or kappa < 0, the corresponding properties are obtained for the system on the sphere S-3, the Euclidean space E-3, or the hyperbolic space H-3, respectively

Generalized virial theorem for the Liénard-type systems

A geometrical description of the virial theorem (VT) of statistical mechanics is pre- sented using the symplectic formalism. The character of the Clausius virial function is determined for second-order differential equations of the Lie´nard type. The explicit dependence of the virial function on the Jacobi last multiplier is illustrated. The latter displays a dual role, namely, as a position-dependent mass term and as an appropriate measure in the geometrical context

Sundman transformation and alternative tangent structures

A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman transformation is shown. Among other applications of such theory we study the linearisability of a system of second-order differential equations and in particular the simplest case of a second-order differential equation. The theory is illustrated with several examples

Nilpotent classical mechanics: s-geometry

We introduce specific type of hyperbolic spaces. It is not a general linear covariant object, but of use in constructing nilpotent systems. In the present work necessary definitions and relevant properties of configuration and phase spaces are indicated. As a working example we use a D=2 isotropic harmonic oscillator.Comment: 8 pages, presented at QGIS, June 2006, Pragu

Jacobi-Lie systems: Fundamentals and low-dimensional classification

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest.Comment: 15 pages. Examples, references and comments added. Based on the contribution presented at "The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications", July 07-11, 2014, Madrid, Spain. To appear in the Proceedings of the 10th AIMS Conferenc

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom