Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians

Abstract

The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of functions $\mathcal{W}$, but an arbitrary $t$-dependent function on $\mathcal{W}$. This novel formalism is accomplished through a detailed analysis of related structures, such as momentum maps and generalized distributions, together with the extension of the Poisson coalgebra method to a $t$-dependent frame, in order to systematize the construction of constants of the motion for nonlinear systems. Several relevant relations between nonlinear Lie-Hamilton systems, Lie-Hamilton systems, and collective Hamiltonians are analyzed. The new notions and tools are illustrated with the study of the harmonic oscillator, Hénon-Heiles systems and Painlevé trascendents within a $t$-dependent framework. In addition, the formalism is carefully applied to construct oscillators with a $t$-dependent frequency and Kepler-Coulomb systems with a $t$-dependent coupling constant on the $n$-dimensional sphere, Euclidean and hyperbolic spaces, as well as on some spaces of non-constant curvature.

47 pages

References in corpus (44)

• Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
• Completeness of superintegrability in two-dimensional constant curvature spaces
• Superposition rules, Lie theorem and partial differential equations
• Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$
• Relation of the oscillator and Coulomb systems on spheres and pseudospheres
• A systematic construction of completely integrable Hamiltonians from coalgebras
• Quantum superintegrability and exact solvability in N dimensions
• Position Dependent Mass Oscillators and Coherent States
• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• The quantum harmonic oscillator on the sphere and the hyperbolic plane
• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Superintegrability of the Caged Anisotropic Oscillator
• Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
• Maximal superintegrability on N-dimensional curved spaces
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Lie systems: theory, generalisations, and applications
• A New Superintegrable Hamiltonian
• Bertrand spacetimes as Kepler/oscillator potentials
• Lie-Hamilton systems on the plane: properties, classification and applications
• Integrable deformations of oscillator chains from quantum algebras
• Generalized nonlinear oscillators with quasi-harmonic behaviour: classical solutions
• Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
• From constants of motion to superposition rules for Lie-Hamilton systems
• The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
• Dirac--Lie systems and Schwarzian equations
• Lie--Hamilton systems: theory and applications
• N-dimensional sl(2)-coalgebra spaces with non-constant curvature
• Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• Niederer's transformation, time-dependent oscillators and polarized gravitational waves
• Lie-Hamilton systems on curved spaces: A geometrical approach
• Completeness of the cubic and quartic Hénon-Heiles Hamiltonians
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• Isotropic oscillator in the space of constant positive curvature. Interbasis expansions
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Integrable Henon-Heiles Hamiltonians: a Poisson algebra approach
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• N-dimensional integrability from two-photon coalgebra symmetry
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications