Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
Rutwig Campoamor-Stursberg, Francisco J. Herranz, Javier de Lucas
TL;DR
This work develops nonlinear Lie--Hamilton (LH) systems, a broad generalization of LH theory for $t$-dependent Hamiltonians of the form $h(t,x)=F(t,J(x))$, where $J$ is a momentum map to a finite-dimensional Lie algebra $\mathfrak{g}$. It extends the Poisson coalgebra approach to a $t$-dependent frame, connects nonlinear LH systems to collective Hamiltonians, and provides a geometric framework based on momentum maps and generalized distributions. The authors illustrate the theory with nonlinear $t$-dependent Hénon--Heiles systems, Painlevé Hamiltonians, and then develop a comprehensive construction of oscillators with $t$-dependent frequency and Kepler–Coulomb systems with $t$-dependent couplings on curved spaces, all within the $\mathfrak{sl}(2,\mathbb{R})$-coalgebra setting. Key contributions include a rigorous definition of nonlinear LH systems, a method to obtain constants of motion via autonomization and Casimir invariants, and explicit families of $t$-dependent curved oscillators and KC systems on spaces of constant and non-constant curvature. The results open avenues for analyzing $t$-dependent dynamics in curved geometries and for exploring extensions to gravity, quantum deformations, and other symmetry algebras.
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.