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

TL;DR

This paper extends the Lie-Hamilton framework to nonlinear systems with time-dependent parameters, enabling systematic construction of constants of motion for complex oscillators and Kepler-Coulomb systems on various curved spaces.

Contribution

It introduces a novel formalism for nonlinear Lie-Hamilton systems with time-dependent functions, expanding the analysis tools for complex dynamical systems.

Findings

Developed a $t$-dependent Poisson coalgebra method.

Constructed oscillators with time-dependent frequencies.

Analyzed Kepler-Coulomb systems on curved spaces.

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, H\'enon-Heiles systems and Painlev\'e 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.