Lie and Noether point Symmetries for a Class of Nonautonomous Dynamical Systems

TL;DR

This paper establishes general theorems to determine Lie and Noether point symmetries for nonautonomous dynamical systems in Riemannian spaces with time-dependent potentials, including applications to specific cases like the harmonic oscillator.

Contribution

It provides new theorems that identify symmetries of nonautonomous systems with time-dependent potentials in Riemannian spaces, extending symmetry analysis methods.

Findings

Derived explicit symmetries for time-dependent central potentials

Identified symmetries of the harmonic oscillator with time-dependent parameters

Extended theorems to systems with linear damping

Abstract

We prove two general theorems which determine the Lie and the Noether point symmetries for the equations of motion of a dynamical system which moves in a general Riemannian space under the action of a time dependent potential $W(t,x)=\omega(t)V(x)$. We apply the theorems to the case of a time dependent central potential and the harmonic oscillator and determine all Lie and Noether point symmetries. Finally we prove that these theorems also apply to the case of a dynamical system with linear dumping and study two examples.