Noether Symmetries of Time-Dependent Damped Dynamical Systems: A Geometric Approach

Finding Noether symmetries for time-dependent damped dynamical systems remains a significant challenge. This paper introduces a complete geometric algorithm for determining all Noether point symmetries and first integrals for the general class of Lagrangians $L=A\left(t\right)L_0$, which model motion with general linear damping in a Riemannian space. We derive and prove a central Theorem that systematically links these symmetries to the homothetic algebra of the kinetic metric defined by $L_0$. The power of this method is demonstrated through a comprehensive analysis of the damped Kepler problem. Beyond recovering known results for constant damping, we discover new quadratic first integrals for time-dependent damping $\varphi \left(t\right)=\gamma /t$ with $\gamma =-1$ and $\gamma =-1/3$. We also include preliminary results on the Noether symmetries of the damped harmonic oscillator. Finally, we clarify why a time reparameterization that removes damping yields a physically inequivalent system with different Noether symmetries. This work provides a unified geometric framework for analyzing dissipative systems and reveals new integrable cases.