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(t)L0, 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 L0. 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 ϕ(t)=γ/t with γ=−1 and γ=−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.