We investigate the challenges of incorporating time-dependent mass in classical Lagrangian mechanics, where velocity-dependent terms break time-translation symmetry and complicate energy conservation. Using the 0-Sphere model—a point-like system with thermally modulated mass inspired by Zitterbewegung and thermal oscillations—we demonstrate that a Hamiltonian formulation simplifies the dynamics by eliminating velocity-dependent terms and preserving energy conservation through conserved momentum, despite the Hamiltonian’s explicit time dependence. The model assumes a position-independent thermal potential and oscillatory mass modulation, providing a mathematically consistent framework. We also explore a preliminary quantum extension via the time-dependent Schrödinger equation, suggesting potential applications to thermally driven systems. While the model’s reliance on simplified potentials and the naive quantum approach limit its generality, this work offers a starting point for understanding systems with dynamic inertial properties, with possible relevance to cosmology and quantum mechanics.