Unified Framework for Geodesic Dynamics with Conservative, Dissipative, and GUP Effects
We derive generalized geodesic equations in curved spacetime that include conservative forces, dissipative effects, and quantum-gravity-motivated minimal-length corrections. Conservative interactions are incorporated through external vector potentials, while dissipative dynamics arise from an exponential rescaling of the particle Lagrangian. Phenomenological study of Quantum-gravity effects is introduced via Generalized Uncertainty Principle (GUP) deformed Poisson brackets in the Hamiltonian framework. We show that free-particle geodesics remain unaffected at leading order, but external potentials induce velocity-dependent corrections, implying possible violations of the equivalence principle. As an application, we analyze modified trajectories in Friedmann-Lemaitre-Robertson-Walker (FLRW) universes dominated by dust, radiation, stiff matter, and dark energy. Our results establish a unified approach to conservative, dissipative, and GUP-corrected geodesics, providing a framework to probe the interplay between external forces, spacetime curvature, and Planck-scale physics.