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Stability and safety are critical properties for successful deployment of automatic control systems. As a motivating example, consider autonomous mobile robot navigation in a complex environment. A control design that generalizes to different operational conditions requires a model of the system dynamics, robustness to modeling errors, and satisfaction of safety \NEWZL{constraints}, such as collision avoidance. This paper develops a neural ordinary differential equation network to learn the dynamics of a Hamiltonian system from trajectory data. The learned Hamiltonian model is used to synthesize an energy-shaping passivity-based controller and analyze its \emph{robustness} to uncertainty in the learned model and its \emph{safety} with respect to constraints imposed by the environment. Given a desired reference path for the system, we extend our design using a virtual reference governor to achieve tracking control. The governor state serves as a regulation point that moves along the reference path adaptively, balancing the system energy level, model uncertainty bounds, and distance to safety violation to guarantee robustness and safety. Our Hamiltonian dynamics learning and tracking control techniques are demonstrated on \Revised{simulated hexarotor and quadrotor robots} navigating in cluttered 3D environments.