Safe autonomous navigation in unknown environments is an important problem for mobile robots. This paper proposes techniques to learn the dynamics model of a mobile robot from trajectory data and synthesize a tracking controller with safety and stability guarantees. The state of a rigid-body robot usually contains its position, orientation, and generalized velocity and satisfies Hamilton's equations of motion. Instead of a hand-derived dynamics model, we use a dataset of state-control trajectories to train a translation-equivariant nonlinear Hamiltonian model represented as a neural ordinary differential equation (ODE) network. The learned Hamiltonian model is used to synthesize an energy-shaping passivity-based controller and derive conditions which guarantee safe regulation to a desired reference pose. We enable adaptive tracking of a desired path, subject to safety constraints obtained from obstacle distance measurements. The trade-off between the robot's energy and the distance to safety constraint violation is used to adaptively govern a reference pose along the desired path. Our safe adaptive controller is demonstrated on a simulated hexarotor robot navigating in an unknown environments.