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SUMMARY Hamiltonian MCMC methods sample from a probability distribution by treating its log as a "potential energy" function over the state space, augmenting the space with extra "momentum variables" and their associated "kinetic energy", and evolving the state of the Markov process by integrating the physical Hamiltonian equations of motion of the system. Each step of the Markov chain is accomplished by numerically integrating the Hamiltonian equations forward in time. However, if the energy function is non-differentiable, the integral is not well-defined. The rejection step that is used to counteract numerical inaccuracies in the integration also accounts for such non-differentiable regions, but at the cost of slowing down the mixing rate of the Markov chain. This paper suggests physically-inspired "reflections" and "refractions" of the trajectory of the system that occur whenever the state crosses a discontinuity in the energy function. It applies to target distributions that are differentiable everywhere except on the boundaries of certain polytopes; the reflection or refraction occurs whenever the trajectory of the system crosses such a boundary.