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Billiards are a class of dynamical systems in which a point particle moves uniformly in a domain $D\subset \mathbb{R}^d$ except for mirror-like reflections from the boundary. Varying $D$ leads to examples satisfying many ergodic properties. Billiards enhance visual explanations of dynamical concepts to students and the general public. There are many applications in physics and image processing. The free motion and/or reflection rule may be generalized.

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Projecting phase space Liouville tori to configuration space in integrable systems

Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards