When transitioning from studying Euclidean space to more Riemannian manifolds, one must first unlearn many special properties of the flat world. The same is true in physics: while one can make sense of classical physics on an arbitrary curved background space, many seemingly foundational concepts (like the center of mass) turn out to have no place in the general theory.  Freed from the constraints such properties induce, classical physics on a curved background space has many surprises in store.  In this talk I will share some stories related to joint work with Brian Day and Sabetta Matsumoto on understanding and simulating such situations, focusing on hyperbolic space when convenient.  To give a taste, here are two such surprises:
(1) there is no Galilean relativity: inside a sealed box in hyperbolic geometry it is possible to perform an experiment which detects your precise velocity.  And (2): it's possible to ‘swim’ in the vacuum in hyperbolic space - to move your arms and legs in a specific pattern that causes you to translate along a geodesic with no external forces.  The arguments for the former are readily accessible to beginning graduate students in geometry, and the latter illustrates a use of gauge theory in classical mechanics, following work of Wilczek and Montgomery.