We will show that some well known but counterintuitive phenomena arising due to high frequency vibration in dynamical systems have a very simple geometrical explanation. We will describe how the curvature and some non--holonomic mechanics play the role in systems such as (1) an inverted pendulum stabilized by the vibration of its hinge; (2) the Paul trap, used to suspend charged particles; (3) a forced sine-Gordon equation exhibiting so-called --kinks; (4) a composition of symplectic matrices, and others. Bifurcational, numerical and topological questions (some still without answers) arising in this study will be mentioned as well.

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