Models of $p$-adic mechanics

We segregate the class of ultrametric ($p$-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincaré recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum $p$-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.