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This book consists of a set of lecture notes on graduate courses in Analytical Mechanics and Statistical Mechanics which the author successively gave at the University of Miami, and at the University and Polytechnic of Turin over the past decade. The book centers on the idea that stochasticity can come out of nonlinearities even in the case of a few degrees of freedom, and on how this bears on the known methods of classical statistical mechanics and its link with thermodynamics.