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Non scaling Fixed-Field Alternating Gradient (FFAG) accelerators have an unprecedented potential for muon acceleration, as well as for medical purposes based on carbon and proton hadron therapy. They also represent a possible active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting from first principle the Hamiltonian formalism for the description of the dynamics of particles in non-scaling FFAG machines has been developed. The stationary reference (closed) orbit has been found within the Hamiltonian framework. The dependence of the path length on the energy deviation has been described in terms of higher order dispersion functions. The latter have been used subsequently to specify the longitudinal part of the Hamiltonian. It has been shown that higher order phase slip coefficients should be taken into account to adequately describe the acceleration in non-scaling FFAG accelerators. A complete theory of the fast (serpentine) acceleration in non-scaling FFAGs has been developed. An example of the theory is presented for the parameters of the Electron Machine with Many Applications (EMMA), a prototype electron non-scaling FFAG to be hosted at Daresbury Laboratory., 1 Introduction Fixed-Field Alternating Gradient (FFAG) accelerators were proposed half century ago [1-4], when acceleration of electrons was first demonstrated. These machines, which were intensively studied in the 1950s and [...]