Generalized clocks in timeless canonical formalism

Hamiltonian dynamics is recast in a timeless formalism in which parameter time a is derived from the generalized coordinates, the Hamiltonian invariance on trajectories, and the Maupertuis principle. In order to define a time variable T in macroscopic systems, the cyclicity in the phase space replaces the self consistent assumption of time periodicity generally adopted for real clocks. Generalized clocks are defined in physical systems of sufficient complexity. Under suitable assumptions, physical systems can be separated in a subsystem to be dynamically described, and another cyclic subsystem which has the role of generalized clock. The latter provides a discrete approximation of the parameter time, called metric time. The stability prescription of generalized clocks guarantees that dynamics is expressed by the same equations of motion parametrized by the parameter time, in terms of metric time at the desired approximation. The timeless Hamiltonian framework, together with the definition of generalized clock, provide a ground to account the fundamental timelessness of nature, and the experimental evidence of time evolution in macroscopic systems experienced by the observers.