Most classical mechanical systems are based on dynamical variables whosevalues are real numbers. Energy conservation is then guaranteed if thedynamical equations are phrased in terms of a Hamiltonian function, which thenleads to differential equations in the time variable. If these real dynamicalvariables are instead replaced by integers, and also the time variable isrestricted to integers, it appears to be hard to enforce energy conservationunless one can also derive a Hamiltonian formalism for that case. We here showhow the Hamiltonian formalism works here, and how it may yield the usualHamilton equations in the continuum limit. The question was motivated by theauthor's investigations of special quantum systems that allow for adeterministic interpretation. The 'discrete Hamiltonian formalism' appears toshed new light on these approaches.
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