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GUP is a phenomenological model aimed for a description of a minimal lengthin quantum and classical systems. However, the analysis of problems inclassical physics is usually approached preferring a different formalism thanthe one used for quantum systems, and vice versa. Potentially, the twoapproaches can result in inconsistencies. Here, we eliminate suchinconsistencies proposing particular meanings and relations between thevariables used to describe physical systems, resulting in a precise form of theLegendre transformation. Furthermore, we introduce two different sets ofcanonical variables and the relative map between them. These two sets allow fora complete and unambiguous description of classical and quantum systems.

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length