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

The generalized uncertainty principle is a phenomenological model whose purpose is to account for a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached using a different formalism than the one used for quantum systems, and vice versa. Potentially, the two approaches can result in inconsistencies. Here, we eliminate such inconsistencies by proposing particular meanings and relations between the variables used to describe physical systems, resulting in a precise form of the Legendre transformation. Furthermore, we introduce two different sets of canonical variables and the relative map between them. These two sets allow for a complete and unambiguous description of classical and quantum systems.