  In this work, we show that it is possible to define a classical system
associated with a Generalized Uncertainty Principle (GUP) theory via the
implementation of a consistent symplectic structure. This provides a solid
framework for the classical Hamiltonian formulation of such theories and the
study of the dynamics of physical systems in the corresponding deformed phase
space. By further characterizing the functions that govern non-commutativity in
the configuration space using the algebra of angular momentum, we determine a
general form for the rotation generator in these theories and crucially, we
show that, under these conditions, unlike what has been previously found in the
literature at the quantum level, this requirement does not lead to the
superselection of GUP models at the classical level. Finally, we demonstrate
that a properly defined GUP theory can be correctly interpreted classically if
and only if the corresponding quantum commutators satisfy the Jacobi
identities.
