  In this article we analyze several mathematical models with singularities
where the classical cotangent model is replaced by a $b$-cotangent model. We
provide physical interpretations of the singular symplectic geometry underlying
in $b$-cotangent bundles featuring two models: the canonical (or non-twisted)
model and the twisted one. The first one models systems on manifolds with
boundary and the twisted model represents Hamiltonian systems where the
singularity of the system is in the fiber of the bundle. The twisted cotangent
model includes (for linear potentials) the case of fluids with dissipation. We
relate the complexity of the fluids in terms of the Reynolds number and the
(non)-existence of cotangent lift dynamics. We also discuss more general
physical interpretations of the twisted and non-twisted $b$-symplectic models.
These models offer a Hamiltonian formulation for systems which are dissipative,
extending the horizons of Hamiltonian dynamics and opening a new approach to
study non-conservative systems.
