  We examine the non-extensive approach to the statistical mechanics of
Hamiltonian systems with $H=T+V$ where $T$ is the classical kinetic energy. Our
analysis starts from the basics of the formalism by applying the standard
variational method for maximizing the entropy subject to the average energy and
normalization constraints. The analytical results show (i) that the
non-extensive thermodynamics formalism should be called into question to
explain experimental results described by extended exponential distributions
exhibiting long tails, i.e. $q$-exponentials with $q>1$, and (ii) that in the
thermodynamic limit the theory is only consistent in the range $0\leq q\leq1$
where the distribution has finite support, thus implying that configurations
with e.g. energy above some limit have zero probability, which is at variance
with the physics of systems in contact with a heat reservoir. We also discuss
the ($q$-dependent) thermodynamic temperature and the generalized specific
heat.
