On the Hojman conservation quantities in Cosmology
  We discuss the application of the Hojman's Symmetry Approach for the
determination of conservation laws in Cosmology, which has been recently
applied by various authors in different cosmological models. We show that
Hojman's method for regular Hamiltonian systems, where the Hamiltonian function
is one of the involved equations of the system, is equivalent to the
application of Noether's Theorem for generalized transformations. That means
that for minimally-coupled scalar field cosmology or other modified theories
which are conformally related with scalar-field cosmology, like $f(R)$ gravity,
the application of Hojman's method provide us with the same results with that
of Noether's theorem. Moreover we study the special Ansatz. $\phi\left(
t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced for
a minimally-coupled scalar field, and we study the Lie and Noether point
symmetries for the reduced equation. We show that under this Ansatz, the
unknown function of the model cannot be constrained by the requirement of the
existence of a conservation law and that the Hojman conservation quantity which
arises for the reduced equation is nothing more than the functional form of
Noetherian conservation laws for the free particle. On the other hand, for
$f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation
laws which provide us with the special function form of $f(T)$ functions, but
the requirement that the reduced second-order differential equation admits a
Jacobi Last multiplier, while the new conservation law is nothing else that the
Hamiltonian function of the reduced equation.
