  We consider the Lagrangian formulation with duplicated variables of
dissipative mechanical systems. The application of Noether theorem leads to
physical observable quantities which are not conserved, like energy and angular
momentum, and conserved quantities like the Hamiltonian, that generate symmetry
transformations and do not correspond to observables. We show that there are
simple relations among the equations satisfied by these two types of
quantities. In the case of the damped harmonic oscillator, from the quantities
obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky.
Further, if we consider the whole dynamics, the degrees of freedom separate
into a physical and an unphysical sector. We analyze several cases, with linear
and nonlinear dissipative forces; the physical consistency of the solutions is
ensured observing that the unphysical sector has always the trivial solution.
