  We investigate by thorough Molecular Dynamics simulations the thermodynamic
scaling (TS) of a polymer melt. Two distinct models, with strong and weak
virial-energy correlations, are considered. Both evidence the joint TS with the
same characteristic exponent $\gamma_{ts}$ of the fast mobility - the mean
square amplitude of the picosecond rattling motion inside the cage -, and the
much slower structural relaxation and chain reorientation. If the cage effect
is appreciable, the TS master curves of the fast mobility are nearly linear,
grouping in a bundle of approximately concurrent lines for different
fragilities. An expression of the TS master curve of the structural relaxation
with one adjustable parameter less than the available three-parameters
alternatives is derived. The novel expression fits well with the experimental
TS master curves of thirty-four glassformers and, in particular, their slope at
the glass transition, i.e. the isochoric fragility. For the glassformer OTP the
isochoric fragility allows to satisfactorily predict the TS master curve of the
fast mobility with no adjustments.
