  The Eckart co-rotating frame is used to analyze the dynamics of star polymers
under shear flow, either in melt or solution and with different types of bonds.
This formalism is compared with the standard approach used in many previous
studies on polymer dynamics, where an apparent angular velocity $\omega$ is
obtained from relation between the tensor of inertia and angular momentum. A
common mistake is to interpret $\omega$ as the molecular rotation frequency,
which is only valid for rigid-body rotation. The Eckart frame, originally
formulated to analyze the infrared spectra of small molecules, dissects
different kinds of displacements: vibrations without angular momentum, pure
rotation, and vibrational angular momentum (leading to a Coriolis cross-term).
The Eckart frame co-rotates with the molecule with an angular frequency
$\Omega$ obtained from the Eckart condition for minimal coupling between
rotation and vibration. The standard and Eckart approaches are compared with a
straight description of the star's dynamics taken from the time autocorrelation
of the monomers positions moving around the molecule's center of mass. This is
an underdamped oscillatory signal, which can be described by a rotation
frequency $\omega_R$ and a decorrelation rate $\Gamma$. We consistently find
that $\Omega$ coincides with $\omega_R$, which determines the characteristic
tank-treading rotation of the star. By contrast, the apparent angular velocity
$\omega < \Omega$ does not discern between pure rotation and molecular
vibrations. We believe that the Eckart frame will be useful to unveil the
dynamics of semiflexible molecules where rotation and deformations are
entangled, including tumbling, tank-treading motions and breathing modes.
