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The Eckart co-rotating frame is used to analyze the dynamics of star polymersunder shear flow, either in melt or solution and with different types of bonds.This formalism is compared with the standard approach used in many previousstudies on polymer dynamics, where an apparent angular velocity $\omega$ isobtained from relation between the tensor of inertia and angular momentum. Acommon mistake is to interpret $\omega$ as the molecular rotation frequency,which is only valid for rigid-body rotation. The Eckart frame, originallyformulated to analyze the infrared spectra of small molecules, dissectsdifferent kinds of displacements: vibrations without angular momentum, purerotation, 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 betweenrotation and vibration. The standard and Eckart approaches are compared with astraight description of the star's dynamics taken from the time autocorrelationof the monomers positions moving around the molecule's center of mass. This isan underdamped oscillatory signal, which can be described by a rotationfrequency $\omega_R$ and a decorrelation rate $\Gamma$. We consistently findthat $\Omega$ coincides with $\omega_R$, which determines the characteristictank-treading rotation of the star. By contrast, the apparent angular velocity$\omega < \Omega$ does not discern between pure rotation and molecularvibrations. We believe that the Eckart frame will be useful to unveil thedynamics of semiflexible molecules where rotation and deformations areentangled, including tumbling, tank-treading motions and breathing modes.

Application of the Eckart frame to soft matter: rotation of star polymers under shear flow