  The Rouse-Ham model is a simple yet useful dynamics model for an unentangled
branched polymer. In this work, we study the normal modes of the Rouse-Ham type
coarse-grained symmetric star polymer model. We model a star polymer by
connecting multiple arm beads to a center bead by harmonic springs. In the
Rouse-Ham model, the dynamics of the bead positions can be decomposed into the
normal modes, which are chosen to be orthogonal to each other. Due to the
existence of degenerate eigenvalues, the eigenmodes do not directly correspond
to the normal modes. We propose several methods to construct the normal modes
for the coarse-grained symmetric star polymer model. We show that we can
construct the normal modes by using a simple permutation or the Hadamard
matrix. These methods give symple and highly symmetric orthogonal modes, but
work just for a special number of arms. We also show that we can construct the
normal modes by using the discrete Fourier transform (DFT) matrix. This method
is applicable for an arbitrary number of arms.
