  We derive generalized Langevin equations (GLEs) for single beads in linear
elastic networks. In particular, the derivations of the GLEs are conducted
without employing normal modes, resulting in two distinct representations in
terms of resistance and mobility kernels. The fluctuation-dissipation relations
are also confirmed for both GLEs. Subsequently, we demonstrate that these two
representations are interconnected via Laplace transforms. Furthermore, another
GLE is derived by utilizing a projection operator method, and it is shown that
the equation obtained through the projection scheme is consistent with the GLE
with the resistance kernel. Finally, as the simplest example, the present
framework is applied to the Rouse model, and the GLEs with the resistance and
mobility kernels are explicitly derived for arbitrary positions of the tagged
bead in the Rouse chain.
