Casimir friction at zero and finite temperatures

The Casimir friction problem for dielectric plates that move parallel to each other is treated by assuming one of the plates to be at rest. The other performs a closed loop motion in the longitudinal direction. Therewith by use of energy dissipation the formalism becomes more manageable and transparent than in the conventional setting where uniform sliding motion is assumed from $t=-\infty$ to $t=+\infty$. One avoids separating off a reversible interparticle force (independent of friction) from the total force. Moreover, the cases of temperatures $T=0$ and finite $T$ are treated on the same footing. For metal plates we find the friction force to be proportional to $v^3$ at $T=0$ while at finite $T$ it is proportional to $v$ for small $v$ as found earlier. Comparisons with earlier results of Pendry (1997, 2010) and Barton (2011) are made.