  The dynamics of sliding friction is mainly governed by the frictional force.
Previous studies have shown that the laboratory-scale friction is well
described by an empirical law stated in terms of the slip velocity and the
state variable. The state variable is a function of time, representing the
physicochemical details of the sliding interface. Since this law is purely
empirical, there has been no unique equation for time evolution of the state
variable. Major equations known to date have their own merits and drawbacks. To
shed light on this problem from a new aspect, here we investigate the
feasibility of periodic motion without the help of radiation damping. Assuming
a patch on which the slip velocity is perturbed from the rest of the sliding
interface, we prove analytically that three major evolution laws fail to
reproduce periodic motion without radiation damping. Furthermore, we propose
two new evolution equations that can produce periodic motion without radiation
damping. These two equations are scrutinized from the viewpoint of experimental
validity and the relevance to slow earthquakes.
