Breakdown of a conservation law in incommensurate systems

We show that invariance properties of the Lagrangian of an incommensurate system, as described by the Frenkel-Kontorova model, imply the existence of a generalized angular momentum that is an integral of motion if the system remains floating. The behavior of this quantity can therefore monitor the character of the system as floating (when it is conserved) or locked (when it is not). We find that, during the dynamics, the nonlinear couplings of our model cause parametric phonon excitations that lead to the appearance of Umklapp terms and to a sudden deviation of the generalized momentum from a constant value, signaling a dynamical transition from a floating to a pinned state. We point out that this transition is related but does not coincide with the onset of sliding friction, which can take place when the system is still floating.