Rotation of Trajectories of Lipschitz Vector Fields

We prove that in finite time a trajectory of a Lipschitz vector field in $\hbox{\bbbb R}^{\hbox{\tmm n}}$ can not have infinite rotation around a given point. This result extends to the mutual rotation of two trajectories of a field in $\hbox{\bbbb R}^{\hbox{\tmm 3}}$: this rotation is bounded from above on any finite time interval. The bounds we give are only in terms of the Lipschitz constant of the field and the length of the time interval.