  Using super Hamiltonian formalism, we study the motion of particles whose
dispersion relations are modified to incorporate Ho\v{r}ava -- Lifshitz type
anisotropic scaling symmetry. We find the following as consequences of this
modified dispersion relation: (i) The speed of a charged particle under a
constant electric field grows without bound and diverges. (ii) The speed of a
particle falling towards the horizon also grows without bound and diverges as
the particle approaches the horizon. (iii) This particle reaches the horizon in
a finite coordinate time, in contrast to the standard case where it requires
infinite time.
