  The hypothesis that the Lorentz transformations may be modified at Planck
scale energies is further explored. We present a general formalism for theories
which preserve the relativity of inertial frames with a non-linear action of
the Lorentz transformations on momentum space. Several examples are discussed
in which the speed of light varies with energy and elementary particles have a
maximum momenta and/or energy. Energy and momentum conservation are suitably
generalized and a proposal is made for how the new transformation laws apply to
composite systems. We then use these results to explain the ultra high energy
cosmic ray anomaly and we find a form of the theory that explains the anomaly,
and leads also to a maximum momentum and a speed of light that diverges with
energy. We finally propose that the spatial coordinates be identified as the
generators of translation in Minkowski spacetime. In some examples this leads
to a commutative geometry, but with an energy dependent Planck constant.
