  L\'evy walks are continuous time random walks with spatio-temporal coupling
of jump lengths and waiting times, often used to model superdiffusive spreading
processes such as animals searching for food, tracer motion in weakly chaotic
systems, or even the dynamics in quantum systems such as cold atoms. In the
simplest version L\'evy walks move with a finite speed. Here, we present an
extension of the L\'evy walk scenario for the case when external force fields
influence the motion. The resulting motion is a combination of the response to
the deterministic force acting on the particle, changing its velocity according
to the principle of total energy conservation, and random velocity reversals
governed by the distribution of waiting times. For the fact that the motion
stays conservative, that is, on a constant energy surface, our scenario is
fundamentally different from thermal motion in the same external potentials. In
particular, we present results for the velocity and position distributions for
single well potentials of different steepness. The observed dynamics with its
continuous velocity changes enriches the theory of L\'evy walk processes and
will be of use in a variety of systems, for which the particles are externally
confined.
