  Billiard systems offer a simple setting to study regular and chaotic
dynamics. Gravitational billiards are generalizations of these classical
billiards which are amenable to both analytical and experimental
investigations. Most previous work on gravitational billiards has been
concerned with two dimensional boundaries. In particular the case of linear
boundaries, also known as the wedge billiard, has been widely studied. In this
work, we introduce a three dimensional version of the wedge; that is, we study
the nonlinear dynamics of a billiard in a constant gravitational field
colliding elastically with a linear cone of half angle $\theta$. We derive a
two-dimensional Poincar\'{e} map with two parameters, the half angle of the
cone and $\ell$, the $z$-component of the billiard's angular momentum. Although
this map is sufficient to determine the future motion of the billiard, the
three-dimensional nature of the physical trajectory means that a periodic orbit
of the mapping does not always correspond to a periodic trajectory in
coordinate space. We demonstrate several integrable cases of the parameter
values, and analytically compute the system's fixed point, analyzing the
stability of this orbit as a function of the parameters as well as its relation
to the physical trajectory of the billiard. Next, we explore the phase space of
the system numerically. We find that for small values of $\ell$ the conic
billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian
systems with a discontinuity, and the dynamics is qualitatively similar to that
of the wedge billiard, although the correspondence is not exact. As we increase
$\ell$ the dynamics becomes on the whole less chaotic, and the correspondence
with the wedge billiard is lost.
