  Pendulums are simple mechanical systems that have been studied for centuries
and exhibit many aspects of modern dynamical systems theory. In particular, the
double pendulum is a prototypical chaotic system that is frequently used to
illustrate a variety of phenomena in nonlinear dynamics. In this work, we
explore the existence and implications of codimension-1 invariant manifolds in
the double pendulum, which originate from unstable periodic orbits around
saddle equilibria and act as separatrices that mediate the global phase space
transport. Motivated in part by similar studies on the three-body problem, we
are able to draw a direct comparison between the dynamics of the double
pendulum and transport in the solar system, which exist on vastly different
scales. Thus, the double pendulum may be viewed as a table-top benchmark for
chaotic, saddle-mediated transport, with direct relevance to energy-efficient
space mission design. The analytical results of this work provide an existence
result, concerning arbitrarily long itineraries in phase space, that is
applicable to a wide class of two degree of freedom Hamiltonian systems,
including the three-body problem and the double pendulum. This manuscript
details a variety of periodic orbits corresponding to acrobatic motions of the
double pendulum that can be identified and controlled in a laboratory setting.
