  We prove the existence of chaotic motions in a planar restricted four body
problem, establishing that the system is not integrable. The idea of the proof
is to verify the hypotheses of a topological forcing theorem. The forcing
theorem applies to two freedom Hamiltonian systems where the stable and
unstable manifolds of a saddle-focus equilibrium intersect transversally in the
energy level set of the equilibrium. We develop a mathematically rigorous
computer assisted argument which verifies the hypotheses of the forcing
theorem, and we implement our approach for the restricted four body problem.
The method is constructive and works far from any perturbative regime and for
non-equal masses. Being constructive, our argument results in useful byproducts
like information about the locations of transverse connecting orbits,
quantitative information about the invariant manifolds, and upper/lower bounds
on transport times.
