Let $M$ be a closed connected Riemannian manifold, and let $\alpha$ be a homotopy class of free loops in $M$. Then, for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle which is sufficiently large over the zero section, we prove the existence of a $1$-periodic orbit whose projection to $M$ represents $\alpha$. The proof shows that the Biran-Polterovich-Salamon capacity of the open unit disk cotangent bundle relative to the zero section is finite. If $M$ is not simply connected, this leads to an existence result for noncontractible periodic orbits on level hypersurfaces corresponding to a dense set of values of any proper Hamiltonian on $T^*M$ bounded from below, whenever the levels enclose $M$. This implies a version of the Weinstein conjecture including multiplicities; we prove existence of closed characteristics—one associated to each nontrivial $\alpha$—on every contact-type hypersurface in $T^*M$ enclosing $M$