Lagrangian control systems that are differentially flat with flat outputs that depend only on configuration variables are said to be configuration flat. We provide a complete characterization of configuration flatness for systems with n degrees of freedom and n-1 controls whose range of control forces only depends on configuration and whose Lagrangian has the form of kinetic energy minus potential. The method presented allows us to determine if such a system is configuration flat and, if so, provides a constructive method for finding all possible configuration flat outputs. Our characterization relates configuration flatness to Riemannian geometry. We illustrate the method with two examples.