  If a variational problem comes with no boundary conditions prescribed
beforehand, and yet these arise as a consequence of the variation process
itself, we speak of a free boundary values variational problem. Such is, for
instance, the problem of finding the shortest curve whose endpoints can slide
along two prescribed curves. There exists a rigorous geometric way to formulate
this sort of problems on smooth manifolds with boundary, which we review here
in a friendly self-contained way. As an application, we study a particular free
boundary values variational problem, the free-sliding Bernoulli beam.
