  In this work we consider variational properties of exact symplectic twist
maps $T$ that act on the cotangent bundle of a torus, or on a ball bundle over
a sphere. An example of such a map is the well-known Birkhoff billiard map
corresponding to smooth convex hypersurfaces. In this work we will focus on the
important class $\mathcal{M}$ of orbits of $T$ which are locally maximizing
with respect to the variational principle associated with a generating function
of the symplectic twist map. Our first goal is to give geometric and
variational characterization for orbits in the class $\mathcal M$.
  The billiard map is known to have two different generating functions, which
brings forth a natural question: to compare the properties of these two
generating functions. While our motivation comes from billiards, we will work
in general, and assume that a general twist map $T$ has two generating
functions. Thus we consider the orbits of $T$ which are locally maximizing with
respect to either of the generating functions. We formulate a geometric
criterion that guarantees that two generating functions of the same twist map
have the same class of locally maximizing orbits, and we will show that the two
generating functions for the Birkhoff billiard map do, in fact, satisfy this
criterion. The proof of this last property will rely on the Sinai-Chernov
formula from geometric optics and billiard dynamics.
