  A complete solution to the multiplier version of the inverse problem of the
calculus of variations is given for a class of hyperbolic systems of
second-order partial differential equations in two independent variables. The
necessary and sufficient algebraic and differential conditions for the
existence of a variational multiplier are derived. It is shown that the number
of independent variational multipliers is determined by the nullity of a
completely algebraic system of equations associated to the given system of
partial differential equations. An algorithm for solving the inverse problem is
demonstrated on several examples. Systems of second-order partial differential
equations in two independent and dependent variables are studied and systems
which have more than one variational formulation are classified up to contact
equivalence.
