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A complete solution to the multiplier version of the inverse problem of thecalculus of variations is given for a class of hyperbolic systems ofsecond-order partial differential equations in two independent variables. Thenecessary and sufficient algebraic and differential conditions for theexistence of a variational multiplier are derived. It is shown that the numberof independent variational multipliers is determined by the nullity of acompletely algebraic system of equations associated to the given system ofpartial differential equations. An algorithm for solving the inverse problem isdemonstrated on several examples. Systems of second-order partial differentialequations in two independent and dependent variables are studied and systemswhich have more than one variational formulation are classified up to contactequivalence.

The inverse problem of the calculus of variations for systems of second-order partial differential equations in the plane