Equivalence of higher-order lagrangians: III. new invariant differential equations

The solution to the equivalence problem for a higher-order Lagrangian leads to new differential equations which are invariantly associated with the variational functional. We derive explicit expressions for these equations in the case of second-order particle Lagrangians on the line under fibre-preserving, point and contact transformations. A geometrical interpretation of these equations based on the Poincaré-Cartan form is discussed.