Summary: A projection is defined such that a second-order Lagrangian density factors through this projection modulo contact forms if and only if it is parameter invariant. In this way, a geometric interpretation of the parameter invariance conditions is obtained. The above projection is then used to prove the strict factorization of the Poincaré-Cartan form attached to a parameter-invariant variational problem thus leading us to state the Hamilton-Cartan formalism, the complete description of symmetries and regularity for such problems. The case of the squared curvature Lagrangian in the plane is analysed especially.