Muñoz Masqué, JaimePozo Coronado, Luis Miguel2023-06-202023-06-201998Abate M and Patrizio G 1994 Finsler Metrics—A Global Approach (Lecture Notes in Mathematics 1591)(New York: Springer) Atiyah M F and MacDonald I G 1969 Introduction to Commutative Algebra (Reading, MA: Addison-Wesley)[3] Bao D and Chern S S 1993 On a notable connection in Finsler geometry Houston J. Math. 19 135–80 Batlle C, Gomis J, Pons J M and Rom´an-Roy N 1988 Lagrangian and Hamiltonian constraints for secondorder singular Lagrangians J. Phys. A: Math. Gen. 21 2693–703 Blaschke W 1930 Vorlesungenuber Differentialgeometrie vol I 3rd edn (Berlin: Springer) Bryant R and Griffiths P 1986 Reduction for constrained variational problems and R k2ds Am. J. Math. 108 525–70 Constantelos G C 1984 On the Hamilton–Jacobi theory with derivatives of higher order Riv. Nuovo Cimento B 84 91–101 Giaquinta M and Hildebrandt S 1996 Calculus of Variations II: The Hamiltonian Formalism (Berlin: Springer) Godbillon C 1969 Geometrie Differentielle et Mecanique Analytique (Paris: Hermann) Goldschmidt H 1967 Integrability criteria for systems of non-linear partial differential equations J. Diff.Geom. 1 269–307 Goldschmidt H and Sternberg S 1973 The Hamilton–Cartan formalism in the calculus of variations Ann. Inst.Fourier 23 203–67 Grifone J 1972 Structure presque-tangente et connections II Ann. Inst. Fourier Grenoble 22 291–338 Guggenheimer H W 1963 Differential Geometry (New York:McGraw-Hill)0305-447010.1088/0305-4470/31/29/014https://hdl.handle.net/20.500.14352/57772A projection is defined such that a second-order Lagrangian density factors through this projection module 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 Poincare-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.engjournal articlehttp://iopscience.iop.org/0305-4470/31/29/014/pdf/0305-4470_31_29_014.pdfhttp://iopscience.iop.orgrestricted access514.7Second-order Lagrangian densityParameter invariancePoincare-Cartan formSquared curvature LagrangianGeometría diferencial1204.04 Geometría Diferencial