From variational principles to geometry

A method to construct a geometric structure with the same solutions as a given variational principle is presented. The method applies to large families of variational principles. In particular, the known results that assign cosymplectic geometry to Hamilton’s principle and cocontact geometry to Herglotz’s principle for regular Lagrangians are recovered. The unified Lagrangian–Hamiltonian formalism is also recovered via the absorption of the holonomy conditions. The method is applied to singular time-dependent Lagrangians, proving that they can always be described with a (pre)cosymplectic structure, although it is not always given by the Lagrangian 2-form. When applied to singular action-dependent Lagrangians, the method does not always lead to (pre)cocontact geometry. In these cases, the resulting geometry associated with Herglotz’s variational principle is new.