  New geometric structures that relate the lagrangian and hamiltonian
formalisms defined upon a singular lagrangian are presented. Several vector
fields are constructed in velocity space that give new and precise answers to
several topics like the projectability of a vector field to a hamiltonian
vector field, the computation of the kernel of the presymplectic form of
lagrangian formalism, the construction of the lagrangian dynamical vector
fields, and the characterisation of dynamical symmetries.
