  We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions,
expressed in terms of a semi-basic 1-form, that characterize when a semispray
is locally Lagrangian. We also discuss the relation between these Helmholtz
conditions and their classic formulation written using a multiplier matrix.
When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two
(one) of the Helmholtz conditions are consequences of the other ones. These two
special cases correspond to two inverse problems in the calculus of variation:
Finsler metrizability for a spray, and projective metrizability for a spray.
