  In this paper we provide generalized Helmholtz conditions, in terms of a
semi-basic 1-form, which characterize when a given system of second order
ordinary differential equations is equivalent to the Lagrange equations, for
some given arbitrary non-conservative forces. For the particular cases of
dissipative or gyroscopic forces, these conditions, when expressed in terms of
a multiplier matrix, reduce to those obtained in [18]. When the involved
geometric structures are homogeneous with respect to the fibre coordinates, we
show how one can further simplify the generalized Helmholtz conditions. We
provide examples where the proposed generalized Helmholtz conditions, expressed
in terms of a semi-basic 1-form, can be integrated and the corresponding
Lagrangian and Lagrange equations can be found.
