Complete separability of time-dependent second-order ordinary differential equations

Extending previous work on the geometric characterization of separability in the autonomous case, necessary and sufficient conditions are established for the complete separability of a system of time-dependent second-order ordinary differential equations. In deriving this result, extensive use is made of the theory of derivations of scalar and vector-valued forms along the projection pi: J(1) E --> E of the first jet bundle of a fibre bundle E --> R. Two illustrative examples are discussed, which fully demonstrate all aspects of the constructive nature of the theory.