  We present a new class of solutions for the inverse problem in the calculus
of variations in arbitrary dimension $n$. This is the problem of determining
the existence and uniqueness of Lagrangians for systems of $n$ second order
ordinary differential equations. We also provide a number of new theorems
concerning the inverse problem using exterior differential systems theory
(EDS). Concentrating on the differential step of the EDS process, our new
results provide a significant advance in the understanding of the inverse
problem in arbitrary dimension. In particular, we indicate how to generalise
Jesse Douglas's famous solution for $n=2$. We give some non-trivial examples in
dimensions 2,3 and 4. We finish with a new classification scheme for the
inverse problem in arbitrary dimension.
