A lie group structure on the space of time-dependent vector fields

In this note we use a banal relation between flows of vector fields, a sort of time-dependent Campbell—Baker—Hausdorff formula, to obtain a non-linear version of the variation of constants formula. We employ this formula to calculate the tangent to the mapping which assigns to each time-dependent vector fieldV with compact support the diffeomorphism _t (V), where _t (V) is the global flow ofV. Afterwards we use these results to give a non-commutative Lie group structure to the space of time-dependent vector fields with compact support. We also treat of the Lie algebra of this group and we calculate the adjoint action and the exponential mapping. The setting is the same of 1] and we refer the reader to this book for notations.