Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττ_k:T ^k+1(M)=T(T ^k(M))→T ^k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT ⁰(M)=M. LetV represent a possibly nonstationary vector field overT ^k(M), and letQ be a subset/submanifold inT ^k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f ^[k], wheref ^[k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail. To the memory of my teacher and friend M. Kuga (1928–1990).