Higher order differential lift equations over manifolds

Abstract

LetM be a differentiable manifold modeled on a Banach space overK=R or C. LetT ^k(M) be thekth iterated tangential extension ofM, and letk ^M be thekth Bowman (=restricted tangential) extension ofM. It is shown that there is an embedding ϕ_k:k ^{→T k}(M), and that such embeddings constitute a natural transformation of functors. LetQ be a subset/submanifold inT ^k(M), and letV:Q→T(Q) be a differentiable vector field. CallV k-suitable if everyK-curveg inQ satisfyingg′=V° g has the formg=f ^[k], wheref ^[k] denotes thekth iterated differential lift of aK-curvef inM. It is shown thatV isk-suitable if and only if: (a)$Q = \varphi _k \left( {\overline Q } \right)$, where$\overline Q$ is a subset/submanifold ink ^M, and (b)$V = T\left( {\varphi _k } \right)^\circ \bar V^\circ \varphi _k^{ - 1}$, where$\overline V :\overline Q \to T\left( {\overline Q } \right)$ isk-suitable relative to restricted tangentialK-curve liftsf ^(k). Interpretive consequences for motion problems are discussed.