Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 559Motion and Controllability in Affine ConnectionsIf we start with the local Riemannian metric form g ↦−→ g ij (q) dq i dq j ,then we have a kinetic energy Lagrangian L(q, v) = g ij (q) ˙q i ˙q j , and consequentlythe Euler–Lagrangian equations read(ddt ∂ ˙q iL−∂ q iL ≡ g ij ¨q j + ∂ q kg ij − 1 )2 ∂ q ig jk ˙q j ˙q k = u a Fi a , (i = 1, ..., n).Now multiply this by g li and take the symmetric part of the coefficientof ˙q j ˙q k to get ¨q l + Γ l jk ˙qj ˙q k = u a Ya, l (l = 1, ..., n,), where Γ i jkare theChristoffel symbols (3.143) for the Levi–Civita connection ∇ (see (3.10.1.1)above). So, the equations of motion an be rewritten∇ ˙γ(t) ˙γ(t) = u a (t) Y a (γ(t)) ,(a = 1, ..., m),where Y a = (F a ) ♯ , while ♯ : T ∗ M → T M is the ‘sharp’–isomorphism associatedwith the Riemannian metric g.Now, there is nothing to be gained by using a Levi–Civita connection,or by assuming that the vector–fields come from 1−forms. At this point,perhaps the generalization to an arbitrary affine connection seems like asenseless abstraction. However, as we shall see, this abstraction allows usto include another large class of mechanical control systems. So we willstudy the control system∇ ˙γ(t) ˙γ(t) = u a (t) Y a (γ(t)) [+Y 0 (γ(t))] , (4.56)with ∇ a general affine connection on M, and Y 1 ..., Y m linearly independentvector–fields on M. The ‘optional’ term Y 0 = Y 0 (γ(t)) in (4.56) indicateshow potential energy may be added. In this case Y 0 = − grad V (however,one looses nothing by considering a general vector–field instead of agradient) [Lewis (1998)].A solution to (4.56) is a pair (γ, u) satisfying (4.56) where γ : [0, T ] → Mis a curve and u : [0; T ] → R m is bounded and measurable.Let U be a neighborhood of q 0 ∈ M and denote by R U M (q 0, T ) thosepoints in M for which there exists a solution (γ, u) with the following properties:(1) γ(t) ∈ U for t ∈ [0, T ];(2) ˙γ(0) = 0 q ; and(3) γ(T ) ∈ T q M.