Ivancevic_Applied-Diff-Geom

572 Applied Differential Geometry: A Modern IntroductionThe flow of f0T measures how the integral curves of f 0 change as we changethe initial condition in the direction of v x .Now, perhaps we can understand Z T on T M with f 0 = Z in the discussionof tangent lift. Let γ(t) be a geodesic. By varying the initial conditionfor the geodesic we generate an ‘infinitesimal variation’ which satisfies theextended Jacobi equation,∇ 2 ẏ(t)ξ(t) + R(ξ(t), ẏ(t)) ẏ(t) + ∇ẏ(t) (T (ξ(t), ẏ(t))) = 0. (4.64)To make the ‘connection’ between Z T and the Jacobi equation, we performconstructions on the tangent bundle using the spray Z. ∇ comes from alinear connection on M which induces an Ehresmann connection on τ M :T M → M. Thus we may write T vq T M ≃ T q M ⊕ T q M. Now, if I M :T T M → T T M is the canonical involution then I ∗ M ZT is a spray. We useI ∗ M ZT to induce an Ehresmann connection on τ T M : T T M → T M. Thus,T Xvq T T M ≃ T vq T M ⊕ T vq T M ≃T q M ⊕ T q M} {{ }⊕ T q M ⊕ T q M} {{ }.geodesic equations variation equationsOne represents Z T in this splitting and determines that the Jacobi equationsits ‘inside’ one of the four components. Now one applies similar constructionsto T ∗ T M and Z T ∗to derive a 1−form version of the Jacobi equation(4.64), the so–called adjoint Jacobi equation [Lewis (2000b)]:∇ 2 ẏ(t)λ(t) + R ∗ (λ(t), ẏ(t)) ẏ(t) − T ∗ ( ∇ẏ(t) λ(t), ẏ(t) ) = 0, (4.65)where we have used 〈R ∗ (α, u)v; ω〉 = 〈α; R(ω, u)v〉, and 〈T ∗ (α, u); ω〉 =〈α; T (ω, u)〉 .The adjoint Jacobi equation forms the backbone of a general statementof the PMP for affine connection control systems. When objective functionis the Lagrangian L(u, v q ) = 1 2 g(v q, v q ), when ∇ is the Levi–Civita connectionfor the Riemannian metric g, and when the system is fully actuated,then we recover the equation of [Noakes et al. (1989)]∇ 3 ẏ(t) ẏ(t) + R ( ∇ẏ(t) ẏ(t), ẏ(t) ) = 0.Therefore, the adjoint Jacobi equation (4.65) captures the interestingpart of the Hamiltonian vector–field Z T ∗ , which comes from the PMP, interms of affine geometry, i.e., from Z T ∗follows∇ẏ(t) ẏ(t) = 0, ∇ 2 ẏ(t)λ(t) + R ∗ (λ(t), ẏ(t)) ẏ(t) − T ∗ ( ∇ẏ(t) λ(t), ẏ(t) ) = 0.