Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 77of the vector–field, such that all the vectors from the vector–field are tangentto integral curves at different representative points x i ∈ U. In this way,through every representative point x i ∈ U passes both a curve from the flowand its tangent vector from the vector–field. Geometrically, vector–field isdefined as a cross–section of the tangent bundle T M, the so–called velocityphase–space. Its geometrical dual is the 1–form–field, which represents afield of one–forms (see Figure 2.1), defined at the same representative pointsx i ∈ U. Analytically, 1–form–field is defined as an exterior differentialsystem, an algebraic dual to the autonomous set of ODEs. Geometrically,it is defined as a cross–section of the cotangent bundle T ∗ M, the so–calledmomentum phase–space. Together, the vector–field and its corresponding1–form–field define the scalar potential field (e.g., kinetic and/or potentialenergy) at the same movable region U ⊂ M.Next, we need to formulate the internal acceleration vector–field, a i ≡a i (x i , ẋ i , t), acting in all movable joints, and at the same time generalizingthe Newtonian 3D acceleration vector a.According to Newton, acceleration is a rate–of–change of velocity. But,from the previous subsections, we know that a i ≠ ˙v i . However,a i := ˙¯v i = ˙v i + Γ i jkv j v k = ẍ i + Γ i jkẋ j ẋ k . (2.30)Once we have the internal acceleration vector–field a i= a i (x i , ẋ i , t),defined by the set of ODEs (2.30) (including Levi–Civita connections Γ i jkof the Riemannian configuration manifold M), we can finally define theinternal force 1–form field, F i = F i (x i , ẋ i , t), as a family of force one–forms, half of them rotational and half translational, acting in all movablejoints,F i := mg ij a j = mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), (2.31)where we have used the simplified material metric tensor, mg ij , for thesystem (considering, for simplicity, all segments to have equal mass m),defined by its Riemannian kinetic energy formT = 1 2 mg ijv i v j .Equation F i = mg ij a j , defined properly by (2.31) at every representativepoint x i of the system’s configuration manifold M, formulates thesought for covariant force law, that generalizes the fundamental Newtonianequation, F = ma, for the generic physical or engineering system. Its