Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 535Now, let us see the mechanical interpretation of these geometrical objects.Consider a nonholonomic mechanical system corresponding to a Riemannianmanifold (M, g), where g is a metric defined by the system’s kineticenergy [Dragovic and Gajic (2003)]. Suppose that the distribution Vis defined by (n − m) one–forms ω α ; in local coordinates q = (q 1 , ..., q n ) onMω ρ (q)( ˙q) = a ρi (q) ˙q i = 0,(ρ = m + 1, . . . , n; i = 1, . . . , n).A virtual displacement is a vector–field X on M, such that ω ρ (X) = 0, i.e.,X belongs to the differential system N(V ).Differential equations of motion of a given mechanical system followfrom the D’Alambert–Lagrangian principle: trajectory γ of the given systemis a solution of the equation〈∇ ˙γ ˙γ − Q, X〉 = 0, (4.40)where X is an arbitrary virtual displacement, Q a vector–field of internalforces, and ∇ is the affine Levi–Civita connection for the metric g.The vector–field R(x) on M, such that R(x) ∈ Vx ⊥ , Vx⊥ ⊕ V x = T x M, iscalled reaction of ideal nonholonomic connections. (4.40) can be rewrittenas∇ ˙γ ˙γ − Q = R, ω α ( ˙γ) = 0. (4.41)If the system is potential, by introducing L = T − U, where U is thepotential energy of the system (Q = − grad U), then in local coordinates qon M, equations (4.41) becomes the forced Lagrangian equation:ddt L ˙q − L q = ˜R, ω α ( ˙q) = 0.Now ˜R is a one–form in (V ⊥ ), and it can be represented as a linear combinationof one–forms ω m+1 , . . . , ω n which define the distribution, ˜R = λ α ω α .Suppose e 1 , . . . , e n are the vector–fields on M, such that e 1 (x), . . . , e n (x)form a base of the vector space T x M at every point x ∈ M, and e 1 , . . . , e mgenerate the differential system N(V ). Express them through the coordinatevector–fields:e i = A j i (q)∂ q j ,(i, j = 1, . . . , n).Denote by p a projection p : T M → V orthogonal according to themetric g. Corresponding homomorphism of C ∞ −modules of sections of