Ivancevic_Applied-Diff-Geom

64 Applied Differential Geometry: A Modern Introduction2.1.3.6 3D Curve Geometry: Frenet–Serret FormulaeGiven three unit vectors: tangent τ i , principal normal β i , and binormalν i , as well as two scalar invariants: curvature K and torsion T, of a curveγ(s) = γ[x i (s)], the so–called Frenet–Serret formulae are valid 1˙¯τ i ≡ ˙τ i + Γ i jkτ j ẋ k = Kβ i ,˙¯β i ≡ ˙β i + Γ i jkβ j ẋ k = −(Kτ i + T ν i ),˙¯ν i ≡ ˙ν i + Γ i jkν j ẋ k = T β i .2.1.3.7 Mechanical Acceleration and ForceIn modern analytical mechanics, the two fundamental notions of accelerationand force in general curvilinear coordinates are substantially differentfrom the corresponding terms in Cartesian coordinates as commonly usedin engineering mechanics. Namely, the acceleration vector is not an ordinarytime derivative of the velocity vector; ‘even worse’, the force, which isa paradigm of a vector in statics and engineering vector mechanics, is nota vector at all. Proper mathematical definition of the acceleration vectoris the absolute time derivative of the velocity vector, while the force is adifferential one–form.To give a brief look at these ‘weird mathematical beasts’, consider a materialdynamical system described by n curvilinear coordinates x i = x i (t).First, recall from section 2.1.3.3 above, that an ordinary time derivative ofthe velocity vector v i (t) = ẋ i (t) does not transform as a vector with respectto the general coordinate transformation (2.2). Therefore, a i ≠ ˙v i . So, weneed to use its absolute time derivative to define the acceleration vector(with i, j, k = 1, ..., n),a i = ˙¯v i ≡ Dvidt= v i ;kẋ k ≡ ˙v i + Γ i jkv j v k ≡ ẍ i + Γ i jkẋ j ẋ k , (2.12)which is equivalent to the l.h.s of the geodesic equation (2.10). Only inthe particular case of Cartesian coordinates, the general acceleration vector(2.12) reduces to the familiar engineering form of the Euclidean accelerationvector 2 , a = ˙v.1 In this paragraph, the overdot denotes the total derivative with respect to the lineparameter s (instead of time t).2 Any Euclidean space can be defined as a set of Cartesian coordinates, while anyRiemannian manifold can be defined as a set of curvilinear coordinates. Christoffel’ssymbols Γ i jk vanish in Euclidean spaces defined by Cartesian coordinates; however, theyare nonzero in Riemannian manifolds defined by curvilinear coordinates.