Ivancevic_Applied-Diff-Geom

170 Applied Differential Geometry: A Modern IntroductionA geodesic in M is a parameterized curve γ : I → M whose acceleration¨γ is everywhere orthogonal to M; that is, ¨γ(t) ∈ Mα(t) ⊥ for all t ∈ I ⊂ R.Thus a geodesic is a curve in M which always goes ‘straight ahead’ in thesurface. Its acceleration serves only to keep it in the surface. It has nocomponent of acceleration tangent to the surface. Therefore, it also has aconstant speed ˙γ(t).Let v ∈ M m be a vector on M. Then there exists an open interval I ⊂ Rcontaining 0 and a geodesic γ : I → M such that:(1) γ(0) = m and ˙γ(0) = v; and(2) If β : Ĩ → M is any other geodesic in M with β(0) = m and ˙β(0) = v,then Ĩ ⊂ I and β(t) = γ(t) for all t ∈ Ĩ.The geodesic γ is now called the maximal geodesic in M passing throughm with initial velocity v.By definition, a parameterized curve γ : I → M is a geodesic of M iffits acceleration is everywhere perpendicular to M, i.e., iff ¨γ(t) is a multipleof the orientation N(γ(t)) for all t ∈ I, i.e., ¨γ(t) = g(t) N(γ(t)), whereg : I → R. Taking the scalar product of both sides of this equation withN(γ(t)) we find g = − ˙γṄ(γ(t)). Thus γ : I → M is geodesic iff it satisfiesthe differential equation¨γ(t) + Ṅ(γ(t)) N(γ(t)) = 0.This vector equation represents the system of second–order componentODEsẍ i + N i (x + 1, . . . , x n ) ∂N j∂x k (x + 1, . . . , xn ) ẋ j ẋ k = 0.The substitution u i = ẋ i reduces this second–order differential system (inn variables x i ) to the first–order differential systemẋ i = u i ,˙u i = −N i (x + 1, . . . , x n ) ∂N j∂x k (x + 1, . . . , xn ) ẋ j ẋ k(in 2n variables x i and u i ). This first–order system is just the differentialequation for the integral curves of the vector–field X in U × R (U openchart in M), in which case X is called a geodesic spray.Now, when an integral curve γ(t) is the path a mechanical system Ξfollows, i.e., the solution of the equations of motion, it is called a trajectory.In this case the parameter t represents time, so that (3.36) describes motionof the system Ξ on its configuration manifold M.