Ivancevic_Applied-Diff-Geom

70 Applied Differential Geometry: A Modern IntroductionSince the coordinates q i are independent this equation is true for all variationsδq i and we get as a final result the covariant Lagrangian equations ofmotion,( )d ∂Ekindt ∂ ˙q i − ∂E kin∂q i = F i .If the force system is conservative and E pot is the system’s potential energygiven byF i = − ∂E pot∂q i ,then, using (2.25) the Lagrangian equations take the standard form( )d ∂Ldt ∂ ˙q i = ∂L∂q i , (2.26)where the Lagrangian function L = L(q, ˙q) of the system is given by L =E kin − E pot (since E pot does not contain ˙q i ).Now, the kinetic energy E kin of the system, given by quadratic form(2.22), is always positive except when ˙q i is zero in which case E kin vanishes.In other words, the quadratic form (2.22) is positive definite. Consequently,we can always find the line (or arc) element, defined byds 2 = G ij dq i dq j . (2.27)A manifold in which ds 2 is given by relation of the type of (2.27), geometricallywith g ij instead of G ij , is called a Riemannian manifold.2.1.4.1 Riemannian Curvature TensorEvery Riemannian manifold is characterized by the Riemann curvature tensor.In physical literature (see, e.g., [Misner et al. (1973)]) it is usuallyintroduced through the Jacobi equation of geodesic deviation, showing theacceleration of the relative separation of nearby geodesics (the shortest,straight lines on the manifold). For simplicity, consider a sphere of radiusa in R 3 . Here, Jacobi equation is pretty simple,d 2 ξ+ Rξ = 0,ds2 where ξ is the geodesic separation vector (the so–called Jacobi vector–field),s denotes the geodesic arc parameter given by (2.27) and R = 1/a 2 is theGaussian curvature of the surface.