Ivancevic_Applied-Diff-Geom

836 Applied Differential Geometry: A Modern Introduction(5.95), we get the well–known formula for a Jacobi vector–field u along ageodesic c:∫ ba(gλµ ( ˙q α ∇ α u λ )( ˙q β ∇ β u µ ) + R λµαν u λ u α ˙q µ ˙q ν) dt+ g λµ ˙q α ∇ α u λ u ′µ | t=a − g λµ ˙q α ∇ α u λ u ′µ | t=b = 0.Therefore, the following assertions also remain true [Kobayashi and Nomizu(1963/9)]: (i) if the sectional curvature R λµαν u λ u α ˙q µ ˙q ν is positiveon a geodesic c, this geodesic has no conjugate points; (ii) if the sectionalcurvature R λµαν u λ u α v µ v ν , where u and v are arbitrary unit vectors ona Riemannian manifold Q less than k < 0, then, for every geodesic, thedistance between two consecutive conjugate points is at most π/ √ k.For example, let us consider a 1D motion described by the LagrangianL = 1 2 ( ˙q1 ) 2 − φ(q 1 ),where φ is a potential. The corresponding Lagrangian equation is equivalentto the geodesic one on the 2D Euclidean space R 2 with respect to theconnection ˜K whose non–zero component is ˜K 1 0 0 = −∂ 1 φ. The curvatureof ˜K has the non–zero componentR 1010 = ∂ 1 ˜K010 = −∂ 2 1φ.Choosing the particular Riemannian metric g with componentswe come to the formulag 11 = 1, g 01 = 0, g 00 = 1,∫ ba[( ˙q µ ∂ µ u 1 ) 2 − ∂ 2 1φ(u 1 ) 2 ]dt = 0for a Jacobi vector–field u, which vanishes at points a and b. Then we getthat, if ∂1φ 2 < 0 at points of c, this motion has no conjugate points. Inparticular, let us consider the oscillator φ = k(q 1 ) 2 /2. In this case, thesectional curvature is R 0101 = −k, while the half–period of this oscillatoris exactly π/ √ k.5.6.6 ConstraintsRecall that symplectic and Poisson manifolds give an adequate Hamiltonianformulation of classical and quantum conservative mechanics. This