Ivancevic_Applied-Diff-Geom

446 Applied Differential Geometry: A Modern IntroductionUsing the fact that X j ẋ i ∇ i ẋ j = X j ∇ ˙γ ˙γ j = 0, as well as the auto–parallelism of the geodesics, this can be rewritten as∂ s (X i ẋ i ) = 1 2ẋj ẋ i (∇ i X j + ∇ j X i ),(i, j = 1, ..., N).This means that the conservation of X i ẋ i along a geodesic, i.e., ∂ s (X i ẋ i ) =0, is guaranteed by (see [Clementi and Pettini (2002)])∇ (i X j) ≡ ∇ i X j + ∇ j X i = 0. (3.255)If such a field exists on a manifold, it is the Killing vector–field. Recallthat (3.255) is equivalent to L X g = 0, where L is the Lie derivative. Onthe biodynamical manifolds (M, g), being the unit vector ˙q i – tangent to ageodesic – proportional to the canonical momentum p i = ∂L∂ ˙q= ˙q i , the existenceof a Killing vector–field X implies that the corresponding momentumimap (see subsection 3.12.3.5 above),J(q, p) = X k (q)∂ s q k =1√ X k (q) ˙q k 1= √ X k (q)p k ,2(E − V (q)) 2T (q)(3.256)is a constant of motion along the geodesic flow. Thus, for an NDOF Hamiltoniansystem, a physical conservation law, involving a conserved quantitylinear in the canonical momenta, can always be related with a symmetryon the manifold (M, g) due to the action of a Killing vector–field on themanifold. These are the Noether conservation laws. The equation (3.255)is equivalent to the vanishing of the Poisson brackets,( ∂H ∂J{H, J} =∂q i − ∂H )∂J∂p i ∂p i ∂q i = 0, (3.257)which is the standard definition of a constant of motion J(q, p) (see, e.g.,[Abraham and Marsden (1978)]).However, if a 1–1 correspondence is to exist between conserved physicalquantities along a Hamiltonian flow and suitable symmetries of thebiodynamical manifolds (M, g), then integrability will be equivalent to theexistence of a number of symmetries at least equal to the number of DOF,which is equal to dim(M). If a Lie group G acts on the phase–space manifoldthrough completely canonical transformations, and there exists anassociated momentum map, then every Hamiltonian having G as a symmetrygroup, with respect to its action, admits the momentum map asthe constant of motion. These symmetries are usually referred to as hidden