Ivancevic_Applied-Diff-Geom

448 Applied Differential Geometry: A Modern Introductionout that the equations (3.261) are equivalent to the vanishing of the Poissonbrackets of J(q, p), that is{H, J} = 0 is equivalent to ∇ (j K j1j 2...j r) = 0.Thus, the existence of Killing tensor–fields, obeying (3.261), on a biodynamicalmanifold (M, g) give the rephrasing of integrability of Newtonian equationsof motion or, equivalently, of standard Hamiltonian systems, withinthe Riemannian geometrical framework.The first natural question to address concerns the existence of a Killingtensor–field, on any biodynamical manifold (M, g), to be associated withtotal energy conservation. Such a Killing tensor–field actually exists andcoincides with the metric tensor g, in fact it satisfies by definition (3.261).One of the simplest case of integrable system is represented by a decoupledsystem described by a generic HamiltonianN∑[ ]p2 N∑H = i2 + V i(q i ) = H i (q i , p i )i=1for which all the energies E i of the subsystems H i , i = 1, . . . , N, are conserved.On the associated biodynamical manifold, N second–order Killingtensor–fields exist, they are given byi=1K (i)jk = δ jk{V i (q i )[E − V (q i )] + δ i j[E − V (q i )] 2 }.In fact, these tensor–fields fulfil (3.261), which explicitly reads [Clementiand Pettini (2002)]∇ k K (i)lm + ∇ lK (i)mk + ∇ mK (i)kl= ∂ q kK (i)lm + ∂ q lK(i) mk + ∂ q mK(i) kl− 2Γ j kl K(i) jm − 2Γj km K(i) jl− 2Γ j lm K(i) jk = 0.The conserved quantities J (i) (q, p) are then get by saturation of the tensorsK (i) with the velocities ˙q i ,J (i) (q, p) = K (i)jk ˙qj ˙q k = E i .3.16 Application: Lax–Pair Tensors in GravitationRecall that many problems in general relativity require an understandingof the global structure of the space–time. Currently discussed global problemsinclude the occurrence of naked singularities [Ori and Piran (1987)]and universality in gravitational collapse situations [Choptuik (1993)]. The