Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1073expands into linearized Hamiltonian dynamicsN∑( ∂˙ξ 2 V1 = ξ N+1 , ˙ξN+1 = −ξ∂q 1 ∂q j ,j)q(t)j=1... ... (6.116)˙ξ n = ξ 2N ,N∑( ∂ 2 V˙ξ2N = −ξ∂q N ∂q j .j)q(t)Using (6.115) we can get the analytical expression for the largest Lyapunovexponent[1 ξ 2 1(t) + · · · + ξ 2 N(t) + ˙ξ 2 1(t) + · · · + ˙ξ] 2 1/2N(t)λ 1 = limt→∞ t log [ξ 2 1(0) + · · · + ξ 2 N(0) + ˙ξ 2 1(0) + · · · + ˙ξ] 2 1/2. (6.117)N (0)If there are critical points of V in configuration space, that is pointsq c = [q 1 , . . . , q N ] such that ∇V (q)| q=qc= 0, according to the Morse lemma(see e.g., [Hirsch (1976)]), in the neighborhood of any critical point q c therealways exists a coordinate system ˜q(t) = [q 1 (t), . . . , q N (t)] for whichV (˜q) = V (q c ) − ( q 1) 2− · · · −(qk ) 2+(qk+1 ) 2+ · · · +(qN ) 2, (6.118)where k is the index of the critical point, i.e., the number of negativeeigenvalues of the Hessian of V . In the neighborhood of a critical point,equation (6.118) yieldsj=1∂ 2 V/∂q i ∂q j = ±δ ij ,which, substituted into equation (6.115), gives k unstable directions whichcontribute to the exponential growth of the norm of the tangent vectorξ = ξ(t). This means that the strength of dynamical chaos, measured by thelargest Lyapunov exponent λ 1 , is affected by the existence of critical pointsof V . In particular, let us consider the possibility of a sudden variation,with the potential energy v, of the number of critical points (or of theirindexes) in configuration space at some value v c , it is then reasonable toexpect that the pattern of λ 1 (v) – as well as that of λ 1 (E) since v = v(E)– will be consequently affected, thus displaying jumps or cusps or othersingular patterns at v c .On the other hand, recall that Morse theory teaches us that the existenceof critical points of V is associated with topology changes of thehypersurfaces {M v } v∈R , provided that V is a good Morse function (that is: