Ivancevic_Applied-Diff-Geom

1072 Applied Differential Geometry: A Modern Introductiona diffeomorphism invariant to probe if and how the topology of the M vchanges as a function of v. This is a very challenging task because we haveto deal with high dimensional manifolds. Fortunately a topological invariantexists whose computation is feasible, yet demands a big effort. Recall(from subsection 3.10.1 above) that this is the Euler characteristic, a diffeomorphisminvariant of the system’s configuration manifold, expressingits fundamental topological information.6.4.5.2 Geometry of the Largest Lyapunov ExponentNow, the topological hypothesis has recently been promoted into a topologicalTheorem [Franzosi and Pettini (2004)]. The new Theorem says thatnon–analyticity is the ‘shadow’ of a more fundamental phenomenon occurringin the system’s configuration manifold: a topology change within thefamily of equipotential hypersurfaces (6.114). This topological approach toPTs stems from the numerical study of the Hamiltonian dynamical counterpartof phase transitions, and precisely from the observation of discontinuousor cuspy patterns, displayed by the largest Lyapunov exponent atthe transition energy (or temperature).Recall that the Lyapunov exponents measure the strength of dynamicalchaos and cannot be measured in laboratory experiments, at variancewith thermodynamic observables, thus, being genuine dynamical observablesthey are only measurable in numerical simulations of the microscopicdynamics. To get a hold of the reason why the largest Lyapunov exponentλ 1 should probe configuration space topology, let us first remember that forstandard Hamiltonian systems, λ 1 is computed by solving the tangent dynamicsequation for Hamiltonian systems (see Jacobi equation of geodesicdeviation (3.133)),( ∂ 2 V¨ξi +∂q i ∂q)q(t)j ξ j = 0, (6.115)which, for the nonlinear Hamiltonian system˙q 1 = p 1 , ṗ 1 = −∂ q 1V,... ...˙q N = p N , ṗ N = −∂ q N V,