Ivancevic_Applied-Diff-Geom

860 Applied Differential Geometry: A Modern Introductionwhile the global section h ′ : W → W ′ such that I 0 ◦ h ′ = −H ′ , of the trivialbundle ζ (5.173), gives W with the Hamiltonian formH ′ = I i dφ i − H ′ (I j )dt.The associated vector–field γ H (5.166) is exactly the projection onto Wof the Hamiltonian vector–field γ T (5.172), and takes the same coordinateform. It defines the Hamiltonian equation on W ,I i = const, ˙φi = ∂ i H ′ (I j ).One can think of this equation as being the Hamiltonian equation of a time–dependent Hamiltonian system around the invariant manifold N relative totime–dependent action–angle coordinates.5.6.14 Lyapunov StabilityThe notion of the Lyapunov stability of a dynamical equation on a smoothmanifold implies that this manifold is equipped with a Riemannian metric.At the same time, no preferable Riemannian metric is associated to a first–order dynamical equation. Here, we aim to study the Lyapunov stability offirst–order dynamical equations in non–autonomous mechanics with respectto different (time–dependent) Riemannian metrics.Let us recall that a solution s(t), for all t ∈ R, of a first–order dynamicalequation is said to be Lyapunov stable (in the positive direction) if fort 0 ∈ R and any ε > 0, there is δ > 0 such that, if s ′ (t) is another solutionand the distance between the points s(t 0 ) and s ′ (t 0 ) is inferior to δ, thenthe distance between the points s(t) and s ′ (t) for all t > t 0 is inferior to ε.In order to formulate a criterion of the Lyapunov stability with respect to atime–dependent Riemannian metric, we introduce the notion of a covariantLyapunov tensor as generalization of the well–known Lyapunov matrix.The latter is defined as the coefficient matrix of the variation equation[Gallavotti (1983); Hirsch and Smale (1974)], and fails to be a tensor undercoordinate transformations, unless they are linear and time–independent.On the contrary, the covariant Lyapunov tensor is a true tensor–field, butit essentially depends on the choice of a Riemannian metric. The followingwas shown in [Sardanashvily (2002b)]:(i) If the covariant Lyapunov tensor is negative definite in a tubularneighborhood of a solution s at points t ≥ t 0 , this solution is Lyapunovstable.