Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 867vertical cotangent bundle V ∗ Y of Y → R. With respect to the holonomiccoordinates (t, y k , y k ) on V ∗ Y , it takes the coordinate formg = 1 2 g ij(t, y k )dy i ∨ dy j ,where {dy i } are the holonomic fibre bases for V ∗ Y .Recall that above we have proposed the following: Let λ be a realnumber. Given a dynamical equation defined by a complete vector–field γ(5.175), there exists a Riemannian fibre metric on Y such that the Lyapunovspectrum of any solution of γ is λ. The following example aims to illustratethis fact.Let us consider 1D motion on the axis R defined by the first–orderdynamical equationẏ = y (5.189)on the fibre bundle Y = R × R −→ R coordinated by (t, y). Solutions of theequation (5.189) reads(t) = c exp(t), (with c = const). (5.190)Let e yy = 1 be the standard Euclidean metric on R. With respect to thismetric, the instant–wise distance between two arbitrary solutionsof the equation (5.189) iss(t) = c exp(t), s ′ (t) = c ′ exp(t) (5.191)ρ t (s, s ′ ) e = |c − c ′ | exp(t).Hence, the Lyapunov exponent K(s, s ′ ) (5.188) equals 1, and so is theLyapunov spectrum of any solution (5.190) of the first–order dynamicalequation (5.189).Let now λ be an arbitrary real number. There exists a coordinatey ′ = y exp(−t) on R such that, written relative to this coordinate, thesolutions (5.190) of the equation (5.189) read s(t) = const. Let us choose theRiemannian fibre metric on Y → R which takes the form g y′ y ′ = exp(2λt)with respect to the coordinate y ′ . Then relative to the coordinate y, itreadsg yy = ∂y′ ∂y ′∂y ∂y g y ′ y ′ = exp(2(λ − 1)t). (5.192)