Ivancevic_Applied-Diff-Geom

868 Applied Differential Geometry: A Modern IntroductionThe instant–wise distance between the solutions s and s ′ (5.191) with respectto the metric g (5.192) isρ t (s, s ′ ) = [g yy (s(t) − s ′ (t)) 2 ] 1/2 = |c − c ′ | exp(λt).One at once gets that the Lyapunov spectrum of any solution of the differentialequation (5.189) with respect to the metric (5.192) is λ.5.7 Application: Jets and Multi–Time RheonomicDynamicsRecall that a number of geometrical models in mechanics and physics arebased on the notion of ordinary, autonomous Lagrangian (i.e., a smoothreal function on R × T M). In this sense, we recall that a Lagrangian spaceL n = (M, L(x, y)) is defined as a pair which consists of a real, smooth,nD manifold M with local coordinates x i , (i = 1, ..., n) and a regular LagrangianL : T M → R. The geometry of Lagrangian spaces is now used invarious fields to study natural phenomena where the dependence on position,velocity or momentum is involved [Kamron and Olver (1989)]. Also,this geometry gives a model for both the gravitational and electromagneticfield theory, in a very natural blending of the geometrical structureof the space with the characteristic properties of the physical fields. Again,there are many problems in physics and variational calculus in which time–dependent Lagrangians are involved.In the context exposed in [Miron et. al. (1988); Miron and Anastasiei(1994)], the energy action functional E, attached to a given time–dependentLagrangian,L : R × T M → R, (t, x i , v i ) ↦→ L(t, x i , v i ), (i = 1, ..., n)not necessarily homogenous with respect to the direction {v i }, is of theformE(c) =∫ baL(t, x i (t), ẋ i (t)) dt, (5.193)where [a, b] ⊂ R, and c : [a, b] → M is a smooth curve, locally expressed byt ↦→ x i (t), and having the velocity ẋ = (ẋ i (t)). It is obvious that the non–homogeneity of the Lagrangian L, regarded as a smooth function on theproduct manifold R × T M, implies that the energy action functional E isdependent of the parametrizations of every curve c. In order to remove this