Deviations of stationary curves in the extended fibred Finslerian approach. Characterization and analytical solutions for E.D.G (Q1372460)

The authors study the equations of stationary curves and their deviations, on a vector bundle, using the classical variational principle. For a vector bundle \((E,p,M) =\xi\) endowed with a nonlinear connection \(N\) and an \((h,v)\)-metric \(\mathbb{G}\), one considers an \(N\)-linear connection \(\nabla\) compatible with \(\mathbb{G}\) and the arc length \(L\) of a curve \(c\). The first variation of \(L\) is expressed by means of \(\nabla\) and leads to the equations of stationary curves on \(E\) in an extended fibered Finslerian approach: \({\nabla \nu \over ds} = {\mathcal F}\), where \(\nu\) are \(h\)- and \(v\)-covariant velocities and \({\mathcal F}\) is the force moving of the test body. Some special cases are derived.