  Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler
spaces are applied in fractional calculus with Caputo derivatives and for
elaborating models of fractional gravity and fractional Lagrange mechanics. The
geometric data for such models are encoded into (fractional) bi-Hamiltonian
structures and associated solitonic hierarchies. The constructions yield
horizontal/vertical pairs of fractional vector sine-Gordon equations and
fractional vector mKdV equations when the hierarchies for corresponding curve
fractional flows are described in explicit forms by fractional wave maps and
analogs of Schrodinger maps.
