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Methods from the geometry of nonholonomic manifolds and Lagrange-Finslerspaces are applied in fractional calculus with Caputo derivatives and forelaborating models of fractional gravity and fractional Lagrange mechanics. Thegeometric data for such models are encoded into (fractional) bi-Hamiltonianstructures and associated solitonic hierarchies. The constructions yieldhorizontal/vertical pairs of fractional vector sine-Gordon equations andfractional vector mKdV equations when the hierarchies for corresponding curvefractional flows are described in explicit forms by fractional wave maps andanalogs of Schrodinger maps.

Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics