Fedosov quantization of Lagrange-Finsler and Hamilton-Cartan spaces and Einstein gravity lifts on (co) tangent bundles

We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kahler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian/Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange-and Hamilton-Fedosov manifolds.