Nonlinear splittings on fibre bundles

We introduce the notion of a nonlinear splitting on fibre bundle as generalization an Ehresmann connection. present its basic properties and we pay attention to special cases affine, homogeneous principal splittings. explain where splittings appear in context Lagrangian systems Finsler geometry show their relation Routh symmetry reduction, submersive second-order differential equations unreduction. define curvature map for splitting, indicate this concept appears nonholonomic with affine constraints magnetic type.