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1 The Geometry of tangent manifold 1.1 The manifold TM 1.2 Semisprays on the manifold TM 1.3 Nonlinear connections 1.4 N-linear connections 1.5 Parallelism. Structure equations 2 Lagrange spaces 2.1 The notion of Lagrange space 2.2 Variational problem. Euler-Lagrange equations 2.3 Canonical semispray. Nonlinear connection 2.4 Hamilton-Jacobi equations 2.5 Metrical N-linear connections 2.6 The electromagnetic and gravitational fields 2.7 The almost Kahlerian model of a Lagrange space Ln 2.8 Generalized Lagrange spaces 3 Finsler Spaces 3.1 Finsler metrics 3.2 Geodesics 3.3 Cartan nonlinear connection 3.4 Cartan metrical connection 4 The Geometry of Cotangent Manifold 4.1 Cotangent bundle 4.2 Variational problem. Hamilton-Jacobi equations 4.3 Nonlinear connections 4.4 N-linear connections 4.5 Parallelism, paths and structure equations 5 Hamilton spaces 5.1 Notion of Hamilton space 5.2 Nonlinear connection of a Hamilton space 5.3 The canonical metrical connection of Hamilton space Hn 5.4 Generalized Hamilton Spaces GHn 5.5 The almost Kahlerian model of a Hamilton space 6 Cartan spaces 6.1 Notion of Caftan space 6.2 Canonical nonlinear connection of Ln 6.3 Canonical metrical connection of Ln 6.4 The duality between Lagrange and Hamilton spaces 7 The Geometry of the manifold TkM 7.1 The bundle of acceleration of order k¡Ý1 7.2 The Liouville vector fields 7.3 Variational Problem 7.4 Semisprays. Nonlinear connections 7.5 The dual coefficients of a nonlinear connection 7.6 Prolongation to the manifold TkM of the Riemannian structures given on the base manifold M 7.7 N-linear connections on TkM 8 Lagrange Spaces of Higher-order 8.1 The spaces L(k)n = (M,L) 8.2 Examples of spaces L(k)n 8.3 Canonical metrical N-connection 8.4 The Riemannian (k - 1)n-contact model of the space L(k)n