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This monograph presents the foundations of Hamilton Geometry. The concept of Hamilton Space opens a new domain in differential geometry with large applications in such areas as mechanics, physics, and optimal control. The book consists of 13 chapters. The first three present the topics of the tangent bundle geometry, and Finsler and Lagrange spaces. Subsequent chapters are devoted to the construction of geometry of Hamilton spaces and the duality between these spaces and Lagrange spaces, also, the symplectic transformations of cotangent bundle. The last five chapters present the geometrical theory and applications of Higher-Order Hamilton spaces. In particular, the case of order 2 is presented in detail.
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