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This book covers a new explanation of the origin of Hamiltonian chaos and its quantitative characterization. The author focuses on two mainareas: Reimannian formulations of dynamics using either the Jacobi metric or the Eisenhart metric in extended configuration space; and topological aspects of phase transitions. Other items such as quantumsystems are also discussed in this framework. Mathematicians and physicists working in this area will find this book of interest.