Hamiltonian and Lagrangian Dynamics is useful for solving difficult mechanics problems, and is critical for understanding Quantum Mechanics and Quantum Field Theory. The present text is unique for its emphasis on geometric reasoning and intuition, and for its accessibility. Volume 1 is primarily mathematical, covering Linear and Abstract Algebra, Functional Analysis and Differential Geometry, and Volume 2 is primarily devoted to physics, beginning with a serious approach to Newtonian physics, then Hamiltonian and Lagrangian Dynamics, including classical field theory, the basis of Quantum Field theory. This book is intended to be used by physics undergraduates who have just finished the Calculus sequence and by physics graduate students. Beginning mathematics students will also find it useful, for introducing some modern mathematics in a way that is accessible, and then showing some of its many uses. Engineering students may also be interested in this book.