Hamiltonian and Lagrangian Dynamics (HLD) are two interrelated regimes and sets of techniques that can be used to solve Classical Mechanics problems, like Newtonian Physics does, but HLD is much more powerful and flexible, making manageable the otherwise intractable. In addition, HLD provides intuitive insight and guides approximation techniques. Most importantly, HLD is a foundation for Quantum Mechanics, Quantum Field Theory, Elementary Particle Physics, Solid State Physics and Celestial Mechanics. The book begins with Newtonian Mechanics, and Special Relativity, then HLD, and goes on to cover the usual applications in the physics curriculum. This book emphasizes geometric reasoning in both the exposition and in exercises. The author has also written a book on Linear Algebra for Physicists and Engineers, wherein the necessary mathematics is treated lovingly. I treat rotations, Hermitian matrices, Functional Analysis, with a derivation of the Euler Lagrange equation, Lie Groups, Elliptic Function Theory. The book is intended to be useful for undergraduates and graduate students in Physics and Engineering and Mathematics.