This book is intended for first year physics graduate students who wish to learn about analytical mechanics. Lagrangians and Hamiltonians are extensively treated; particle motion, oscillations, coordinate systems, and rigid bodies are dealt with in far greater detail than in most undergraduate textbooks; and perturbation theory, relativistic mechanics, and two case studies of continuous systems are presented.Each subject is approached at progressively higher levels of abstraction. Lagrangians and Hamiltonians are first presented in an inductive way, leading up to general proofs. Hamiltonian mechanics is expressed in Cartan's notation not too early; there is a self-contained account of the traditional formulation.