Towards Lagrangian Mechanics -- Smooth Manifolds -- Lagrangian Mechanics on Manifolds -- Variational Methods: the Action Principle -- Important Special Systems -- Hamilton Equations -- Symplectic Geometry -- Canonical Transformations -- The Hamilton-Jacobi Theory -- Integrable Systems and All That -- Introduction to Perturbative Methods.

This textbook is based on the author's lecture notes held at Qiuzhen College, Tsinghua University, Beijing, renowned for its rapid scientific growth of its excellent students. The book offers a remarkable combination of characteristics that are both exceptional and seemingly contradictory. It is designed to be entirely self-contained, starting from the basics and building a strong foundation in geometric and algebraic tools. Simultaneously, topics are infused with mathematical elegance and profundity, employing contemporary language and techniques. From a physicist's perspective, the content delves deeply into the physical aspects, emphasizing the underlying principles. This book bridges the gap between students and cutting-edge research, with a special focus on symplectic geometry, integrability, and recent developments in the field. It is designed to engage and captivate the reader. A conscious selection of topics ensures a more relevant and contemporary approach compared to traditional textbooks. The book addresses common misconceptions, offering clarity and precision. In its quest for brevity, this book is tailored for a one-semester course, offering a comprehensive and concise resource. The author's dedication is evident throughout this volume, encapsulating these goals within roughly 300 pages.