The course contains two parts. Part 1 covers the essential and relevant parts of differential geometry/symplectic geometry/Poisson geometry, Lie group/Lie algebra. For example, exterior differentiation, Lie differentiation, covariant differentiation, Schouten-Nijenhuis bracket, connection on principal bundles, co-adjoint orbits, symplectic reduction, and Sternberg phase space. Part 2 covers Hamiltonian mechanics, Lagrangian mechanics, and Tulczyjew mechanics. Students who wish to understand the essence of classical mechanics are strongly advised to take this course. Students should seek the course instructor’s approval to take this course.