This book is an introduction to Lagrangian and Hamiltonian mechanics primarily for mathematics undergraduates. Although the approach is traditional and coordinate based, it incorporates some of the insights and new perspectives of modern geometric treatments of mechanics. The book is intended for advanced undergraduates or graduate students and assumes familiarity with linear algebra, the chain rule for partial derivatives, and (to a lesser extent) three-dimensional vector mechanics. The aims are to give a confident understanding of the chain of argument that leads from Newton's laws through Lagrange's equations and Hamilton's principle to Hamilton's equations and canonical transformations; to confront head-on the points that mathematicians in particular find most awkward and confusing; to give practice in problem solving; and to elucidate the techniques that will reappear in later courses on relativity and quantum theory.