When I began looking for mechanics textbooks, I wanted explicit definitions and proofs alongside the physical arguments. Rudin's Principles of Mathematical Analysis had helped me understand limits after less precise explanations left me confused, and I hoped to find a similar level of mathematical care in mechanics.
That preference does not make physical intuition dispensable. A rigorous deduction tells us what follows from a model; physical reasoning and experiment help decide whether the model describes the system. This note revises my original Chinese recommendation.
Four books with different strengths
Darryl D. Holm, Tanya Schmah, and Cristina Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. This is a route into mechanics through geometry and symmetry, with examples connecting the formal structures to mechanical systems. Its scope and chapter list are available from Oxford University Press. It is one of the books I would consider first for a mathematically structured treatment.
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, second edition. This develops Hamiltonian and Lagrangian mechanics with an emphasis on geometric structure, symmetry, and reduction. It is a substantial text, better approached as a course of study than a quick introduction. The authors' institutional record provides the bibliographic details and available material.
V. I. Arnold, Mathematical Methods of Classical Mechanics. Arnold connects Newtonian, Lagrangian, and Hamiltonian viewpoints and develops their geometric meaning. The exposition is concise and often expects the reader to supply intermediate reasoning. I would use it when I wanted to understand how the formulations fit together, rather than assume it is less rigorous because its style differs from the first two books.
L. D. Landau and E. M. Lifshitz, Mechanics, volume 1 of the Course of Theoretical Physics. This is a compact theoretical-physics treatment, with symmetry and variational reasoning playing central roles. It complements the geometric texts, but its pace and style make it a different choice from a definition–theorem–proof introduction.
Reading all four at once is unnecessary. Choose one main text and use another when its treatment clarifies a specific difficulty.
Preparation should match the book
Start with multivariable calculus, linear algebra, and ordinary differential equations. For geometric mechanics, familiarity with manifolds, tangent and cotangent spaces, and differential forms becomes useful. The required depth depends on the chapters being read, and some concepts can be learned alongside the mechanics.
Rudin's Principles of Mathematical Analysis is an analysis text, not a first course in computational calculus. His Real and Complex Analysis and Functional Analysis are not general prerequisites for beginning classical mechanics. Requiring all three before opening a mechanics book would impose a large and mostly unnecessary detour.
For a freely available bridge into the subject, David Tong's classical dynamics lectures cover the Lagrangian and Hamiltonian formalisms. Work through concrete systems as well as abstract definitions: a pendulum, a central-force problem, and a rigid body reveal what the notation is accomplishing.
For Chinese readers interested in history, my original reading list also included 《分析力学史略》, by 梅凤翔、吴惠彬、李彦敏. I had begun reading it when I wrote the earlier version of this note.