Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.
As a student of pure mathematics, I once found myself on a quest to find extremely rigorous textbooks for studying classical mechanics — something in the Bourbaki style, akin to a “baby Rudin” version of classical mechanics.
Due to the different thinking patterns between mathematics and natural sciences, I often approach natural science through the lens of formal science. Unfortunately, most of the available textbooks didn’t suit my taste. For example, when it comes to introductory calculus, my personal favorite is Baby Rudin (Walter Rudin’s Principles of Mathematical Analysis). Although it’s difficult to digest, I simply cannot accept textbooks that skip the $\varepsilon – \delta$ definition of limits and jump straight to using limits. This used to leave me baffled—questions like “Where does this come from?” and “What justifies this?” plagued my reading. I read painstakingly slow, even attempting to force myself to accept and understand these imprecise definitions and derivations, but I just couldn’t. At that time, I even began to question my intelligence, feeling as though I must be too slow to comprehend what others grasped with ease.
That painful confusion lasted until I saw all the related proofs laid out in Baby Rudin. It was then that the fog lifted entirely.
After this experience, I realized that I can only walk the path of formal sciences, such as pure mathematics and computer science. It’s not that I’m unwilling to learn other subjects — rather, the learning cost for me is too high. My formal science mindset simply does not support my ability to easily accept and understand the common textbooks used in natural science.
Fortunately, after browsing many forums and reading numerous books, I found a few classical mechanics textbooks that emphasize rigor:
- Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions – Darryl D. Holm, Tanya Schmah, and Cristina Stoica
- Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (2nd Edition) – Jerrold E. Marsden and Tudor S. Ratiu
- Mathematical Methods of Classical Mechanics – V. I. Arnold
- Mechanics: Volume 1 – L. D. Landau and E. M. Lifshitz
Among these, I believe the first two are the most suitable for those seeking rigor. The level of formalization in these books, to me, is more than adequate. They are written in a style reminiscent of Bourbaki. The third and fourth books are excellent supplementary materials (though not as rigidly formal, they are still exceptionally good). I highly recommend reading these books together for a more comprehensive understanding of classical mechanics.
However, be aware that these books require a solid foundation in mathematics, including calculus, differential equations, differential geometry, tensor analysis, abstract algebra, and variational calculus. If your mathematical analysis foundation isn’t strong enough, I recommend working through Rudin’s three-part series (Baby Rudin, Father Rudin, Grandpa Rudin) to build a solid groundwork.
Additionally, if you’re interested in the history of classical mechanics, you might find A Brief History of Analytical Mechanics by Fengxiang Mei, Huibin Wu, and Yanmin Li (《分析力学史略》) to be a fascinating read. I’ve recently started reading this book myself.
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