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As a student of physics I have learned solving Euler equations for rigid bodies by solving examples and exercises in self-contained books rather than understanding the proofs of Euler equations (I know this is not an ideal approach to skip theorem-proofs and right away dive into solving problems).

But I like studying solved problems in books and internalize ways to solve new problems I may come across. Eg. I effectively learned how to draw free body diagrams by studying them for some solved examples.

Similarly I have understood applying Lagrange formalization to problems by studying solved problems. Whereas given a new system I can apply the Lagrange principle to find the governing equations of motions easily, I may have to look at a textbook several times to derive Lagrange principle using calculus of variation.

Recently problems in mechanics are solved by taking help of Riemannian geometric and semi-Riemannian tools and also topological abstractions. I am more interested in the application of a theory to concrete problems rather than proving a theory.

Are there books there with several solved and unsolved examples on how to apply all these new techniques to actually solve problems in real life like newton's $F=ma$ does? After searching through several books, it was found most stress only on the rigor of formalization and proof of facts rather than applications.

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  1. Schaum series Differential Geometry will solve part of your problem.
  2. Search "problem book in riemannian geometry" on google and it should bring out something useful.
  3. Also see V.I. Arnold's books.
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