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.