[omnicognate]

It's an old book and I can't vouch for it as I only just discovered it myself, but it appears to be very highly regarded, it focuses on precisely the questions you (and I) have, and just from the preface I like the author already [1]: The Variational Principles of Mechanics, by Cornelius Lanczos.

There's a PDF here: https://pages.jh.edu/rrynasi1/PhysicalPrinciples/literature/...

[1] An appetising quote:

> The author is well aware that he could have shortened his exposition considerably, had he started directly with the Lagrangian equations of motion and then proceeded to Hamilton’s theory. This procedure would have been justified had the purpose of this book been primarily to familiarize the student with a certain formalism and technique in writing down the differential equations which govern a given dynamical problem, together with certain “recipes” which he might apply in order to solve them. But this is exactly what the author did not want to do. There is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, and of Hamilton and Jacobi, which is completely smothered in a purely formalistic treatment, although it cannot fail to be a source of the greatest intellectual enjoyment to every mathematically-minded person. To give the student a chance to discover for himself the hidden beauty of these theories was one of the foremost intentions of the author.

[DarmokJalad1701]

The author of this book is the same Lanczos that the "Lanczos Resampling" algorithm (used in image resizing for example), is named after:

https://en.wikipedia.org/wiki/Lanczos_resampling

[sn9]

Here's a fun book review: https://www.thepsmiths.com/p/review-the-variational-principl...

[in_a_hole]

I had heard about this book and that quote makes me want to read it. Thank you.