Cambridge University Press make their books free until May

Yes, but it is making them "free to access online during the coronavirus outbreak". Don't look a gift horse in the mouth.

> I find this very doubtful.

I'm sorry to say this, but I don't think you understand the meaning of coordinate invariance. Coordinate invariance in physics means that if you go from one set of coordinates, say (q1, q2, ..., qn) to (s1, s2, ..., sn), the form of the equations remain invariant (assuming the transformation is nice and smooth). Newton's laws aren't invariant under a general coordinate transformation of that sort, and this is precisely because the acceleration involves the second derivatives of the basis vectors. E.g., Newton's laws when expressed in polar coordinates would involve centrifugal and Coriolis terms, which are absent in Cartesian coordinates. Equations in analytical mechanics (e.g., Lagrange's or Hamilton's equations) are coordinate invariant however [1].

> The method described in this book makes this kind of problem very straightforward.

Perhaps we have different definitions of "straightforward", but most physicists I know would not consider using Newton's laws in Cartesian coordinates to find the equations of a particle constrained to move on a smooth surface straightforward. At the very least, one should try to introduce a local coordinate system on the surface. It can be done, no doubt, but it's way more cumbersome than writing Lagrange's equations involving the generalized coordinates on that surface.

> Note that both F and g have direction and magnitude in this word problem even though there are no basis vectors to speak of and thus no way we can represent any of this numerically without defining more mathematical objects.

No one is claiming that Newton's laws are invalid in different coordinates! Of course they are valid and F = ma still holds true. The invariance you're talking about is the general invariance of equations involving (polar) vectors. That's obvious since vectors are inherently geometrical objects. But that is in no way the same thing as "coordinate invariance". I'm not sure how the book defines "coordinate free", and I certainly don't have a bone to pick with anyone. But perhaps also look at how physicists define coordinate invariance, since after all it's something they've been doing for a while?

I still stand by the claim that this book is a pretty elementary one. Sure, you might've learned new things from this book, and I can see how many people would benefit from it. But if you think this is what counts for "advanced mechanics", then you're very very mistaken.

[1]: https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises...

> Considering you've spent all this time arguing that it's not straightforward (it is), I really suggest you take a look.

I just downloaded a copy of the book from LibGen. I change my original assessment that it might be a good introductory book. In fact, now I think that this is a terrible book that reinvents the wheel in so many places, and novices should avoid it since it teaches bad practices. E.g., look at Example 3-2 of the book where the task is to find the motion of the particle constrained on a parabola y = r^2/2R (similar to the paraboloid example I asked, but more easier since the constraint manifold is one-dimensional). The solution of that problem (Eq. 3-105) is 4 pages of algebra! The Lagrangian for the system expressed in terms of r is

  L = m/2*(1 + r^2/R^2)*v^2 - (mg/2R)*r^2,

v being dr/dt. Now, to find Eq. 3-105, which the author derives in 4 pages, all it takes is to plug this Lagrangian into the Euler-Lagrange equations, and answer pops out in 2-3 lines of algebra involving some very trivial partial derivatives. Curiously, the author has also reinvented d'Alembert's principle when he does this problem the second time in Example 3-9. I'm also surprised that the author hasn't mentioned d'Alembert's principle (or virtual work for that matter) -- something that engineers make extensive use of -- anywhere in this book.

> Since I'm not invoking the phrase "coordinate invariant" here I don't know what the deal is.

You did mention that this book introduces mechanics in a coordinate-free manner (which this book actually doesn't).

> It's nonconventional and may change your perspective if you really give it a chance.

It's not just unconventional, this book is filled with terrible examples and techniques to solve problems and the author has reinvented the wheel in several places. The reason you found this book challenging was because this book chooses to do problems using the most contrived methods possible. In fact, this is a book to show why one needs analytical mechanics.

There's always libgen if the book you want is behind a paywall. Like it's cousin scihub, it is very easy to use, fast, and has most of the books you could possibly need. Considering the awful practices of most scientific publishers, I have no problems with illegally downloading textbooks.

> It seems like you've not tried reading actual physics books written by real physicists

I find it pretty baffling you think this is the inescapable conclusion. Of course I've taken physics courses, that's where the frustration comes from. Many physics and astronomy books are riddled with errors and that's almost entirely down to how frequently they skip steps. Not only are 90% of underclassman physics texts written lazily at best, when I got into Modern Physics, half the answers in the back of the book were flat-out wrong, while the example problems elided a third of the steps or neglected important edge cases. The book was Tipler / Llewellyn by the way.

> You don't need this book

Apparently I did, considering that none of the other relevant courses I took which included calcs 1 through 3, differential equations, linear algebra, mechanics of materials taught this material. I'm sure it would've come up again if I'd stuck with mechanical engineering but regardless the whole point of recommending the book is to say "this has value" not to say "other things do not have this value."

> haphazard approaches

Compared to the other books I've seen, this is one of the most verbose, consistent, and generally anal-retentive approaches. It does not take shortcuts, and its notation alone is refreshing considering its lack of ambiguous styling aside from minor issue of the bold being used for tensors and vectors. (On the whiteboard, this is resolved with a double-underline being used for tensors and a single-underline for vectors). I would consider most other physics textbooks to be the haphazard ones, and when you compare the rate of errors in the text I have a strong suspicion those other books have much higher rates on average.

Granted, I'm not a physicist and make no claims to be. Of course there is truth out there to be found among the vast sea of physics textbooks, but from my own experience and also that of someone I know well who is postgrad in plasma physics, the textbooks are generally shit and to gain a correct understanding you need to wade through multiple texts that are extremely fragmented in both notation and correctness. So I'm not sure physics is the gold standard here. I'm sure there's a good math textbook out there about coordinate systems, but like I said, I didn't post here to say "everything else is worthless."