Supplement to Classical Mechanics by Goldstein

[shinobi20]

Are there any lecture notes that closely follow Classical Mechanics by Goldstein? I am asking this since I am seeing some comments in this forum that it contains some conceptual errors, e.g. nonholonomic constraints. If there is a book that "closely" follows Goldstein, it will be good too.

 

[hutchphd]

https://ocw.mit.edu/courses/physics...i-fall-2014/lecture-notes/MIT8_09F14_full.pdf

 

[wrobel]

both texts are not for mathematicians

 

[shinobi20]

both texts are not for mathematicians

What do you mean by both text are not for mathematicians?

 

[wrobel]

What do you mean by both text are not for mathematicians?

It is long to explain. I mean that both texts are below the mathematical standards of rigor

 

[vanhees71]

For mathematicians, maybe

Arnold, Mathematical methods of classical mechanics, Springer

is a good choice. For a physicist it's also a good read after he or she is familiar with the physics.

 

[George Jones]

both texts are not for mathematicians

It is long to explain. I mean that both texts are below the mathematical standards of rigor

These two quotes mean different things.

I know a counter-example to the first quote.

I know someone who has a had long career as a university professor teaching and doing research in pure mathematics, and who likes to read theoretical physics books. For example, he loves "Introduction to Particle Physics" by Griffiths and "The Quantum Theory of Fields" by Weinberg. When I told him about mathematically more rigourous books, like "Quantum Theory for Mathematicians" by Hall or the forthcoming quantum field theory book by Talagrand on quantum field theory, he told me that he prefers standard physics books.

So, physics books are for him, yet he would be in complete agreement with the second quote.

 

[wrobel]

I just bring only one example from the text cited in #2.

page 23:

The author suggests to consider variations of this functional in the class of curves with fixed ends in the phase space. It is pointless and it does not correspond to the well-known $\int Ldt$.
Indeed, the Hamiltonian equations are of the first order and if you fix a point in the phase space you also fix a trajectory that passes through this point (existence and uniqueness theorem) . This trajectory is not obliged to pass through the second point you fixed. Moreover if you fix two arbitrary points in the phase space, the probability that these points are appeared to be connected by a solution to the Hamiltonian system, will be equal to zero. It is so even if the points are close to each other.
Thus the functional $J$ (according to this text) does not have stationary paths almost certainly. In the correct statement (see for example cited above Arnold's book) one fixes only $q$ positions at the ends of the path and does not fix impulses $p$. This corresponds to the Lagrangian situation ($\int Ldt$) when we also fix $q$ positions and do not fix velocities $\dot q$.

 

[vanhees71]

Well, that's a severe mistake also for physicists' standards of rigor. It's simply a wrong statement, which I can't remember having encountered in the literature before.

Of course, one really understands why the boundary conditions in phase space are what they are when considering the derivation from the path integral of quantum theory (which is not mathematically rigorous either though).

 

[shinobi20]

If I may ask, what is the relevance of the previous posts to the OP?

 

[Mondayman]

I found these two textbooks.

https://www.amazon.com/dp/0198508026/?tag=pfamazon01-20

https://www.amazon.com/dp/0191001627/?tag=pfamazon01-20

 

[CJ2116]

My favorite is Lemos' Analytical Mechanics - there are a large number of worked examples, clear exposition and a lot of end of chapter problems. There are a few appendices explaining differential forms and why they're a natural tool to describe constraints.

I've honestly spent a fair bit of time trying to learn from Goldstein, but for some reason was never really able to learn much from it - this is by far the best alternative I have found.

 

[shinobi20]

I found these two textbooks.

https://www.amazon.com/dp/0198508026/?tag=pfamazon01-20

https://www.amazon.com/dp/0191001627/?tag=pfamazon01-20

How would you rate Oliver Johns' book?

My favorite is Lemos' Analytical Mechanics - there are a large number of worked examples, clear exposition and a lot of end of chapter problems. There are a few appendices explaining differential forms and why they're a natural tool to describe constraints.

I've honestly spent a fair bit of time trying to learn from Goldstein, but for some reason was never really able to learn much from it - this is by far the best alternative I have found.

So how do you compare Lemos' book with Goldstein's (Is it really that bad)? I've never heard of Lemos before but by looking it seems to follow the standard topics, and it looks promising.

 

[vanhees71]

Well, Goldstein is a standard reference, but it has this really serious flaw concerning the Hamilton principle in connection with non-holonomic constraints. He seems not to have realized that he gets equations that differ from the equations he got from the d'Alembert principle though of course both principles are equivalent.

 

[shinobi20]

Well, Goldstein is a standard reference, but it has this really serious flaw concerning the Hamilton principle in connection with non-holonomic constraints. He seems not to have realized that he gets equations that differ from the equations he got from the d'Alembert principle though of course both principles are equivalent.

Yes, I saw you commenting in another post that he made a mistake regarding non-holonomic constraints but I do not know which part of the book you are pertaining, so now I know it is in the part of Hamilton's principle.

Do you know other flaws in Goldstein that I should watch out? Also, do you know any lecture notes that follow Goldstein's book and which writes correctly the part which Goldstein had written incorrectly?

 

[vanhees71]

No, not that I know. The only thing that's flawed is how non-holonomic constraints are treated using the action principle. A correct treatment is in Landau&Lifshitz vol. 1 (which anyway is a very good book for a 2nd read about analytical mechanics). It's in the usual LL style: Very brief, no unnecessary details and very elegant in its math to come straight away to the interesting physics. Didactically it's not too good and it can be frustrating, particularly if you use it as a first read on a new subject. Nevertheless as a reference theory-physics set of books, it's brillant. My absolute favorites of the series are vol. 2 (classical electrodynamics in the relativity-first approach, particularly the chapter on radiation reaction, including for years the now finally accepted solution of the age-old problem as far as it can be solved within a classical point-particle description), vol. 5 (statistical physics), and vol. 10 (kinetic theory). The only volume I find a bit outdated is vol. 4 (quantum electrodynamics) though it contains a lot of material not covered in other standard textbooks.

 

[weirdoguy]

Do you know other flaws in Goldstein that I should watch out?

A minor flaw, but on page 4 of latest edition he writes that "infinitesimal" work done by a friction force is always positive, which of course is not true. He writes this in context of non-conservativeness of friction, but the argument should be rather that this infinitesimal work does not change its sign, so the integral $\oint \vec{F}\cdot\textsf{d}\vec{s}$ over closed path cannot be $0$.