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> 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 me

> 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%2002.07.08%20(1-3).pdf https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises...

ironmagma 7y ago
I never said it was advanced mechanics, what I said was it's not an introductory book. You shouldn't give this to someone who's never taken a physics course. It's written for use in vehicle and aeronautical/astronautical dynamics and supposes a fairly developed understanding of coordinate systems and vector math. > Perhaps we have different definitions of "straightforward" It is certainly more straightforward than Lagrange. You just set up the constraints, which is done in two steps: (1) set up the kinematics, i.e. setting the position equal to some parameterization of that curve (2) set up the dynamics, i.e. express Newton's laws in terms of the same variables. After that, you can perform any derivatives needed using the transport theorem and set the kinematics and dynamics equal to each other using the common variables. > Coordinate invariance in physics means Since I'm not invoking the phrase "coordinate invariant" here I don't know what the deal is. All I said was that the concepts in this book express various physical laws including friction, Hooke's, and gravitational without use of coordinates. There is no X, Y, Z or 1, 2, 3 of the vectors in these descriptions. Whether to you that means coordinate invariant, is another story but ultimately irrelevant. > I still stand by the claim that this book is a pretty elementary one Considering you've spent all this time arguing that it's not straightforward (it is), I really suggest you take a look. It's nonconventional and may change your perspective if you really give it a chance.
- b215826 7y ago
  > 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.

ironmagma 7y ago

I never said it was advanced mechanics, what I said was it's not an introductory book. You shouldn't give this to someone who's never taken a physics course. It's written for use in vehicle and aeronautical/astronautical dynamics and supposes a fairly developed understanding of coordinate systems and vector math. > Perhaps we have different definitions of "straightforward" It is certainly more straightforward than Lagrange. You just set up the constraints, which is done in two steps: (1) set up the kinematics, i.e. setting the position equal to some parameterization of that curve (2) set up the dynamics, i.e. express Newton's laws in terms of the same variables. After that, you can perform any derivatives needed using the transport theorem and set the kinematics and dynamics equal to each other using the common variables. > Coordinate invariance in physics means Since I'm not invoking the phrase "coordinate invariant" here I don't know what the deal is. All I said was that the concepts in this book express various physical laws including friction, Hooke's, and gravitational without use of coordinates. There is no X, Y, Z or 1, 2, 3 of the vectors in these descriptions. Whether to you that means coordinate invariant, is another story but ultimately irrelevant. > I still stand by the claim that this book is a pretty elementary one Considering you've spent all this time arguing that it's not straightforward (it is), I really suggest you take a look. It's nonconventional and may change your perspective if you really give it a chance.

b215826 7y ago
> 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.