Yes I think that's what Lanczos might be demonstrating the need to do, following his discussion after 16.7. It is cumbersome, and I don't know of the importance of the need to know holonomic systems to this degree to try further to understand it
Lanczos procedure is that if our differential is integrable then 16.7 must be an identity
Both you and I arrive at the equivalent of (using your working just before you rearrange):
-xy-z+y2=-x2
This isn't an identity unless we assume the "answer", the equation that Lanczos has given as the...
Thanks for that solution and confirms what I did above as well. As I say though I expected to come up with the LHS and RHS of 16.7 being an identity, as opposed to finding a solution for z that makes it so. I guess Lanczos is demonstrating his point of the procedure where he says "However it may...
Thanks guys.
I had used Philip Wood's procedure to get the result that 16.7 becomes 1=(-xy+y2-z)/x2. The next step seems to be to solve for z, whereas I was expecting just identity at this stage. Validity of that seems questionable to me...
Anyway it seems that Lanczos is not pedagogical and...
I have started going through Lanczos' book "The Variational Principles of Mechanics", and have got stuck on the first problem he gives at 16.9 to do with holonomic constraints
Investigate the integrability of the following differential relation:
x dz + (y2 - x2 - z) dx + (z - y2 - xy) dy = 0...
Only because that's how distance is defined in cartesian coords. But see below...
But I just find that sort of transformation unconvincing because it is just simple translations within the same coord system, and so too trivial.
Now I'm thinking that the circle example is an unfortunate...
OK, if this is all true and given my lack of grounding in diff geom, I give up and need to mull over it all and do more reading. I think it's worth it if a deeper understanding of GR and things like gauge invariance are of consequence here.
But if you care to answer them here are some...
I didn't say distance function, I said distance as in a distance of 10 metres.
The hole argument _is_ Einstein's argument. You are defending Einstein's hole argument but then saying Einstein was wrong...? Of course the way you solve the hole argument, by always transforming rather than...
No, here is a link to the hole argument as described by Norton:
http://www.pitt.edu/~jdnorton/papers/decades.pdf
See p801 and 802. The transliteration is the last step, g'(x) is supposed to solve the same equation. There's no _transformation_, just _transliteration_. You've got to stop there...
Maybe you meant to quote me as "you forgot to add the physics", if so yes I think it is analogous. The difference being I guess Hurkyl makes the transformation a positive legitimate step to defend the hole argument, a step the hole argument doesn't actually take.
I agree with the second part...
No that is what I had in mind for form. However I am curious as to why you chose a distance formula as an example of something not retaining form...that would seem to be confusing the issue. I'm wondering if that's what you think I was saying, that the formula for distance was generally...
No I don't agree here. They may be solutions to the same empty mathematical form of an equation, but not to some concrete physical field equation. It's trivial that g'(x) is going to solve an equation of the same form as the EFE, but it's a bit hopeful that replacing x' by x using a simple...
Yes I think you and I are in agreement on the sore spot nevertheless. To me it's not a trivial relabelling because it is specifically using the old coord system as the relabel, not just any label. This leads to a bending and breaking of the physics.
Well for me I'm much more satisfied that...