Could anyone indicate me a good and simple material that deals with methods for variational calculus in curved spaces? (It does not have to be, necessarily, basic at the level of a General Relativity course)
You may like this book https://www.degruyter.com/view/product/278657
I have also tried to make a good introduction in chapter 3.9 in the book
http://as.wiley.com/WileyCDA/WileyTitle/productCd-3527408568,subjectCd-PH00.html
!) Researchers in math and theor. physics begin to use the acheivements in pure math that concern their field with a good delay. V.Arnol'd estimated this diffusion of the mathematical knowledge to physics in 50 years. For instance, even now the great magority of physisists think that symmetries and conservation laws are the same things and the only mechanism for finding coservation laws is the Noether theorem, etc.
2) Variational caculus is a natural math construction on smooth manifolds, which is independent on any curvature matters and, therefore, on General Relativity. From the modern point of view it is a part of the theory of C-spectral sequences on infinite jet spaces. This theory embraces both symmetries and conservation laws of (nonlinear) PDEs and many other related or not topics.
3) The concept of "simple" depends on the "observer", as in General Relativity. The C-spectral sequence methods are not difficult for those who is familiar with elementary homological algebra and intorductory infinite jet matters. The recommended book is
"Symmetries and conservation laws for differential equations of mathematical physics",AMS Translation of Mathematical Monographs series, vol 182
The delay is simply enormous. I was working on the theory of cosmological perturbations and discovered that everything what cosmologists where doing until now was rather primitive from mathematical point of view.
Thanks. Indeed, the delay in relation to the advancement of the mathematical arsenal is huge. But even in well-established branches of theoretical physics we can find conserved quantities that are not related to Noether charges. For example, defect structures in field theory are characterized by a topological charge, as well as skyrmions and vortices.
These can be related to the extended concept of the conserved Noether current. A well known example is the integral of the center of mass in N body system. We discussed it in the first book, I have pointed out above, in detail. Effectively, the Noether theorem is more sophisticated than people are used to think.
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