What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?
[Math] reference for Noether’s theorem
classical-mechanicsconservation-lawsdg.differential-geometryreference-request
Related Solutions
The theory of currents is a part of the geometric measure theory. Unfortunately, Federer made the subject completely inaccessible after he wrote his famous monograph:
H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.
The problem is that the book contains `everything' (well, almost) and it is unreadable. After this book was published, people did not dare to write other books on the topic and only the bravest hearts dared to read Federer's Bible.
In my opinion the first accessible book on the subject is
L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.
You can find it as a pdf file in the internet. Note that this book was written 14 years after Federer's book and there was nothing in between.
I would also suggest:
F. Lin, X. Yang, Geometric measure theory—an introduction. Advanced Mathematics (Beijing/Boston), 1. Science Press Beijing, Beijing; International Press, Boston, MA, 2002.
I haven't read it, but it looks relatively elementary (relatively, because by no means the subject is elementary).
The last, but not least is
F. Morgan, Geometric measure theory. A beginner's guide. Fifth edition. Illustrated by James F. Bredt. Elsevier/Academic Press, Amsterdam, 2016.
You will not learn anything form that book as it does not have detailed proofs, but you can read it rather quickly and after that you will have an idea about what it is all about.
In addition to the variational approach based on Noether's theorem, there are other ways to find conservation laws for nonlinear PDE's:
- The symmetry/adjoint symmetry pair method extracts a conservation law from a bilinear skew-symmetric identity. It involves the following steps: (a) Linearize the given system of PDEs; (b) Find the adjoint system of the linearized system; (c) Find solutions of the linearized system, i.e., local symmetries in characteristic (evolutionary) form of the given PDE system; (d) Find solutions of the adjoint system; (e) For any pair, consisting of a solution for the adjoint system and a local symmetry, construct a conservation law.
- The multiplier approach (also known as the variational derivative method), generalizes Noether's approach so that it can be applied whether or not the linearized system is self-adjoint (no Lagrangian formulation is needed).
- Scale invariance can be used to obtain a conservation law from a simple algebraic formula.
- The GeM software package searches for conservation laws of ordinary and partial differential equations without human intervention.
So yes, there are alternatives to just "mucking around until you see something".
Best Answer
There are not too many books that do a proper job regarding Noether's theorem. Some books which are standard references for a differential geometric treatment of theoretical (classical) mechanics, and which deal with it in that language are:
Marsden / Abraham - "Foundations of Mechanics"
Marsden - "Introduction to Mechanics and Symmetry"
Arnold - "Mathematical Methods of Classical Mechanics"
José / Saletán - "Classical Dynamics, A Contemporary Approach"
Dubrovin / Fomenko / Novikov - "Modern Geometry. Part I: Geometry of Surfaces, Transformation Groups and Fields" (as recommended in a previoius comment by Giuseppe)
de León / Rodrigues - "Methods of Differential Geometry in Analytical Mechanics"
As a theoretical physicist who wanted to study these things in the most mathematical way possible, I found those books extremely helpful to bridge the gap between the two settings. For the history of such an important result, this recent book is very interesting:
You may also be interested in the style of mathematical mechanics articles and books developed by:
Sardanashvily - Noether conservation laws in Classical Mechanics"
Sardanashvily - "Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians"
Mangiarotti / Sardanashvili - "Gauge Mechanics"
Mangiarotti / Sardanashvili - "Connections in Classical and Quantum Field Theory"
Giachetta / Mangiarotti / Sardanashvili - "Advanced Classical Field Theory"
Giachetta / Mangiarotti / Sardanashvili - "Geometric Formulation of Classical and Quantum Mechanics"
Giachetta / Mangiarotti / Sardanashvili - "New Lagrangian and Hamiltonian methods in field theory"