Question

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?

Answer 1

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:

• Kosmann-Schwarzbach - "The Noether Theorems"

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"

Answer 2

Javier already gave some very good references. Let me just add one more if you are thinking about classical field theories: Demetrios Christodoulou, Action Principle and Partial Differential Equations.

Answer 3

Another excellent book that does a proper job with Noether's Theorem is Peter Olver's Applications of Lie Groups to Differential Equations.

Answer 4

A fairly modern approach which is usually attributed to Vinogradov (see also the last part of Kosmann-Schwarzbachs "Noether Theorems") can be found in the book Symmetries and Conservation Laws for Differential Equations in Mathematical Physics. The chapter on conservation laws and the Noether theorem is somewhat dense and requires a little familiarity with homological algebra and spectral sequences. So it might be good to complement it with another book (like Olvers).

Answer 5

The simplest, most elegant and strongest version I know is, by far, the one in Aderson's book (see page 106 and ss.) He deals directly with the variational equation, with no explicit mention to the lagrangian.

(By the way, it is surprising why this statement is not mentioned in Schwarzbachs' book!)