Introductory material for jets and jet bundles

Question

A student of mine would like to learn more about jets and jet bundles, and more in general about how to treat derivatives and differential equations in an invariant way. She's also interested in the reformulation of some of the basic concepts of differential geometry from that point of view.

Are there any books, papers, or notes that follow such an approach, and which are suitable for a first approach to the subject (there is motivation, etc)? She knows more than basic differential geometry, just not jet bundles yet.

By the way, she studies physics, so if there are examples taken from physics, even better.

olver's book on symmetries of differential equations – Jürgen Sukumaran Apr 15 '20 at 01:54 — Jürgen Sukumaran, Apr 15 '20 at 01:54

score 7 · Accepted Answer · answered Apr 16 '20 at 18:56

The Geometry of Jet Bundles by D. J. Saunders, Cambridge University Press:

“The purpose of this book is to provide an introduction to the theory of jet bundles for mathematicians and physicists who wish to study differential equations, particularly those associated with the calculus of variations, in a modern geometric way. One of the themes of the book is that first-order jets may be considered as the natural generalisation of vector fields for studying variational problems in field theory, and so many of the constructions are introduced in the context of first- or second-order jets, before being described in their full generality.”

After checking the table of contents, I think the book “Symmetries and Conservation Laws for Differential Equations of Mathematical Physics” mentioned below is more digestible for a physics student, as its title too implies. The book by Saunders might be a harder read. So in my opinion it’s better if your student reads both of the books simultaneously or if she first reads the more physics-oriented one — , Apr 18 '20 at 18:04

score 5 · Answer 2 · answered Apr 15 '20 at 08:39

Volume 182 of the Translations of Mathematical Monographs series of the AMS (1999):

Symmetries and Conservation Laws for Differential Equations of Mathematical Physics

by A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor'kova, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, A. M. Vinogradov (editor).

From the self-description:

The book is rendered as self-contained as possible and includes detailed motivations, extensive examples, and exercises, along with careful proofs of all results. The book is intended for readers who wish to learn the basics on applications of symmetry methods to differential equations of mathematical physics, but will also be useful for the experts because it assembles a lot of results previously dispersed in numerous journal publications. The text is accessible to advanced graduate students in mathematics, applied mathematics, mathematical and theoretic physics, mechanics, etc.

Introductory material for jets and jet bundles

2 Answers2