Geometry, Symmetries, and Classical Physics

A Mosaic

Manousos Markoutsakis

• Publisher: CRC Press
• Year: 2021
• eBook ISBN: 9781000530261
• Edition: 1
• Topic: Biological Sciences
• Subtopic: Geometry
• Index: Biological Sciences

This book provides advanced undergraduate physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Readers, working through the book, will obtain a thorough understanding of symmetry principles and their application in mechanics, field theory, and general relativity, and in addition acquire the necessary calculational skills to tackle more sophisticated questions in theoretical physics.

Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.

Key features:

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• Contains a modern, streamlined presentation of classical topics, which are normally taught separately

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• Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity

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• Focuses on the clear presentation of the mathematical notions and calculational technique

IGeometric Manifolds

1Manifolds and Tensors

DOI: 10.1201/9781003087748-1

1. 1.1 Differentiation in Several Dimensions
2. 1.2 Differentiable Manifolds
3. 1.3 Tangent Structure, Vectors and Covectors
4. 1.4 Vector Fields and the Commutator
5. 1.5 Tensor Fields on Manifolds

In this chapter we introduce the foundational notions of differentiable manifolds, vectors at a point, vector fields, and tensor fields. We begin with a summary of basic facts about differentiation in ℝ^D, since we aim to transfer the known calculational concepts to the case of differentiable manifolds. We provide the general definition of a differentiable manifold, introduce coordinates, and discuss diffeomorphisms. The directional derivative leads us to the algebraic definition of a vector as an element of the tangent space at a point. In the next step, we move from vectors at a point to vector fields defined over the entire manifold. Finally, we generalize to the multilinear structure and introduce general tensor fields on manifolds.

1.1 Differentiation in Several Dimensions

Euclidean Space ${\mathbb{E}}^D$

One of the cornerstones of classical physics is the use of the continuum of real numbers ℝ, or, for higher dimensions, the vector space ℝ^D with integer dimensionality $D=1,2,3,\dots$. The elements of ℝ^D are represented as column vectors

$$\begin{array}{l}x=\left(\begin{array}{c}{x}^1\hfill \\ ⋮\hfill \\ x^D\hfill \end{array}\right),\hfill \end{array}$$(1.1)

or, in abbreviated form, by the component notation x^k. The canonical basis vectors are given by $e_k={(0,\dots ,1,\dots ,0)}^T$, where the entry 1 is at the kth row. The vector space ℝ^D represents also a raw model for space in classical physics, provided we endow it with an additional structure. This structure is a metric, or equivalently, a scalar product, which for any two vectors x, y of ℝ^D...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• Part I: Geometric Manifolds
• Part II: Mechanics and Symmetry
• Part III: Symmetry Groups and Algebras
• Part IV: Classical Fields
• Part V: Riemannian Geometry
• Part VI: General Relativity and Symmetry
• Part VII: Appendices
• Bibliography
• Index