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Covariance and Gauge Invariance in Continuum Physics pp 9–71Cite as

Birkhäuser

Basic Concepts on Manifolds, Spacetimes, and Calculus of Variations

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Part of the book series: Progress in Mathematical Physics ((PMP,volume 73))

Abstract

In the present book, we are interested in continuum physics namely mechanics, gravitation, electromagnetism and their mutual interaction. Most of field equations governing theoretical physics are deduced from a variational principle after defining a suitable Lagrangian density \(\mathscr {L}\) and its arguments. Three steps are considered for deriving the field equations governing their evolution and mutual interaction. The first focus on the continuum geometry and by the way the spacetime with the concept of reference frame where physics happen.

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Notes

  1. 1.

    It is worth noting that covariant components of event hold as (x 0 = ct, x 1 = −x 1, x 2 = −x 2, x 3 = −x 3) in this flat Minkowskian spacetime \(\mathscr {M}\).

  2. 2.

    Given the two groups \(\mathbb {R}^{1,3}\) and \(\mathcal {S}\mathcal {O}^{+} (1,3)\) such \(\mathbb {R}^{1,3} \cap \mathcal {S}\mathcal {O}^{+} (1,3) = \left \{ \mathbb {I} \right \}\), the semi-direct product of \(\mathbb {R}^{1,3}\) and \(\mathcal {S}\mathcal {O}^{+} (1,3)\) is a group denoted and with its element:

    $$\displaystyle \begin{aligned} \mathcal{S}\mathcal{O}^{+} (1,3) \ \ltimes\ \mathbb{R}^{1,3} := \left\{ g = l \ k \in \mathcal{S}\mathcal{O}^{+} (1,3) \ \ltimes\ \mathbb{R}^{1,3}, \mathrm{where}\ { } k \in \mathcal{S}\mathcal{O}^{+} (1,3), l \in \mathbb{R}^{1,3} \right\}. \end{aligned}$$
  3. 3.

    In relativistic mechanics, dust is usually defined as a set of particles forming a perfect fluid with zero pressure and with no interaction between them. Particles move independently each other.

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  1. Institut de Recherche Mathématique, Université de Rennes, Rennes CX, France

    Lalaonirina R. Rakotomanana

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R. Rakotomanana, L. (2018). Basic Concepts on Manifolds, Spacetimes, and Calculus of Variations. In: Covariance and Gauge Invariance in Continuum Physics. Progress in Mathematical Physics, vol 73. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-91782-5_2

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