International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 9

Section 1.1.3.6.3. Examples of the use in physics of different representations of the same quantity

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.6.3. Examples of the use in physics of different representations of the same quantity

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Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it:

  • (i) by the fundamental law of dynamics: [{\bf F} = m {\boldGamma},\quad \hbox{with }F^{i}= m \,\,\hbox{d}^{2}x^{i}/ {\rm d}t^{2},]where m is the mass and [\boldGamma] is the acceleration. The force appears here in a contravariant form.

  • (ii) as the derivative of the energy, W: [F_{i}= \partial W/\partial x^{i}= \partial _{i}W.]

    The force appears here in covariant form. In effect, we shall see in Section 1.1.3.8.1[link] that to form a derivative with respect to a variable contravariant augments the covariance by unity. The general expression of the law of dynamics is therefore written with the energy as follows: [m \,\,{\rm d}^{2}x^{i}/ {\rm d}t^2 = g^{ij}\partial _{j}W.]








































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