Page:The Meaning of Relativity - Albert Einstein (1922).djvu/30

the more important equations of physics from this point of view.

The equations of motion of a material particle are

$m{\frac {d^{2}x_{\nu }}{dt^{2}}}=X_{\nu }$

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$(dx_{\nu })$ is a vector; $dt$, and therefore also ${\frac {1}{dt}}$, an invariant; thus $\left({\frac {dx_{\nu }}{dt}}\right)$ is a vector; in the same way it may be shown that $\left({\frac {d^{2}x_{\nu }}{dt^{2}}}\right)$ is a vector. In general, the operation of differentiation with respect to time does not alter the tensor character. Since $m$ is an invariant (tensor of rank 0), $\left(m{\frac {d^{2}x_{\nu }}{dt^{2}}}\right)$ is a vector, or tensor of rank 1 (by the theorem of the multiplication of tensors). If the force $(X_{\nu })$ has a vector character, the same holds for the difference $\left(m{\frac {d^{2}x_{\nu }}{dt^{2}}}-X_{\nu }\right)$. These equations of motion are therefore valid in every other system of Cartesian co-ordinates in the space of reference. In the case where the forces are conservative we can easily recognize the vector character of $(X_{\nu })$. For a potential energy, $\Phi$, exists, which depends only upon the mutual distances of the particles, and is therefore an invariant. The vector character of the force, $X_{\nu }=-{\frac {\delta \Phi }{\delta x_{\nu }}}$, is then a consequence of our general theorem about the derivative of a tensor of rank 0.

oriented differently. We shall meet with this point of view again in the theories of special and general relativity.