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

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

The equations of motion of a material particle are

$$(dx_\nu)$$ 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 $$\left(\frac{d^2x_\nu}{dt^2}\right)$$ 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), $$\left(m\frac{d^2x_\nu}{dt^2}\right)$$ 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 $$\left(m\frac{d^2x_\nu}{dt^2} - X_\nu\right)$$. 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.