Page:Zur Dynamik bewegter Systeme.djvu/8

$\left(1-{\frac {q^{2}}{c^{2}}}\right)^{\frac {2}{3}}:1$. This result as well as various other related theorems are in line with the conclusions to which we are led by the study of K. Mosengeil,^[1]. Below (in § 15), an even simpler and more direct derivation will be given.

Second Section. Principle of least action and principle of relativity.

§ 2.

In the following, we consider an arbitrary body in a steady state (consisting of a given number^[2] of similar or different types of molecules), determined by the independent variables^[3] V, T and the velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$ of the body along the three axes x, y, z of a linear orthogonal reference frame at rest. The magnitude of the velocity q is then given by:

$q^{2}={\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}$.

If the state of the body is changed in a reversible manner, then according to H. von Helmholtz^[4] the differential equations derived from the principle of least action are given:

${\frac {d}{dt}}{\frac {\partial H}{\partial {\dot {x}}}}={\mathfrak {F}}_{x},\ {\frac {d}{dt}}{\frac {\partial H}{\partial {\dot {y}}}}={\mathfrak {F}}_{y},\ {\frac {d}{dt}}{\frac {\partial H}{\partial {\dot {z}}}}={\mathfrak {F}}_{z}$

(6)

and

${\frac {\partial H}{\partial V}}=p,\ {\frac {\partial H}{\partial T}}=S$.

(7)

There, H is the kinetic potential of the body as a function of the above-mentioned five independent variables, where the velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$ only occur in combination with q, and ${\mathfrak {F}}$ is the external moving force acting on the body.

We can use these five differential equations in the definition of the kinetic potential as well; but as we see, the function H is still not completely defined by them,

1. K. Von Mosengeil, l. c. equation (47) etc.
2. This number can also be zero. Then the body is reduced to cavity radiation, as it was discussed in the previous section.
3. On the existence of a state equation. see Byk, Ann. d. Phys. (4) 19, p. 441, 1906.
4. H. Von Helmholtz, Ges. Abh. (Leipzig, J. A. Barth) III, S. 225, 1895. There, the kinetic potential is defined by the opposite sign.