Page:Zur Dynamik bewegter Systeme.djvu/21

 Now we start from relation (26) and set therein q' = 0. Then it follows with respect to (17) in the recently introduced term:

and thus H is represented as a function of the three variables q, V, and T, if H0 as a function of two variables V and T is known. By H all other physical state variables are determined according to (6) and (7). We at first obtain for the pressure:

If the pressure of the body at rest is known by the usual state equation as a function of volume and temperature, the state equation of the moving body follows immediately. Similarly, the entropy is given by:

Furthermore, the components of the momentum are given by:

$$\mathfrak{G}_{x}=G\frac{\dot{x}}{q},\quad \mathfrak{G}_{y}=G\frac{\dot{y}}{q},\quad \mathfrak{G}_{z}=G\frac{\dot{z}}{q}$$,

where G, the resulting momentum, is according to (38):

$$G=\frac{\partial H}{\partial q}=-\frac{q}{c\sqrt{c^{2}-v^{2}}}H'_{0}+\frac{\sqrt{c^{2}-q^{2}}}{c}\left\{ \left(\frac{\partial H}{\partial V}\right)_{0}^{'}\frac{cqV}{(c^{2}-q^{2})^{\frac{3}{2}}}+\left(\frac{\partial H}{\partial T}\right)_{0}^{'}\frac{cqT}{(c^{2}-q^{2})^{\frac{3}{2}}}\right\}$$

Furthermore, the energy according to (10) is given by:

Considering that $$E_0 = TS_0 H_0$$ and

$$E'_{0}=\frac{cT}{\sqrt{c^{2}-q^{2}}}S'_{0}-H'_{0}$$,

thus we can write:

Finally, the thermal function R is by (30):