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:

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

(38)

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:

$p=p'_{0}$.

(39)

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:

$S=S'_{0}$.

(40)

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):

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

$G={\frac {q}{c^{2}-q^{2}}}Vp'_{0}+{\frac {q}{c^{2}-q^{2}}}TS'_{0}-{\frac {q}{c{\sqrt {c^{2}-q^{2}}}}}H'_{0}$.

(41)

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

$E={\frac {q^{2}}{c^{2}-q^{2}}}Vp'_{0}+{\frac {c^{2}}{c^{2}-q^{2}}}TS'_{0}-{\frac {c}{\sqrt {c^{2}-q^{2}}}}H'_{0}$.

(42)

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:

$E={\frac {c}{\sqrt {c^{2}-q^{2}}}}E'_{0}+{\frac {q^{2}}{c^{2}-q^{2}}}Vp'_{0}$.

(43)

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

$R={\frac {c^{2}}{c^{2}-q^{2}}}Vp'_{0}+{\frac {c^{2}}{c^{2}-q^{2}}}TS'_{0}-{\frac {c}{{\sqrt {c^{2}-}}q^{2}}}H'_{0}$

(44)