Page:Zur Dynamik bewegter Systeme.djvu/29

where, by (49):

$K=-Mc^{2}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}+const$

(51)

The full compatibility of these relations with the formulas of the previous section is most evident, when in equations (6) and (7) of the principle of least action the independent variables V and T are replaced by p and S. Thus they are:

${\frac {d}{dt}}\left({\frac {\partial K}{\partial {\dot {x}}}}\right)_{p,S}={\mathfrak {F}}_{x}$, u.s.w.

(52)

$\left({\frac {\partial K}{\partial p}}\right)_{S}=-V,\ \left({\frac {\partial K}{\partial S}}\right)_{p}=-T$,

(53)

where

$K=H-pV-TS.$

(54)

That these relationships, in fact, are quite equivalent with (6) and (7), can be directly and most easily seen by substituting the value (54) of K in equations (52) and (53), and the differentiation of H by the independent variables p and S is replaced by the differentiation of the independent variables V and T.

When we consider that by (10) and (30):

$K=q{\frac {\partial H}{\partial q}}-pV-E=qG-R$,

it follows by substitution in (46):

$K=-{\frac {\sqrt {c^{2}-q^{2}}}{c}}R'_{0}$.

In order to compare this relation with that previously derived by me (51), we must restrict ourselves to adiabatic isobaric processes, because only for such (51) was derived. However, according to § 6 for an adiabatic-isobaric process $V'={\frac {V}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$ is constant, as well as $T'={\frac {T}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$ is constant, so R'₀ is independent of q. Therefore we write R₀ instead of R'₀, and then we obtain by (48):

$K=-{\frac {\sqrt {c^{2}-q^{2}}}{c}}R_{0}=-Mc{\sqrt {c^{2}-q^{2}}}$

in full accordance with (51).