Page:Zur Dynamik bewegter Systeme.djvu/18

This is by (26), (14) and (13):

${\frac {\partial H'}{\partial {\dot {x}}}}={\frac {\partial }{\partial {\dot {x}}}}\left(H'{\sqrt {\frac {c^{2}-q'^{2}}{c^{2}-q{}^{2}}}}\right)={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {x}}}}+{\frac {vc{\sqrt {c^{2}-v^{2}}}}{(c^{2}-v{\dot {x}})^{2}}}H$ ${\frac {\partial H'}{\partial {\dot {y}}}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {y}}}},\ {\frac {\partial H'}{\partial {\dot {z}}}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {z}}}}$ ${\frac {\partial H'}{\partial V}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial V}},\ {\frac {\partial H'}{\partial T}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial T}}$ ${\frac {\partial {\dot {x}}}{\partial {\dot {x}}'}}={\frac {(c^{2}-v{\dot {x}})^{2}}{c^{2}(c^{2}-v^{2})}},\ {\frac {\partial {\dot {y}}}{\partial {\dot {x}}'}}=-{\frac {v{\dot {y}}(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}},\ {\frac {\partial {\dot {x}}}{\partial {\dot {x}}'}}=-{\frac {v{\dot {z}}(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}$, ${\frac {\partial V}{\partial {\dot {x}}'}}=-{\frac {v(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}V,\ {\frac {\partial T}{\partial {\dot {x}}'}}=-{\frac {v(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}T$.

This is given by substitution with respect to (8) and (7):

${\mathfrak {G}}'_{x'}={\frac {1}{c{\sqrt {c^{2}-v^{2}}}}}\{(c^{2}-v{\dot {x}}){\mathfrak {G}}_{x}+vH-v{\dot {y}}{\mathfrak {G}}_{y}-v{\dot {z}}{\mathfrak {G}}_{z}-vpV-vTS\}$

or from the introduction of the energy E (10):

${\mathfrak {G}}'_{x'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {v(E+pV)}{c^{2}}}\right)$.

(29)

If we introduce instead of the energy E the "thermal function at constant pressure" R by Gibbs:

$R=E+pV,$

(30)

whose variation in isobaric processes describes the supplied heat, then the last relation is simply given by:

${\mathfrak {G}}'_{x'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {v}{c^{2}}}R\right)$.

(31)

§ 11.

By differentiating the equation (29) by time t:

${\frac {d{\mathfrak {G}}'_{x'}}{dt}}={\frac {d{\mathfrak {G}}'_{x'}}{dt'}}\cdot {\frac {dt}{dt}}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left\{{\frac {d{\mathfrak {G}}{}_{x}}{dt}}-{\frac {v}{c^{2}}}\left({\frac {dE}{dt}}+p{\frac {pV}{dt}}+V{\frac {dp}{dt}}\right)\right\}$,

the relation between the x-components of the force ${\mathfrak {F}}$ follows with consideration of (27), (20), (14) and (11), namely:

${\mathfrak {F}}'_{x'}={\mathfrak {F}}_{x}-{\frac {v}{c^{2}-v{\dot {x}}}}({\mathfrak {F}}_{y}{\dot {y}}+{\mathfrak {F}}_{z}{\dot {z}}+V{\dot {p}}+T{\dot {S}})$.

(32)

Comparing this relation with the one found above (21), it follows that those have no general meaning, but only apply if ${\dot {p}}=0$ and ${\dot {S}}=0$, that is, when the process runs isobaric and adiabatic. In fact, this property is