Page:Zur Dynamik bewegter Systeme.djvu/16

Now, by (13) and (14) we have:

${\frac {\partial H'}{\partial {\dot {y}}'}}={\frac {\partial H'}{\partial {\dot {y}}}}{\frac {\partial {\dot {y}}}{\partial {\dot {y}}'}}={\frac {\partial H'}{\partial {\dot {y}}}}{\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}+v{\dot {x}}'}}={\frac {\partial }{\partial {\dot {y}}}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)$

(23)

and:

${\frac {dt'}{dt}}={\frac {c^{2}-v{\dot {x}}}{c{\sqrt {c^{2}-v^{2}}}}}$.

It follows:

$d{\frac {\partial }{\partial {\dot {y}}}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)=d{\frac {\partial H}{\partial {\dot {y}}}}$

and by integration:

${\frac {\partial }{\partial {\dot {y}}}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)={\frac {\partial H}{\partial {\dot {y}}}}$, ebenso: ${\frac {\partial }{\partial {\dot {z}}}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)={\frac {\partial H}{\partial {\dot {z}}}}$.

(24)

The constant of integration, an absolute constant, vanishes because only q' = q H' goes over into H.

§ 9.

Now, the four equations (19) and (24) give by integration:

$H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}=H+const$.

The constant does not depend on V, T, ${\dot {y}},{\dot {z}}$; but it can still depend on ${\dot {x}}$, or by (14), ${\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}$. We therefore write:

$H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}=H+f\left({\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}\right)$

and determine the most general expression of the function f.

At first, we have:

${\frac {H'}{\sqrt {c^{2}-q'^{2}}}}-{\frac {H}{\sqrt {c^{2}-q{}^{2}}}}={\frac {1}{\sqrt {c^{2}-q^{2}}}}\cdot f\left({\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}\right)$.

(25)

Since the function H only depends on q, V and T, and since V' and T' are only connected to V and T by the relations (17), then the right-hand side of the equation as well as the left-hand side, are of the form:^[1]

${\frac {1}{\sqrt {c^{2}-q^{2}}}}\cdot f\left({\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}\right)=Q'-Q$,

1. This can be seen in the most simple way, when we take an arbitrary value q" and sum up the three expressions${\frac {H'}{\sqrt {c^{2}-q'^{2}}}}-{\frac {H}{\sqrt {c^{2}-q{}^{2}}}},{\frac {H''}{\sqrt {c^{2}-q''^{2}}}}-{\frac {H'}{\sqrt {c^{2}-q'^{2}}}}$ and ${\frac {H}{\sqrt {c^{2}-q^{2}}}}-{\frac {H''}{\sqrt {c^{2}-q''{}^{2}}}}$.