Page:Zur Dynamik bewegter Systeme.djvu/17

where Q depends only on q. It necessarily follows:

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

if C is an absolute constant.

This substituted into (25) gives the desired relation between H' and H:

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

Since the function H-C satisfies exactly the same differential equations (6) and (7) as the function H, we may easily imagine to set in all previous equations the function H-C instead of H, and we want from now on denote H-C simply by H. Then it is found:

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

(26)

In other words: If we set the constant C = 0, then this represents no physical limitation, but a useful supplement to the definition of the kinetic potential, which is not completely determined by the differential equations (6) and (7), as it was pointed out there already.

§ 10.

Having found the general relation between H' and H now, the relation of those values of any physical quantity of the two reference frames is directly given from the differential equations of the principle of least action. Consider first the momentum, whose components in the primed frame are:

${\mathfrak {G}}'_{x'}={\frac {\partial H'}{\partial {\dot {x}}'}},\quad {\mathfrak {G}}'_{y'}={\frac {\partial H'}{\partial {\dot {y}}'}},\quad {\mathfrak {G}}'_{z'}={\frac {\partial H'}{\partial {\dot {z}}'}}$.

(27)

While the connection of the y and z-components of momentum is directly given from the comparison with (8) and (13):

${\mathfrak {G}}'_{y'}={\mathfrak {G}}{}_{y},\quad {\mathfrak {G}}'_{z'}={\mathfrak {G}}{}_{z}$,

(28)

while the connection between the x-components ${\mathfrak {G}}'_{x'}$ and ${\mathfrak {G}}_{x}$ is of a much more complicated nature.

From (27) we obtain for this connection in an easily understandable description:

${\mathfrak {G}}'_{x'}={\frac {\partial H'}{\partial {\dot {x}}}}{\frac {\partial {\dot {x}}}{\partial {\dot {x}}'}}+{\frac {\partial H'}{\partial {\dot {y}}}}{\frac {\partial {\dot {y}}}{\partial {\dot {x}}'}}+{\frac {\partial H'}{\partial {\dot {z}}}}{\frac {\partial {\dot {z}}}{\partial {\dot {x}}'}}+{\frac {\partial H'}{\partial V}}{\frac {\partial V}{\partial {\dot {x}}'}}+{\frac {\partial H'}{\partial T}}{\frac {\partial T}{\partial {\dot {x}}'}}$,