Page:Zur Dynamik bewegter Systeme.djvu/9

because with certain ${\mathfrak {F}}$ and p and S, there remains in H an additive constant having no physical meaning and which can be arbitrarily determined. Further below (in § 9) we will give a proper disposal of these constant, and hence we will make the necessary complement as to the completion of H.

The momentum of the body is then given by the components:

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

(8)

or by the resulting momentum:

$G={\frac {\partial H}{\partial q}}$

(9)

and the total energy of the body by:

$E=q{\frac {\partial H}{\partial q}}+T{\frac {\partial H}{\partial T}}-H={\dot {x}}{\mathfrak {G}}_{x}+{\dot {y}}{\mathfrak {G}}_{y}+{\dot {z}}{\mathfrak {G}}_{z}+TS-H$,

(10)

whence the equation for the energy principle is given:

$dE={\mathfrak {F}}_{x}dx+{\mathfrak {F}}_{y}dy+{\mathfrak {F}}_{z}dz-pdV+TdS$,

(11)

which on its right side contains the translation work, the compression work, and the heat supplied from the outside.

All these relations are also valid, of course, for the special case of pure cavity radiation as discussed in the previous section, as one can easily convince himself if one substitutes in the above equations the value for the kinetic potential:

$H={\frac {ac^{4}T^{4}V}{3(c^{2}-q^{2})^{2}}}$

(12)

So far, in the application to ponderable bodies it was always proceeded (also by H. von Helmholtz) in such a way that the kinetic potential H was split into two parts:

$H={\frac {1}{2}}Mq^{2}-F$,

and it was assumed that the mass of the body M is constant, while the free energy of the body F was assumed to be independent of q. Then equations (6) goes over into the equations of ordinary mechanics, and equations (7) into those of ordinary thermodynamics.

However, as shown by the example of cavity radiation, which has been elaborated above in the introduction, such a decomposition, strictly speaking, cannot be allowed in any single case: for every ponderable body contains in its interior radiant energy