Page:Zur Dynamik bewegter Systeme.djvu/8

 $$\left(1-\frac{q^{2}}{c^{2}}\right)^{\frac{2}{3}}:1$$. This result as well as various other related theorems are in line with the conclusions to which we are led by the study of K.,. Below (in § 15), an even simpler and more direct derivation will be given.

Second Section. Principle of least action and principle of relativity.
In the following, we consider an arbitrary body in a steady state (consisting of a given number of similar or different types of molecules), determined by the independent variables V, T and the velocity components $$\dot{x},\dot{y},\dot{z}$$ of the body along the three axes x, y, z of a linear orthogonal reference frame at rest. The magnitude of the velocity q is then given by:

If the state of the body is changed in a reversible manner, then according to H. the differential equations derived from the principle of least action are given:

and

There, H is the kinetic potential of the body as a function of the above-mentioned five independent variables, where the velocity components $$\dot{x},\dot{y},\dot{z}$$ only occur in combination with q, and $$\mathfrak{F}$$ is the external moving force acting on the body.

We can use these five differential equations in the definition of the kinetic potential as well; but as we see, the function H is still not completely defined by them,