Page:Das Prinzip der Relativität und die Grundgleichungen der Mechanik.djvu/5

These equations contain the solution of the problem, they form that generalization of Newton's equations of motion 2), which is required by the principle of relativity.

Comparing them with the Lagrangian equations of motion:

${\frac {d}{dt}}\left({\frac {\partial H}{\partial {\dot {x}}}}\right)=X,\ {\mathsf {etc}}.$

7)

where H represents the kinetic potential, we obtain:

$H=-mc^{2}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}+const$

8)

The expression for the kinetic force is obtained, if the equations 7) are respectively multiplied and summed up by ${\dot {x}}\ dt,{\dot {y}}\ dt,{\dot {z}}\ dt$. Then it follows:

$d\left({\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H\right)=X\ dx+Y\ dy+Z\ dz,$

and from this relation the expression of the kinetic force L of the mass point is developed:

$L={\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H={\frac {mc^{2}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}+const.$

The equations of motion 7) can also be represented in the form of the Hamiltonian principle:

$\int _{t_{0}}^{t_{1}}(\delta H+A)dt=0,$

whereby the time t, as well as the start and end position, remain unchanged and the virtual work is designated by A:

$A=X\delta x+Y\delta y+Z\delta z.$

At last we set up the Hamiltonian canonical equations of motion. For this task, the "momentum coordinates" ξ, η, ζ are introduced, where:

$\xi ={\frac {\partial H}{\partial {\dot {x}}}}={\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$, etc.