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## 2.3 Proofs of the conservation laws

### 2.3.1 Conservation of momentum

In quantino theory, the electric force of a point charge $q_s$ at the coordinate origin onto a second point charge $q_d$ at the place $\vec{r}$ with the velocity $\vec{w}$ is given by equation (2.2.1.4).The force of a point charge $q_i$ at the location $\vec{r}_i$ with the speed $\dot{\vec{r}}_i$ onto another point charge $q_j$ at the location $\vec{r}_j$ with the speed $\dot{\vec{r}}_j$ is thus

$$\vec{F}_{ij} = \vec{F}_R(\vec{r}_j-\vec{r}_i,\dot{\vec{r}}_j-\dot{\vec{r}}_i).$$ | (2.3.1.1) |

$$\vec{F}_{ji} = \vec{F}_R(\vec{r}_i-\vec{r}_j,\dot{\vec{r}}_i-\dot{\vec{r}}_j).$$ | (2.3.1.2) |

$$\vec{F}_{ji} = -\vec{F}_R(\vec{r}_j-\vec{r}_i,\dot{\vec{r}}_j-\dot{\vec{r}}_i)$$ | (2.3.1.3) |

$$\vec{F}_{ij} = -\vec{F}_{ji}.$$ | (2.3.1.4) |

$$\dot{\vec{p}}_i = \sum\limits_{j=1}^{n} \vec{F}_{ji}$$ | (2.3.1.5) |

$$\sum_{i=1}^{n} \dot{\vec{p}}_i = \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \vec{F}_{ji}.$$ | (2.3.1.6) |

$$\sum_{i=1}^{n} \dot{\vec{p}}_i = 0.$$ | (2.3.1.8) |

### 2.3.2 Conservation of angular momentum

The angular momentum $\vec{L}$ is defined as a cross product of the vector $\vec{r}$ and the momentum $\vec{p}$. For a point charge $q_d$ at the place $\vec{r}_d$ therefore equation$$\vec{L} := \vec{r}_d \times \vec{p}_d$$ | (2.3.2.1) |

$$\dot{\vec{L}} = \dot{\vec{r}}_d \times \vec{p}_d + \vec{r}_d \times \dot{\vec{p}}_d.$$ | (2.3.2.2) |

$$\dot{\vec{L}} = \vec{r}_d \times \dot{\vec{p}}_d.$$ | (2.3.2.3) |

$$\dot{\vec{L}} = \vec{r}_d \times \vec{F}_R.$$ | (2.3.2.4) |

$$\dot{\vec{L}} = 0.$$ | (2.3.2.5) |

### 2.3.3 Conservation of energy

Note: This proof is wrong. The correction can be found here.

The force given by formula (2.2.1.4) depends not only on the distance vector of the point charges $\vec{r}$ to each other, but also on the relative speed $\dot{\vec{r}}$. Furthermore, the force is a central force, but not radially symmetric and therefore not vortex-free. For these reasons, Equation (2.2.1.4) does not represent a conservative force in the original sense. Furthermore, it is not possible to determine the force by means of gradient calculation.
Nonetheless, it is possible to define a formula for the potential energy $V_R$ of a point charge $q_d$ with the inertial mass $m_d$ in the field of a point charge $q_s$. The equation is

In order to show that equation (2.3.3.1) is actually the potential energy of the charge $q_d$ in the field of the charge $q_s$, we take advantage of the fact that the conservation of angular momentum has already been proven. This makes it clear that the movement of the point charge $q_d$ always takes place in the same plane. For this reason, the task is inherently two-dimensional, which makes it possible to express $\vec{r}$ in cylinder coordinates

$$\vec{r} = \left(\begin{matrix}\rho\,\cos\left(\varphi\right) \\ \rho\,\sin\left(\varphi\right)\end{matrix}\right).$$ | (2.3.3.2) |

$$r = \rho,\quad\dot{r}^2 = \rho^2\,\dot{\varphi}^2 + \dot{\rho}^2\quad\text{and}\quad\dot{\vec{r}}\,\vec{r}=\dot{\rho}\rho.$$ | (2.3.3.3) |

$$\dot{V}_R = -\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0}\left(\frac{1}{\rho^2} + \frac{\dot{\varphi}^2}{c^2}-\frac{\dot{\rho}^2}{2\,c^2\,\rho^2}\right)\,\dot{\rho}.$$ | (2.3.3.8) |

$$\dot{V}_R = -\vec{F}_R\,\dot{\vec{r}}$$ | (2.3.3.9) |

$$\vec{F}_R = m_d\,\ddot{\vec{r}},$$ | (2.3.3.10) |

$$\dot{V}_R = -m_d\,\ddot{\vec{r}}\,\dot{\vec{r}} = -\dot{T}$$ | (2.3.3.11) |

$$T(\dot{\vec{r}}) = \frac{1}{2}\,m_d\,\dot{\vec{r}}\,\dot{\vec{r}} = \frac{1}{2}\,m_d\,\dot{r}^2.$$ | (2.3.3.12) |

$$\dot{V}_R + \dot{T} = 0.$$ | (2.3.3.13) |

$$\dot{\mathcal{E}} = \dot{V}_R + \dot{T} = 0$$ | (2.3.3.14) |