Electromagnetism

Electromagnetism is one of the four fundamental forces of nature. It is carried by virtual photons, massless particles travelling at the speed of light. The force is produced by electric charge.

Lagrangian
In pilitron dynamics, the electromagnetic lagrangian is given by:


 * $$\mathcal{L} = -\frac{q_1 q_2 \cos \theta}{4\pi\epsilon_0\mu_0 d}$$

Where:
 * $$q_1$$ and $$q_2$$ are the electrical charges of 2 particles.
 * $$\theta$$ is the angle between the particles.
 * $$\epsilon_0$$ is the vacuum permittivity.
 * $$\mu_0$$ is the vacuum permeability.
 * $$d$$ is the distance between the two objects.

The distribution of possible Lagrangians in an electromagnetic field is given by:


 * $$\mathsf{L} = \sum -\frac{q_1 q_2 \cos \mathsf{A}}{4\pi\epsilon_0\mu_0 \mathsf{D}}$$

Where:
 * $$\mathsf{A}$$ is the distribution of possible angles.
 * $$\mathsf{D}$$ is the distribution of possible distances.

Field
The electromagnetic field is generated by electrical charge, and is described by the following equation:


 * $$\frac{\partial \mathbf{E}\left(q, \mathbf{r}\right)}{\partial \mathbf{r}} = -\frac{q}{4 \pi \epsilon \mu |\mathbf{r}|^2}\hat{\mathbf{r}}$$

Where:
 * $$q$$ is the charge of the source.
 * $$\mathbf{r}$$ is the vector that points from the given position in the field to the source.
 * $$\epsilon$$ is the permittivity of the medium.
 * $$\mu$$ is the permeability of the medium.

The field expands from the charge and propagates at the speed of light by means of photons. The energy carried by each photon is virtual, and is given by:


 * $$E = |\mathbf{E}(q, r)| i$$

The Lagrangian, in terms of the field, is:


 * $$\mathcal{L} = pv + q_1|\mathbf{E}\left(q_2, \mathbf{r}\right)|$$