# Coupled differential equations for charged particles

• SgrA*
In summary, the conversation discusses studying the behavior of electrons in a spatially bounded system. The system involves a larger number of electrons and is represented by a system of coupled equations. However, the person is unsure of how to solve it and is seeking advice on which method to use. It is mentioned that the three-body problem does not have an analytical solution.
SgrA*
Hello,

I wanted to study the behaviour of electrons in a spatially bounded system. I want to have a larger number of electrons, but I took 3 to start with and arrived at this system of coupled equations:
\begin{align}\begin{bmatrix} \mathbf{\ddot{x_{1}}}\\ \\ \mathbf{\ddot{x_{2}}}\\ \\ \mathbf{\ddot{x_{3}}} \end{bmatrix} = \frac{1}{4\pi\epsilon_0} \begin{bmatrix} \frac{q_1 q_2}{m_1} & \frac{q_1 q_3}{m_1} \\ \\ \frac{q_2 q_1}{m_2} & \frac{q_2 q_3}{m_2} \\ \\ \frac{q_3 q_1}{m_3} & \frac{q_3 q_2}{m_3} \\ \end{bmatrix} \begin{bmatrix} \frac{\mathbf{r_{12}}}{|r_{12}^{3}|} & \frac{\mathbf{r_{21}}}{|r_{21}^{3}|} & \frac{\mathbf{r_{31}}}{|r_{31}^{3}|}\\ \\ \frac{\mathbf{r_{13}}}{|r_{13}^{3}|} & \frac{\mathbf{r_{23}}}{|r_{23}^{3}|} & \frac{\mathbf{r_{32}}}{|r_{32}^{3}|} \end{bmatrix}\end{align}
I'm not sure how to solve it: I've only solved the coupled mass problem by diagonalization, but I had a 2x2 matrix there. What method can I use to solve this system?

Thanks!

Are you trying to solve the three-body problem? It does not have an analytical solution.

## 1. What are coupled differential equations for charged particles?

Coupled differential equations for charged particles are a type of mathematical model used to describe the motion and interactions of charged particles, such as electrons or protons, in an electric or magnetic field. These equations take into account factors such as the particle's charge, mass, and velocity, as well as the strength and direction of the electric or magnetic field.

## 2. How are coupled differential equations for charged particles used in scientific research?

Coupled differential equations for charged particles are commonly used in fields such as physics, astronomy, and plasma physics to study the behavior of charged particles in various environments. They can also be used to predict and analyze the behavior of particles in particle accelerators and other high-energy experiments.

## 3. Can coupled differential equations for charged particles be solved analytically?

In some cases, coupled differential equations for charged particles can be solved analytically, meaning that an exact, closed-form solution can be found. However, in many cases, numerical methods must be used to approximate the solution due to the complexity of the equations.

## 4. What are some real-world applications of coupled differential equations for charged particles?

Coupled differential equations for charged particles have a wide range of applications in various industries and technologies. They are used in the design and optimization of high-voltage equipment, such as power transformers and particle accelerators. They are also used in plasma physics research and in the development of electric propulsion systems for spacecraft.

## 5. How do coupled differential equations for charged particles relate to Maxwell's equations?

Coupled differential equations for charged particles are derived from Maxwell's equations, which are a set of fundamental equations that describe the behavior of electric and magnetic fields. The equations for charged particles are often referred to as the "particle-in-field" equations, as they describe the motion of particles in response to the electric and magnetic fields described by Maxwell's equations.

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