Theory

Step 1: Particle Characteristics

Each particle is treated as a point mass, and each point mass has three intrinsic characteristics:

  1. a mass \(m_i\),
  2. a position vector \(\vec{r}_i=\left[\begin{array}{ccc}x_i & y_i & z_i\end{array}\right]\), and
  3. a velocity vector \(\vec{v}_i=\left[\begin{array}{ccc}u_i & v_i & w_i\end{array}\right]\).

Step 2: Force Calculation

According to Newton’s law of universal gravitation, given two distinct point masses in space, the gravitational force acting on the objects takes the form:

$$F=G\frac{m_1m_2}{r^2}$$

where:

  • \(F\) is the common force between the masses,
  • \(G\) is the gravitational constant,
  • \(m_1\) is the first mass,
  • \(m_2\) is the second mass, and
  • \(r\) is the distance between the centers of the masses.

However, this formula is only applicable to isolated two-body systems. Therefore, in order to consider \(N\) number of point masses, Newton’s law of universal gravitation must be updated. This formulation is called the \(N\)-body problem.

From Wikipedia: The \(N\)-body problem considers \(N\) point masses \(m_i,i=1,2,…,N\) in an inertial reference frame in three-dimensional space \(\mathbb{R}^3\) moving under the influence of mutual gravitational attraction. In other words, the \(N\) body problem expands Newton’s law of universal gravitation to work with any amount of point masses.

The gravitational force felt on mass \(m_i\) by a single other mass \(m_j\) is:

$$\vec{F}_i=G\frac{m_im_j}{\left|\vec{r}_j-\vec{r}_i\right|^2}⋅\frac{\left(\vec{r}_j-\vec{r}_i\right)}{\left|\vec{r}_j-\vec{r}_i\right|}=G\frac{m_im_j\left(\vec{r}_j-\vec{r}_i\right)}{\left|\vec{r}_j-\vec{r}_i\right|^3}$$

where:

  • \(F_i\) is the force felt on mass \(m_i\) due to mass \(m_j\),
  • \(G\) is the gravitational constant,
  • \(m_i\) is the first mass,
  • \(m_j\) is the second mass, and
  • \(\left|\vec{r}_j-\vec{r}_i\right|\) is the magnitude of the distance between \(\vec{r}_j\) and \(\vec{r}_i\) induced by the \(l_2\) norm.

In order to obtain the equations of motion for each particle in the system, the previous equation must be summed for a total of \(N\) times per particle:

$$\vec{F}i=G\sum{\substack{j=1\j\not{}i}}^N\frac{m_im_j\left(\vec{r}_j-\vec{r}_i\right)}{\left|\vec{r}_j-\vec{r}_i\right|^3}$$

Step 3: Acceleration Calculation

Given the summation form of Newton’s law of gravitation for the \(N\)-body problem, solving for the acceleration of a particle is a trivial task. The discretized vector form of Newton’s second law for a mass \(m_i\) can be written as:

$$\vec{F}_i=m_i\vec{a}_i→\vec{a}_i=\frac{\vec{F}_i}{m_i}$$

Therefore:

$$\vec{a}i=G\sum{\substack{j=1\j\not{}i}}^N\frac{m_j\left(\vec{r}_j-\vec{r}_i\right)}{\left|\vec{r}_j-\vec{r}_i\right|^3}$$

Numerical Implementation

1

x = 1