# How to simulate motion of 4 particles moving towards each other?

## Question:

I have a problem that I need to simulate where there are four particles located at the corners of a square room with known lenght `a`. They are starting to move simultaneously and are supposed to move towards the closest neighbour until one of them reaches some other. They all have constant velocity `V`. I am supposed to use `scipy.integrate.RK45` function.

I am not sure how to describe mathematically the fact that they are all moving towards the closest neightbour. I suppose the function should look something like this (for one particles for now):

``````def motion_eq(t, y):
???
``````
``````r = np.array([0, 0], 'float')
v = np.array([0, 1], 'float')

dt = 0.1
a = 2

motion_solver = scipy.integrate.RK45(motion_eq, 0, np.hstack([r, v]),
t_bound = np.inf, first_step = dt, max_step = dt)

particle, *_ = plt.plot(*r.T, 'o')
plt.gca().set_aspect(1)
plt.xlim([-a, a])
plt.ylim([-a, a])
def update():
motion_solver.step()
r = motion_solver.y[0:2]
particle.set_data(*r.T)
plt.draw()
timer = plt.gcf().canvas.new_timer(interval = 50)
timer.start()

plt.show()
``````

Could somebody please give a hint of how to do this? Thanks in advance.

## Notes

• We consider a 2D plane for the calculation of the equation of motion.
• We assume four particles, each of which has three neighboring particles, in this case.
• We find the average position (`pos`) of the four particles (assumed random positions for each particle initially).

Other calculations are in this image:

### Code

``````
import numpy as np
import scipy.integrate
import matplotlib.pyplot as plt

def closest_neighbor_dir(r, a):
all_ptls = np.array([[a, a], [a, -a], [-a, -a], [-a, a]])
ptls = np.delete(all_ptls, np.where((all_ptls == r).all(axis=1)), axis=0)
pos = np.mean(ptls, axis=0)
dir_vec = pos - r
return dir_vec / np.linalg.norm(dir_vec)

def motion_eq(t, y, a, V):
r, v = y[:2], y[2:]
accel = V * closest_neighbor_dir(r, a) - v
return np.hstack([v, accel])

def update():
for i in range(4):
motion_solver[i].step()
r[i] = motion_solver[i].y[0:2]
particles.set_data(r[:, 0], r[:, 1])
plt.draw()

r = np.array([[1.1, 1.7], [-1, 1.3], [-1.2, -1.2], [1.4, -1.7]], dtype=float)
v = np.array([[0, 0], [0, 0], [0, 0], [0, 0]], dtype=float)

dt, a, V = 0.1, 2, 0.2

motion_solver = [scipy.integrate.RK45(lambda t, y: motion_eq(t, y, a, V), 0, np.hstack([r[i], v[i]]),
t_bound=np.inf, first_step=dt, max_step=dt) for i in range(4)]

particles, *_ = plt.plot(*r.T, 'o')
plt.gca().set_aspect(1)
plt.xlim([-a, a])
plt.ylim([-a, a])

timer = plt.gcf().canvas.new_timer(interval=50)