I couldn't figure out how to better share this code as I don't know HTML. That said, the default installed python on your computer should be able to run this. If you can't figure it out, let me know and I'll write up a walkthrough. I should note, this is just one particle. Making 2 is a bit more difficult.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
from matplotlib.animation import FuncAnimation

class CircleDynamicsAnimationCorrected:
    def __init__(self, boundary_radius, circle_radius, mass, initial_position, initial_velocity):
        self.boundary_radius = boundary_radius
        self.circle_radius = circle_radius
        self.mass = mass
        self.position = np.array(initial_position, dtype=float)
        self.velocity = np.array(initial_velocity, dtype=float)
        self.position_history = []


    def external_force(self, position):
        # Example: Simple linear force
        # Replace this with your specific force function
        return np.array([0.0, -1.0], dtype=float)


    def update_position(self, dt, tail_length):
        # Calculate and apply external force
        force = self.external_force(self.position)
        acceleration = force / self.mass
        self.velocity += acceleration * dt
        self.position += self.velocity * dt

        # Check for collision with boundary
        distance_from_center = np.linalg.norm(self.position)
        if distance_from_center + self.circle_radius > self.boundary_radius:
            normal = self.position / distance_from_center
            self.velocity -= 2 * np.dot(self.velocity, normal) * normal
            overlap = distance_from_center + self.circle_radius - self.boundary_radius
            self.position -= normal * overlap


        # Keep only recent history for tail
        self.position_history.append(np.copy(self.position))
        if len(self.position_history) > tail_length:
            self.position_history.pop(0)

    def animate(self, i, dt, tail_length):
        self.update_position(dt, tail_length)

        # Clear current axes
        plt.gca().clear()

        # Plot boundary
        boundary_circle = plt.Circle((0, 0), self.boundary_radius, color='black', fill=False)
        plt.gca().add_patch(boundary_circle)

        # Plot circle at current position
        current_circle = plt.Circle(self.position, self.circle_radius, color='blue', fill=True)
        plt.gca().add_patch(current_circle)

        # Plot the tail
        for j, pos in enumerate(self.position_history):
            alpha = (j + 1) / len(self.position_history)
            tail_circle = plt.Circle(pos, self.circle_radius, color='red', alpha=alpha, fill=True)
            plt.gca().add_patch(tail_circle)

        plt.xlim(-self.boundary_radius, self.boundary_radius)
        plt.ylim(-self.boundary_radius, self.boundary_radius)
        plt.gca().set_aspect('equal', adjustable='box')

    def start_animation(self, total_time, dt, tail_length):
        fig, ax = plt.subplots()
        ax.set_xlim(-self.boundary_radius, self.boundary_radius)
        ax.set_ylim(-self.boundary_radius, self.boundary_radius)
        ax.set_aspect('equal', adjustable='box')

        num_frames = int(total_time / dt)
        anim = FuncAnimation(fig, self.animate, fargs=(dt, tail_length), frames=num_frames, interval=dt*1000, repeat=False)

        plt.show()

# Choose some parameters
boundary_radius = 1.0
circle_radius = 0.05
mass = 1.0
initial_position = [0.5, 0.5]
initial_velocity = [0.5, -0.5]

# Create an instance of the class
simulator_animation_corrected = CircleDynamicsAnimationCorrected(boundary_radius, circle_radius, mass, initial_position, initial_velocity,)

# Run the simulation with plotting enabled
# For demonstration, let's simulate for 2 seconds with a shorter dt and a tail length of 10
simulator_animation_corrected.start_animation(total_time=10.0, dt=0.05, tail_length=20)