# MOD HW 1 # Question 1 # Tessany Schou # Install ffmpeg (allows us to create videos) !apt update -y !apt install ffmpeg -y # Normal Python Mathy Code # Import Libraries: import numpy as np import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle # Parameters m = 1 # kg g = 9.8 # m/s^2 # We need an array of time points from 0 to 10 in increments of 0.001 seconds dt = 0.001 t_vec = np.arange(0,10,dt) sim_length = len(t_vec) # Initialize a vector of zeros x_vec = np.zeros(len(t_vec)) dx_vec = np.zeros(len(t_vec)) y_vec = np.zeros(len(t_vec)) lam_vec = np.zeros(len(t_vec)) # Set our initial condition x_vec[0] = -1 # initial x position (m) dx_vec[0] = 0 # initial x velocity (m/s) # Loop through time # Euler's Method (approximately integrates the differential equation) for i in range(1, len(t_vec)): x_vec[i] = x_vec[i-1] + dx_vec[i-1]*dt dx_vec[i] = dx_vec[i-1] + (-2*x_vec[i-1]*(2*dx_vec[i-1]**2+g))/(4*x_vec[i-1]**2+1)*dt lam_vec[i] = (2*m*dx_vec[i-1]**2+g*m)/(4*x_vec[i-1]**2+1) # calculating y positions for i in range(0, len(t_vec)): y_vec[i] = x_vec[i]**2 # Plotting x positon plt.xlabel('t') plt.ylabel('x') plt.title('x(t)') plt.plot(t_vec,x_vec) plt.show() # Plotting y position plt.xlabel('t') plt.ylabel('y') plt.title('y(t)') plt.plot(t_vec,y_vec) plt.show() # Plotting conservative forces plt.xlabel('t') plt.ylabel('λ') plt.title('Conservative Forces') plt.plot(t_vec,lam_vec) plt.show() # Setup Figure: Initialize Figure / Axe Handles fig, ax = plt.subplots() p, = ax.plot([], [], color='cornflowerblue') ax.axis('equal') ax.set_xlim([-2.5, 2.5]) ax.set_ylim([-0.5, 1.5]) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Question 1 Simulation:') video_title = "Q1simulation" # Now we want to plot stuff on those axes c = Circle((-1,1), radius=0.1, color='cornflowerblue') ax.add_patch(c) # Setup Animation Writer: FPS = 20 sample_rate = int(1 / (dt * FPS)) dpi = 300 writerObj = FFMpegWriter(fps=FPS) x_path = np.linspace(-1.5, 1.5, 1000) y_path = np.zeros(len(x_path)) for i in range(0, len(x_path)): y_path[i] = x_path[i]**2 fig1 = plt.plot(x_path,y_path) plt.show() # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): for i in range(0, sim_length, sample_rate): # Update Pendulum Patch: patch_center = x_vec[i], y_vec[i] c.center = patch_center # Update Drawing: fig.canvas.draw() # Update the figure with the new changes # Grab and Save Frame: writerObj.grab_frame() from IPython.display import Video Video("/work/"+video_title+".mp4", embed=True, width=640, height=480)