import numpy as np
from sklearn.linear_model import LinearRegression

# Generate synthetic data for a pendulum
g, L = 9.81, 1.0
theta = np.linspace(-np.pi, np.pi, 100).reshape(-1, 1)  # angles from -π to π
acc_true = -(g/L) * np.sin(theta)                        # true angular acceleration

# Model 1: Linear regression on raw angle (theta)
model_raw = LinearRegression().fit(theta, acc_true)
pred_raw = model_raw.predict(theta)

# Model 2: Linear regression on sin(theta) as feature
X_feat = np.sin(theta)                                   # physics-inspired feature
model_feat = LinearRegression().fit(X_feat, acc_true)
pred_feat = model_feat.predict(X_feat)

# Evaluate at a sample angle
test_angle = np.array([[50 * np.pi / 180]])  # 50 degrees in radians
print("True acceleration:", -(g/L) * np.sin(test_angle))
print("Linear model (raw theta) prediction:", model_raw.predict(test_angle))
print("Linear model (sin(theta)) prediction:", model_feat.predict(np.sin(test_angle)))