Euler-Lagrange Equation

The calculus of variations asks: which function $y(x)$ extremizes the functional

$S[y] = \int_{x_1}^{x_2} L(x, y, y')\, dx$

where $L$ is the Lagrangian density? The answer is the Euler-Lagrange equation:

$\frac{d}{dx}\!\left(\frac{\partial L}{\partial y'}\right) - \frac{\partial L}{\partial y} = 0$

Classic Examples

Shortest path: $L = \sqrt{1 + y'^2}$. The E-L equation gives $y'' = 0$, i.e., a straight line.

Brachistochrone: The curve of fastest descent under gravity is a cycloid, found by minimizing the travel time functional.

Simple pendulum: The Lagrangian $L = T - V = \tfrac{1}{2}ml^2\dot\theta^2 - mgl(1-\cos\theta)$ gives the equation of motion:

$\ddot\theta + \frac{g}{l}\sin\theta = 0$

In the small-angle approximation ($\sin\theta \approx \theta$), this becomes simple harmonic motion with angular frequency $\omega = \sqrt{g/l}$.

Pendulum Period

The small-angle period is:

$T = 2\pi\sqrt{\frac{l}{g}}$

The small-angle solution is $\theta(t) = \theta_0 \cos(\omega t + \phi)$.

Numerical Action

Given discrete sample points $(x_i, y_i)$, approximate $S[y]$ numerically:

1. Estimate $y'_i \approx (y_{i+1} - y_i) / (x_{i+1} - x_i)$ (forward difference).
2. Evaluate $L$ at the midpoint of each interval.
3. Sum: $S \approx \sum_i L(x_{\text{mid}}, y_{\text{mid}}, y'_i)\, \Delta x_i$.

Your Task

Implement action(L_func, y_values, x_values) using the midpoint-rectangle rule above. Implement pendulum_period_s(l_m, g=9.81) for the small-angle period. Implement pendulum_angle_rad(t_s, theta0_rad, l_m, g=9.81) for the small-angle trajectory $\theta_0 \cos(\omega t)$.

All constants must be computed inside the function bodies.