Euler–Lagrange Lab

An interactive environment exploring analytical mechanics and the Principle of Stationary Action. Discover the fundamental equations that govern the paths of nature.

$$S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$$

$$\delta S = 0$$

The Derivation Pathway

Step 1: Path Variation +

$$q(t) \to q(t) + \epsilon \eta(t)$$

Assume a true path $q(t)$ that minimizes action. We introduce a small variation $\eta(t)$ scaled by parameter $\epsilon$. The endpoints are fixed.

Step 2: Action Functional +

$$S(\epsilon) = \int L(q + \epsilon\eta, \dot{q} + \epsilon\dot{\eta}) \, dt$$

Substitute the varied path into the action integral. The action $S$ is now a function of the scalar parameter $\epsilon$.

Step 3: First Order Expansion +

$$\frac{dS}{d\epsilon} \bigg|_{\epsilon=0} = \int \left( \frac{\partial L}{\partial q}\eta + \frac{\partial L}{\partial \dot{q}}\dot{\eta} \right) dt = 0$$

For stationary action, the derivative with respect to $\epsilon$ evaluated at $\epsilon=0$ must vanish. We use the chain rule to expand the Lagrangian.

Step 4: Integration by Parts +

$$\int \frac{\partial L}{\partial \dot{q}} \dot{\eta} \, dt = \left[ \frac{\partial L}{\partial \dot{q}} \eta \right] - \int \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) \eta \, dt$$

Apply integration by parts to the second term to factor out $\eta(t)$. This shifts the time derivative onto the momentum term.

Step 5: Euler-Lagrange Equation +

$$\frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) = 0$$

Since $\eta(t)$ is zero at the boundaries, the surface term vanishes. Because the remaining integral must be zero for any arbitrary variation $\eta(t)$, the term in brackets must be identically zero.

Action Minimization (Gravity Field)

ACTION PHASE SPACE (S vs ε)

VARIATION ε →

← VARIATION -ε

TRUE PATH (CYAN) / VARIED PATH (VIOLET)

ACTION S = 0.000

Observe a particle in a uniform gravitational field. Modify the variation amplitude $\epsilon$ to see how arbitrary wavy paths increase the total Action compared to the natural parabolic path.

Variation Amplitude ($\epsilon$) 20.0

Mass ($m$) 1.0

Gravity ($g$) 9.8

Initial Velocity ($v_0$) 40

Classical Systems

Lagrangian: $$L = \frac{1}{2} m \dot{x}^2$$

E-L Eq: $$\frac{d}{dt}(m \dot{x}) - 0 = 0$$

EOM: $$\ddot{x} = 0 \implies v = \text{const}$$

Lagrangian: $$L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2$$

E-L Eq: $$\frac{d}{dt}(m \dot{x}) - (-kx) = 0$$

EOM: $$\ddot{x} = -\frac{k}{m}x$$

Lagrangian: $$L = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l \cos(\theta)$$

E-L Eq: $$\frac{d}{dt}(m l^2 \dot{\theta}) - (-m g l \sin(\theta)) = 0$$

EOM: $$\ddot{\theta} = -\frac{g}{l}\sin(\theta)$$

Numerical Workspace

Runge-Kutta 4 (JS)

Euler Method (Python)

// RK4 Integrator for 2nd Order ODE: x'' = f(x, v, t)
function rk4_step(x, v, t, dt, force_func) {
    const k1_v = force_func(x, v, t);
    const k1_x = v;

    const k2_v = force_func(x + 0.5*dt*k1_x, v + 0.5*dt*k1_v, t + 0.5*dt);
    const k2_x = v + 0.5*dt*k1_v;

    const k3_v = force_func(x + 0.5*dt*k2_x, v + 0.5*dt*k2_v, t + 0.5*dt);
    const k3_x = v + 0.5*dt*k2_v;

    const k4_v = force_func(x + dt*k3_x, v + dt*k3_v, t + dt);
    const k4_x = v + dt*k3_v;

    const new_x = x + (dt / 6.0) * (k1_x + 2*k2_x + 2*k3_x + k4_x);
    const new_v = v + (dt / 6.0) * (k1_v + 2*k2_v + 2*k3_v + k4_v);
    
    return { x: new_x, v: new_v };
}

# Basic Euler Integration for Harmonic Oscillator
def simulate_ho(k, m, x0, v0, dt, steps):
    x = x0
    v = v0
    
    history = [(x, v)]
    
    for _ in range(steps):
        # EOM: a = - (k/m) * x
        a = -(k / m) * x
        
        # Update state
        v_new = v + a * dt
        x_new = x + v * dt
        
        x = x_new
        v = v_new
        history.append((x, v))
        
    return history

Phase Space Geometry

The state of a system is fully described by coordinates $q$ and momenta $p$. For a conservative system like the Harmonic Oscillator, energy conservation manifests as closed orbits in phase space.

Passion of Physics ... A Journey Through Space-Time ... By KAZ

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