Physic Labs

Analytical mechanics

The Euler–Lagrange equations

Use a frictionless pendulum to see the Euler–Lagrange equation ddt∂L∂θ˙−∂L∂θ=0\frac{d}{dt}\frac{\partial L}{\partial\dot\theta}-\frac{\partial L}{\partial\theta}=0 in action: verify θ¨=−(g/l)sin⁡θ\ddot\theta=-(g/l)\sin\theta and energy conservation on the phase portrait.

Undergraduate

Equipment

  • Pendulum animation plus phase-plane plot $(\theta,\dot\theta)$
  • Slider “Góc thả θ₀” (release angle)
  • Slider “Chiều dài l” (string length)
  • Energy and equal-energy contour display on the phase plot

Procedure

  1. Release the pendulum

    Set a small release angle θ₀ (under ~10°) and let it run. The phase-plane trace is nearly elliptical — the small-angle regime where sin⁡θ≈θ\sin\theta\approx\theta gives harmonic motion with ω=g/l\omega=\sqrt{g/l}.

  2. Verify energy conservation

    Watch the orbit follow one equal-energy contour (green-blue line): E=12ml2θ˙2+mgl(1−cos⁡θ)E=\tfrac12ml^2\dot\theta^2+mgl(1-\cos\theta) stays constant since the Lagrangian has no explicit time dependence. Pause and compare the trace with the contour.

  3. Push to large angles

    Raise θ₀ toward 90° and beyond: the phase curve deforms away from an ellipse, and the period grows since the true restoring acceleration is −(g/l)sin⁡θ-(g/l)\sin\theta, weaker than linear at large θ. Measure the period at two amplitudes and compare.

  4. Change the length

    Sweep the length slider l and predict the new period from T=2πl/gT=2\pi\sqrt{l/g} before timing it — quadrupling l should double T. Confirm on the phase plot that the energy level and orbit shape rescale with l.

Simulation

Experiment history

Leonhard Euler developed the variational calculus in the 1740s, building on the brachistochrone problems posed by the Bernoullis and Pierre de Maupertuis's principle of least action (1746). Joseph-Louis Lagrange, still a teenager, wrote to Euler in 1755 with the systematic recipe that became the Euler–Lagrange equations. Lagrange's Mécanique analytique (1788) carried out the program proudly “without figures”: choose coordinates qiq_i, form L=T−VL=T-V, and the equations of motion fall out for any system — pendulum, planet, or field. The formulation later proved essential to quantum mechanics through Feynman's path integrals and to field theory through the least-action principle.

Related physicists

Isaac NewtonLeonhard EulerJoseph-Louis LagrangePierre-Louis Moreau de Maupertuis

Related library topics

The Euler–Lagrange equations