Applications: pendulum, Atwood machine, central forces

Lagrangian and Hamiltonian formulations · Physics

Simple pendulum

Interactive 3D

Release the bob and watch true small-angle SHM. The period depends only on the string length.

Controls

Drag the sliders or type a value — the simulation updates live.

String lengthm

0.3m2m

Release angle°

5°45°

Study notes

Worked example - Atwood with m1 = 3 kg, m2 = 2 kg. Step 1: coordinate x = downward displacement of m1. T = (1/2)(3+2) x_dot^2 = 2.5 x_dot^2. V = -3g x - 2g(const - x) = -g x + const' (taking g = 9.8). Step 2: L = T - V = 2.5 x_dot^2 + 9.8 x. Step 3: d/dt(dL/dx_dot) = 5 x_ddot; dL/dx = 9.8. Euler-Lagrange: 5 x_ddot = 9.8, so a = 1.96 m/s^2 downward for m1. Check with Newton: a = (3-2)x9.8/5 = 1.96 m/s^2. Matches - and we never solved for the tension (which would be 2x3x2x9.8/5 = 23.52 N).