Lagrangian Mechanics
Reformulating Newtonian mechanics via the principle of stationary action.
Define the Lagrangian $L = T - V$. The action
$$S[q] = \int_{t_1}^{t_2} L(q,\dot q, t)\, dt$$is stationary on physical trajectories, giving the Euler–Lagrange equations:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0.$$The formalism is coordinate-free, absorbs constraints, and links symmetries to conservation laws via Noether's theorem: every continuous symmetry of the action yields a conserved current. Time-translation invariance gives energy conservation, spatial translation gives momentum, rotational symmetry gives angular momentum.
Interactive: double pendulum
Two coupled pendula — one of the simplest chaotic systems. Tiny changes in initial conditions produce wildly different trajectories.