Noether's Theorem: Symmetry & Conservation Laws

Emmy Noether's theorem is the backbone of every field theory, including the quantum field theory used to build the Standard Model: a continuous symmetry of the Lagrangian always implies a conserved quantity. This simulator makes the link visible with the simplest possible mechanical system — a particle orbiting a 2D potential well rendered as a 3D surface. Dial in an anisotropy ε and the circular, rotation-symmetric bowl stretches into an ellipse; angular momentum, exactly conserved while the bowl is circular, visibly drifts the moment that symmetry breaks, and the orbit itself blooms into a precessing rosette. Independently, flipping on a time-periodic drive breaks time-translation symmetry instead, and total energy — flat as long as the potential has no explicit clock in it — starts climbing or falling every cycle. Live numeric readouts and scrolling sparklines for L_z(t) and E(t) let you watch each conservation law hold or fail in real time, tied to the exact symmetry that produces it.