Noether's Theorem: Symmetry and Conserved Quantities

Every continuous symmetry of the action carries its own conserved quantity. Worked case: N particles on a line with Cartesian positions xᵢ, masses mᵢ, and Lagrangian L = Σᵢ ½mᵢẋᵢ² − V(x₁ − x₂, …, x_{N−1} − x_N), where the potential depends only on separations. Shift every coordinate by the same small amount, xᵢ → xᵢ + ε. Each kinetic term is unchanged because it uses velocities, and each separation is unchanged, so L is invariant for every ε.

The change in L is ε times the sum of ∂L/∂xᵢ, which vanishes for every ε, so that sum is zero. The unforced Euler-Lagrange equation turns each ∂L/∂xᵢ into d/dt(∂L/∂ẋᵢ), so the sum of the conjugate momenta pᵢ = ∂L/∂ẋᵢ = mᵢẋᵢ is constant: total momentum is conserved. Internal forces only move momentum between parts of the system, so the total stays constant whenever the net external force is zero. When a football player hits the goalpost, the player bounces back and the Earth recoils by an immeasurably small amount, keeping the total unchanged.

Rotation about a fixed axis gives that component of angular momentum, and time shift symmetry gives energy: in an isolated system the total energy stays constant while converting between kinetic, potential, and thermal forms. Time symmetry means the action is unchanged under any continuous shift t → t + ε, not merely under a one-hour delay. If the symmetry is only approximate, a small outside torque τ gives dL/dt = τ, so a small rate accumulates over a long enough time. For constant torque use ΔL = τ Δt, converting minutes to seconds by multiplying by 60.