Symmetry & conservation laws
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You should be comfortable with the Euler–Lagrange equation and the idea of a Lagrangian L(q, q̇, t) — see L06.
Why this matters
Conservation laws feel like separate rules you memorise: energy is conserved, momentum is conserved, charge is conserved. Noether's insight is that they are not separate at all.
Every conservation law is the shadow of a symmetry. If the physics does not care where you are, momentum is conserved. If it does not care which way you face, angular momentum is conserved. And if it does not care when you run the experiment — if the laws are the same today as tomorrow — then energy is conserved. This lecture builds that last link: time-translation symmetry gives you conservation of energy.
Symmetry as a machine
A symmetry is any change you can make to a system that leaves its action unchanged. Feed such a continuous symmetry into the machinery of the Lagrangian and out comes a quantity whose value never changes along the true motion — a conserved current. The recipe is mechanical: identify the transformation, check the Lagrangian is invariant (or shifts by a total derivative), and read off the constant.
The derivation
Energy from time-translation symmetry
If the Lagrangian has no explicit time dependence (∂L/∂t = 0), the Hamiltonian is conserved along the motion.
Reading the result
The conserved quantity H is what we call energy. Notice the logic runs one way: it is the absence of explicit time-dependence in L that guarantees conservation, not the other way round. A system driven by a time-varying external field has ∂L/∂t ≠ 0, and its energy genuinely is not conserved — the bookkeeping leaks into the driver.
"Energy conserved" and "Hamiltonian conserved" are not always the same statement. When the coordinate transformation to q is itself time-dependent (a rotating frame, say), H can be conserved while differing from the intuitive kinetic-plus-potential energy — and vice versa. The Noether charge is H; whether it equals "energy" depends on how you defined your coordinates. C
Three quick tests
1. A pendulum whose pivot is being shaken vertically at fixed frequency — is energy conserved? 2. Which symmetry gives momentum? 3. Why does "the laws are the same tomorrow" translate to ∂L/∂t = 0?
Problem set · PU-201
Apply Noether's recipe to translation, rotation, and time-shift symmetries. Graded set with worked solutions in the unit.
Derivations cited: D-002 (energy from time-translation symmetry). Threads: symmetry, energy.