Noether's Theorem and Conservation Laws

Attempt these from memory before reading.

• State Noether’s theorem in one sentence.
• Which symmetry gives conservation of momentum? Of angular momentum? Of energy?
• What does “up to a boundary term” mean in the statement?
• Is Noether’s theorem about symmetries of the equations, or of the Lagrangian?

Before Noether (1918), conservation laws were empirical facts, each proved separately: momentum conservation from Newton’s third law, energy conservation from the work-energy theorem, angular momentum from central forces. Nothing connected them.

Noether’s theorem says they are all the same phenomenon. A continuous symmetry of the action produces a conserved quantity, and each familiar conservation law is the conserved quantity for one particular symmetry:

Reading the table backwards is the useful direction. If you can guess a symmetry, you get a conservation law without solving anything. That is how conserved quantities are found in particle physics, where the Lagrangian is written down from symmetry requirements rather than from forces — and it is why Emmy Noether’s theorem is arguably the most important result in theoretical physics.

Theorem 5.1 (Noether’s Theorem). For every continuous symmetry of the action, there is a Corresponding conserved quantity.

More precisely: if the action is invariant (up to a boundary term) under the infinitesimal transformation $q_j \to q_j + \epsilon f_j(q, \dot{q}, t)$ Then

$$Q = \sum_j \frac{\partial L}{\partial \dot{q}_j} f_j$$

Is a constant of motion.

Theorem 5.2 (Noether’s Theorem --- Full Proof). Suppose the Lagrangian transforms under an infinitesimal transformation $q_j \to q_j + \epsilon \delta q_j$ as:

$$L \to L + \epsilon \frac{dF}{dt}$$

For some function $F(q, t)$. Then the quantity

$$Q = \sum_j p_j\, \delta q_j - F$$

Is conserved.

Proof. The variation of the action is:

$$\delta S = \int_{t_1}^{t_2} \left[\sum_j \left(\frac{\partial L}{\partial q_j}\delta q_j + \frac{\partial L}{\partial \dot{q}_j}\delta\dot{q}_j\right)\right] dt = \int_{t_1}^{t_2} \frac{dF}{dt}\, dt$$

Where the second equality uses the assumption that the action changes by at most a boundary term. Using the Euler-Lagrange equations $\frac{\partial L}{\partial q_j} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_j}$:

$$\delta S = \int_{t_1}^{t_2} \sum_j \left[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right)\delta q_j + \frac{\partial L}{\partial \dot{q}_j}\frac{d}{dt}\delta q_j\right] dt$$ $$= \int_{t_1}^{t_2} \frac{d}{dt}\left(\sum_j p_j\, \delta q_j\right) dt$$

Setting this equal to $\int_{t_1}^{t_2} \frac{dF}{dt}\, dt$:

$$\frac{d}{dt}\left(\sum_j p_j\, \delta q_j - F\right) = 0$$

Therefore $Q = \sum_j p_j\, \delta q_j - F$ is constant. $\blacksquare$

Problem. Show that spatial translation invariance implies conservation of linear momentum.

Solution

Consider an infinitesimal translation $x \to x + \epsilon$ I.e., $\delta x = 1$, $\delta y = 0$, $\delta z = 0$.

For a free particle, $L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)$ Which is invariant ($\delta L = 0$ So $F = 0$).

By Noether’s theorem:

$$Q = p_x \cdot 1 + p_y \cdot 0 + p_z \cdot 0 - 0 = p_x = \mathrm{const}$$

This is conservation of the $x$-component of linear momentum. Translation invariance in all three directions gives conservation of the full momentum vector $\mathbf{p}$. $\blacksquare$

Problem. Show that rotational invariance implies conservation of angular momentum.

Solution

Consider an infinitesimal rotation by angle $\epsilon$ about the $z$-axis:

$$\delta x = -\epsilon y, \quad \delta y = \epsilon x, \quad \delta z = 0$$

For a free particle, $\delta L = m(\dot{x}\,\delta\dot{x} + \dot{y}\,\delta\dot{y}) = m(\dot{x}(-\epsilon\dot{y}) + \dot{y}(\epsilon\dot{x})) = 0$ So $F = 0$.

By Noether’s theorem:

$$Q = p_x(-y) + p_y(x) - 0 = x\, p_y - y\, p_x = L_z$$

This is the $z$-component of angular momentum. Full rotational invariance gives conservation of the entire angular momentum vector $\mathbf{L} = \mathbf{r} \times \mathbf{p}$. $\blacksquare$

Problem. Show that time translation invariance implies conservation of energy.

Solution

Consider an infinitesimal time translation $t \to t + \epsilon$. The coordinates transform as $q_j(t) \to q_j(t + \epsilon) \approx q_j(t) + \epsilon \dot{q}_j(t)$ So $\delta q_j = \dot{q}_j$.

If $L$ does not depend explicitly on time, then:

$$\delta L = \sum_j \left(\frac{\partial L}{\partial q_j}\dot{q}_j + \frac{\partial L}{\partial \dot{q}_j}\ddot{q}_j\right)\epsilon = \frac{dL}{dt}\epsilon = \frac{d}{dt}\left(\epsilon L\right)$$

So $F = \epsilon L$ Giving $F = L$ (per unit $\epsilon$).

By Noether’s theorem:

$$Q = \sum_j p_j \dot{q}_j - L = h$$

This is the energy function, which equals $T + V$ for natural systems. $\blacksquare$

Noether’s theorem is the universe’s most elegant bookkeeping principle. Every symmetry you can identify is a guarantee that something is conserved. If the laws of physics are the same here as they are on the other side of the room, then linear momentum is conserved. If the laws are the same now as they were yesterday, then energy is conserved. If the laws do not care which way you point your coordinate axes, then angular momentum is conserved. Think of it like a conservation bank: every symmetry deposits a conserved quantity that you can withdraw later to solve problems. The deep insight is that conservation laws are not separate accidents of nature but consequences of the underlying symmetry structure. When you discover a new symmetry, you automatically gain a new conservation law.

Problem. A particle moves in a central potential $V(r)$. Show that angular momentum is conserved.

Solution. In spherical coordinates $(r, \theta, \phi)$ with $V = V(r)$:

$$L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2\sin^2\theta\,\dot{\phi}^2) - V(r)$$

Since $L$ does not depend on $\phi$ (rotational symmetry about the $z$-axis):

$$p_\phi = \frac{\partial L}{\partial \dot{\phi}} = mr^2\sin^2\theta\,\dot{\phi} = \mathrm{const}$$

This is the $z$-component of angular momentum. By Noether’s theorem, the full angular momentum vector Is conserved for any central potential. $\blacksquare$

Mistake 1: Assuming every symmetry implies a conservation law without checking continuity Noether’s theorem applies only to continuous symmetries. Discrete symmetries like parity or time reversal do not yield conserved quantities via Noether’s theorem. For example, a crystal lattice has discrete translational symmetry but does not conserve crystal momentum in the same sense as continuous translation invariance conserves linear momentum.

Mistake 2: Confusing the conserved quantity with the symmetry generator The conserved quantity $Q = \sum_j p_j \delta q_j - F$ is not the same as the infinitesimal generator of the transformation. The generator is the vector field $\delta q_j$, while $Q$ is the corresponding momentum map. For time translation, the generator is $\dot{q}_j$ but the conserved quantity is the Hamiltonian $H = \sum_j p_j \dot{q}_j - L$.

Mistake 3: Applying Noether’s theorem to systems with explicit time dependence in the Lagrangian When the Lagrangian depends explicitly on time, the action is not invariant under time translations, so energy is not conserved. The quantity $h = \sum_j p_j \dot{q}_j - L$ is still well-defined but is not conserved. Students often assume $h$ is always the energy, but it only equals the conserved energy when $\partial L/\partial t = 0$.

Boost symmetry is the case where the naive statement is wrong, and it is worth working through.

A free particle has $L = \frac12 m \dot x^2$, which is not invariant under a Galilean boost $x \to x + vt$: the kinetic energy picks up $m v \dot x + \frac12 m v^2 t$. But those extra terms are a total time derivative, $\frac{d}{dt}(m v x + \frac12 m v^2 t)$, and Noether’s theorem is stated “up to a boundary term” exactly for this reason.

The conserved quantity is then

$$Q = m\dot x - m v,$$

which is the momentum in the boosted frame. The boost symmetry gives conservation of the velocity of the centre of mass, not a new independent quantity. Getting this wrong — expecting a new conserved quantity from every transformation — is the standard error.

The first row is the one most often asserted wrongly. It is continuous symmetries only, because the proof differentiates with respect to the transformation parameter. Parity is a symmetry and charge conjugation is a symmetry, and neither produces a Noether current.

• Particle physics. The Standard Model Lagrangian is written down by requiring invariance under $SU(3) \times SU(2) \times U(1)$. Each gauge symmetry gives a conserved current, and the associated charges are electric charge, weak isospin and colour.
• General relativity. Killing vectors of a spacetime metric play the role of symmetry generators; a Killing vector gives a conserved quantity along geodesics, which is how energy and angular momentum are defined for orbits around a black hole.
• Integrable systems. The solitons of the Korteweg-de Vries equation carry infinitely many conserved quantities, each from a symmetry of a Lax pair — the reason that equation is exactly solvable.

Mix these with material from Lagrangian mechanics, Hamiltonian mechanics and the symplectic structure page.

1. (Lagrangian.) Show that for a central potential $V(r)$ the Lagrangian is invariant under rotations, and recover conservation of $L_z$ from Noether’s theorem.
2. (Hamiltonian.) Show that if $G$ generates a symmetry then $\{G, H\} = 0$, connecting the Poisson-bracket statement of conservation to Noether’s.
3. (Relativity.) Show that the Lagrangian of a free relativistic particle, $L = -mc^2\sqrt{1 - v^2/c^2}$, is invariant under Lorentz boosts up to a boundary term, and identify the conserved quantity.
4. (Fields.) For a complex scalar field with $L = |\partial\phi|^2 - m^2|\phi|^2$, show that invariance under $\phi \to e^{i\alpha}\phi$ gives a conserved current $j^\mu$, and compute the charge.
5. (Symplectic structure.) Show that a symmetry generated by $G$ preserves the symplectic form, connecting Noether’s theorem to the geometry of phase space.

• Hamiltonian Mechanics: The Hamiltonian formalism expresses conserved quantities as functions of phase space variables.

• Central Force Problems: Central potentials exhibit rotational symmetry, leading to angular momentum conservation via Noether’s theorem.

• Calculus

• Linear Algebra

• Classical Field Theory
• Generalised Coordinates and Constraints
• Hamiltonian Mechanics
• Hamiltonian Mechanics: Advanced Topics