Symmetry and conservation: Noether's theorem

Why is momentum conserved? Why angular momentum, and why energy? In PHYS-244 each conservation law was derived from Newton's laws for a particular kind of force. In 1918 Emmy Noether, working in Göttingen with David Hilbert and Felix Klein on problems arising from general relativity, proved a theorem that answers all three questions at once and explains where conservation laws come from: every continuous symmetry of the action produces a conserved quantity. Translation symmetry gives momentum, rotation symmetry gives angular momentum, and symmetry under shifts in time gives energy. The theorem has become one of the organizing principles of physics, from mechanics to particle physics.

Symmetries of a Lagrangian

A transformation of the coordinates is a symmetry if it leaves the Lagrangian unchanged (or, more generally, changes it only by a total time derivative, which does not affect the equations of motion). We are interested in continuous families of transformations that depend on a parameter \(\varepsilon\) and reduce to the identity at \(\varepsilon = 0\), such as translations by a distance \(\varepsilon\) or rotations by an angle \(\varepsilon\). For small \(\varepsilon\) such a transformation moves each coordinate by

where \(K_j\) describes the direction of the change. For a translation of a single particle along \(x\), \(K = (1, 0, 0)\); for a rotation about the \(z\) axis, \(\delta x = -\varepsilon y\), \(\delta y = \varepsilon x\), so \(K = (-y, x, 0)\).

The theorem

Suppose \(L\) is unchanged to first order: \(\delta L = \sum_j\big(\frac{\partial L}{\partial q_j}\varepsilon K_j + \frac{\partial L}{\partial\dot q_j}\varepsilon\dot K_j\big) = 0\). On a solution of Lagrange's equations replace \(\partial L/\partial q_j\) by \(\frac{d}{dt}\frac{\partial L}{\partial\dot q_j}\). The two terms combine into a single derivative by the product rule:

So the Noether charge

is constant along every solution. If the Lagrangian changes by a total derivative, \(\delta L = \varepsilon\,\frac{d\Lambda}{dt}\), the same argument gives the conserved quantity \(\sum_j p_jK_j - \Lambda\). The theorem converts a symmetry, a statement about what the Lagrangian does not depend on, into a conservation law, a statement about the motion.

The simplest case is a cyclic coordinate. If \(L\) does not depend on \(q_k\), shifting \(q_k\) by \(\varepsilon\) is a symmetry with \(K_k = 1\) and all other \(K_j = 0\), and the conserved quantity is \(p_k\). That is the case-1 first integral of lesson 2. Noether's theorem generalizes it to symmetries that are not simply the shift of one coordinate, which matters because the coordinates in which a symmetry becomes a shift are not always the ones we want to use.

Momentum, angular momentum and energy

Translation. For a system of particles with \(L = \sum_i\tfrac12 m_i\dot{\mathbf r}_i^2 - V\), where \(V\) depends only on differences \(\mathbf r_i - \mathbf r_k\), translating every particle by \(\varepsilon\hat{\mathbf n}\) leaves \(L\) unchanged. Then \(K_i = \hat{\mathbf n}\) for every particle, and \(Q = \sum_i m_i\dot{\mathbf r}_i\cdot\hat{\mathbf n}\), the component of total momentum along \(\hat{\mathbf n}\). Momentum is conserved because space is homogeneous: the laws do not care where the system is.

Rotation. If \(V\) depends only on distances, rotating every particle about an axis \(\hat{\mathbf n}\) by \(\varepsilon\) gives \(\delta\mathbf r_i = \varepsilon\,\hat{\mathbf n}\times\mathbf r_i\), and \(Q = \sum_i m_i\dot{\mathbf r}_i\cdot(\hat{\mathbf n}\times\mathbf r_i) = \hat{\mathbf n}\cdot\sum_i\mathbf r_i\times m_i\dot{\mathbf r}_i\), the component of total angular momentum along \(\hat{\mathbf n}\). Angular momentum is conserved because space is isotropic: the laws do not care which way the system faces.

Time translation. Time is not one of the coordinates, so this case needs a slightly different argument, which we have already met. If \(L\) has no explicit dependence on \(t\), then along a solution

so \(h = \sum_j p_j\dot q_j - L\) is conserved. When the kinetic energy is a quadratic form in the velocities, as it is whenever the coordinates are related to Cartesian ones without explicit time dependence, Euler's theorem on homogeneous functions gives \(\sum_j\dot q_j\,\partial T/\partial\dot q_j = 2T\), and \(h = 2T - (T - V) = T + V\), the energy. Energy is conserved because the laws do not change with time.

Partial symmetries

Most real systems have only some of these symmetries, and Noether's theorem tells you precisely which conservation laws survive. A projectile near the Earth's surface has \(V = mgz\): translations in \(x\) and \(y\) are symmetries, translations in \(z\) are not, so \(p_x\) and \(p_y\) are conserved but \(p_z\) is not. Rotations about the vertical are a symmetry, so \(L_z\) is conserved, but rotations about horizontal axes are not. A charged particle in a uniform magnetic field along \(z\) has a subtler set: in a suitable gauge the Lagrangian depends on \(x\) but changes only by a total derivative under a shift of \(x\), and the conserved quantity is a combination of momentum and position that fixes the center of the circular orbit.

A planet in the Sun's gravity, treated as a fixed center, has full rotational symmetry but not translational symmetry: angular momentum and energy are conserved, momentum is not. In the two-body problem, where both bodies move, translation symmetry returns and the total momentum is conserved; this is what lets PHYS-244 separate the center-of-mass motion.

A numerical experiment

The program below follows a particle moving in a plane under the isotropic oscillator potential \(V = \tfrac12 r^2\), which is symmetric under rotations and time shifts. It monitors the energy and the angular momentum \(L_z\). Then it breaks each symmetry in turn: an extra \(\varepsilon x^3\) term, which singles out the \(x\) direction, and a forcing term \(Ax\cos t\), which depends on time.

import math

def rk4(f, t, y, dt):
    k1 = f(t, y); k2 = f(t + dt / 2, [a + dt / 2 * b for a, b in zip(y, k1)])
    k3 = f(t + dt / 2, [a + dt / 2 * b for a, b in zip(y, k2)]); k4 = f(t + dt, [a + dt * b for a, b in zip(y, k3)])
    return [a + dt / 6 * (p + 2 * q + 2 * r + s) for a, p, q, r, s in zip(y, k1, k2, k3, k4)]

# A particle of unit mass moving in the plane under V = r^2/2 + eps * x^3 + A x cos(t).
#   eps = 0, A = 0: rotation symmetry and time symmetry -> L_z and E conserved
#   eps > 0:       rotation symmetry broken            -> L_z changes, E still conserved
#   A > 0:         time symmetry broken                -> E changes
def run(eps, A, T=30.0, dt=1e-3):
    def f(t, y):
        x, yy, vx, vy = y
        return [vx, vy, -x - 3 * eps * x * x - A * math.cos(t), -yy]
    def E(t, y):
        x, yy, vx, vy = y
        return 0.5 * (vx * vx + vy * vy) + 0.5 * (x * x + yy * yy) + eps * x ** 3 + A * x * math.cos(t)
    Lz = lambda y: y[0] * y[3] - y[1] * y[2]
    y = [1.0, 0.0, 0.0, 0.8]; t = 0.0
    E0, L0 = E(t, y), Lz(y); dE = dL = 0.0
    while t < T - 1e-9:
        y = rk4(f, t, y, dt); t += dt
        dE = max(dE, abs(E(t, y) - E0)); dL = max(dL, abs(Lz(y) - L0))
    return dE, dL
for eps, A, label in ((0.0, 0.0, "rotation and time symmetric"),
                      (0.1, 0.0, "x^3 term breaks rotation   "),
                      (0.0, 0.1, "forcing breaks time shifts ")):
    dE, dL = run(eps, A)
    print(f"{label}: largest change in E = {dE:.1e}, largest change in L_z = {dL:.1e}")

# Noether charge for a translation: two particles with an interaction depending only on x1 - x2
def two(y):
    x1, x2, v1, v2 = y
    F = -4 * (x1 - x2) ** 3                       # force on particle 1 from V = (x1 - x2)^4
    return [v1, v2, F / 1.0, -F / 3.0]            # masses 1 and 3
y = [0.0, 1.5, 2.0, -0.5]; P0 = 1.0 * y[2] + 3.0 * y[3]; worst = 0.0
for _ in range(20000):
    y = rk4(lambda t, s: two(s), 0, y, 1e-3)
    worst = max(worst, abs(1.0 * y[2] + 3.0 * y[3] - P0))
print(f"translation symmetry: total momentum {P0:.4f} kg m/s, largest change {worst:.1e}")

rotation and time symmetric: largest change in E = 8.2e-15, largest change in L_z = 6.9e-15
x^3 term breaks rotation   : largest change in E = 1.2e-14, largest change in L_z = 8.5e-01
forcing breaks time shifts : largest change in E = 1.1e+00, largest change in L_z = 4.0e-02
translation symmetry: total momentum 0.5000 kg m/s, largest change 1.6e-14

With both symmetries intact, energy and \(L_z\) are conserved to about \(10^{-14}\), limited only by rounding. Adding the \(x^3\) term destroys rotational symmetry: \(L_z\) wanders by 0.85, a large fraction of its starting value of 0.8, while the energy is still conserved to \(10^{-14}\), because the potential does not depend on time. Adding the time-dependent forcing destroys conservation of energy (it changes by 1.1, the forcing pumps energy in), and since the forcing also points along \(x\), it breaks rotational symmetry too and \(L_z\) drifts as well. Finally, two particles interacting through a potential that depends only on \(x_1 - x_2\) keep their total momentum fixed to \(10^{-14}\), even though each particle's own momentum swings back and forth.

Beyond mechanics

Noether's theorem extends directly to fields, where a continuous symmetry gives a conserved current and a local conservation law, such as \(\partial\rho/\partial t + \nabla\cdot\mathbf J = 0\) for electric charge. Charge conservation follows from a symmetry of the quantum description of charged particles under a change of phase. In particle physics, conservation of baryon number and of the various kinds of charge are understood the same way, and conversely the discovery of a conserved quantity is taken as a hint of a hidden symmetry.

Noether's theorem also clarifies the status of energy in general relativity. In an expanding universe there is no symmetry under shifts in time, and there is no general law of conservation of total energy: light from distant galaxies loses energy as it is redshifted, and that energy is not stored anywhere. This is not a paradox once one sees energy conservation as a consequence of a symmetry that the expanding universe does not have.

Emmy Noether was not allowed to hold a paid position at Göttingen for years because she was a woman; Hilbert famously objected that a university faculty was not a bathhouse. She received the right to lecture in 1919, became a central figure in modern algebra, and after being dismissed by the Nazi government in 1933 moved to Bryn Mawr College in Pennsylvania, where she died in 1935.