Noether’s Theorem: Symmetry, Conservation, and Equivariant Networks
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The statement, and why it is not obvious
Conservation laws look like separate empirical facts: energy is conserved, momentum is conserved, angular momentum is conserved. Noether (1918) showed they are the same fact three times, each attached to a symmetry of the action.
Take the Lagrangian formulation and suppose the transformation $q \to q + \varepsilon K(q)$ leaves $L$ unchanged to first order in $\varepsilon$. Expanding,
Along a solution, Euler–Lagrange lets us substitute $\partial L/\partial q = \frac{d}{dt}\big(\partial L/\partial\dot q\big)$, and the two terms assemble into a single derivative:
So $Q = \dfrac{\partial L}{\partial \dot q}K(q) = p\,K(q)$ is conserved. That is the whole theorem for the simplest case: an invariance of $L$ plus the equation of motion equals a constant of the motion.
The three canonical cases
Space translation $\to$ momentum. Suppose $L$ does not depend on $x$ (free space looks the same everywhere). Then $K = 1$ for the shift $x \to x + \varepsilon$, and $Q = \partial L/\partial\dot x = p$. Linear momentum is conserved. Equivalently, $\partial L/\partial x = 0$ makes $\dot p = 0$ directly.
Rotation $\to$ angular momentum. An infinitesimal rotation about the axis $\hat n$ acts as $\delta\mathbf r = \varepsilon\,\hat n\times\mathbf r$, so $K = \hat n\times\mathbf r$. Then
\[Q = \mathbf p\cdot(\hat n\times\mathbf r) = \hat n\cdot(\mathbf r\times\mathbf p) = \hat n\cdot\mathbf L,\]using the cyclic property of the scalar triple product. The component of angular momentum along the symmetry axis is conserved — only that component, which is why a system symmetric about one axis conserves one component and not the vector.
Time translation $\to$ energy. This case needs its own argument because the transformation shifts $t$ rather than $q$. If $\partial L/\partial t = 0$ then
\[\frac{dL}{dt} = \frac{\partial L}{\partial q}\dot q + \frac{\partial L}{\partial \dot q}\ddot q = \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q}\right)\dot q + \frac{\partial L}{\partial \dot q}\ddot q = \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q}\dot q\right),\]again using Euler–Lagrange in the middle step. Rearranging, $\frac{d}{dt}\big(p\dot q - L\big) = 0$. The conserved quantity is the Hamiltonian $H = p\dot q - L$ — the energy.
| symmetry of the action | conserved quantity |
|---|---|
| $t \to t + \varepsilon$ | energy $H = p\dot q - L$ |
| $x \to x + \varepsilon$ | linear momentum $p = \partial L/\partial\dot x$ |
| rotation about $\hat n$ | angular momentum $\hat n\cdot(\mathbf r\times\mathbf p)$ |
Gauge symmetry, briefly
A gauge symmetry is not a symmetry relating distinct physical states; it is a redundancy in how states are labelled. The electromagnetic potential can be shifted by \(A_\mu \to A_\mu + \partial_\mu\lambda(x)\) with an arbitrary function $\lambda$ that varies from point to point, and the observable fields $\mathbf E$ and $\mathbf B$ do not change at all. Two configurations related by a gauge transformation are the same physical situation described twice.
The productive move is to demand such a local invariance and see what it forces. Requiring invariance under a position-dependent phase rotation of a charged field forces the introduction of a connection field to compensate the derivative — and that field is the electromagnetic potential. Interactions are generated by insisting on a local redundancy. The group-theoretic scaffolding behind all of this is developed in symmetry and groups.
Equivariance in neural networks
A map $f$ is equivariant to a group $G$ acting on inputs and outputs if
with $\rho$ the relevant representations. Invariance is the special case where \(\rho_{\text{out}}\) is trivial. Convolution is equivariant to translation; graph networks to node permutation; E(3)-equivariant networks for molecules to rotation, translation and reflection.
The benefit is structural. An unconstrained model must spend capacity and data learning that a rotated molecule has the same energy; an equivariant model cannot represent a function that violates this, so it effectively learns on the quotient of the input space by the group. Elesedy & Zaidi (2021) prove a strict generalisation gain for equivariant models in a linear setting, quantifying the intuition that the symmetry directions have been removed from the hypothesis space rather than merely discouraged.
The same logic gives energy-conserving dynamics models. Hamiltonian Neural Networks (Greydanus et al., 2019) parameterise $H$ with a network and obtain the vector field from $\dot q = \partial H/\partial p$, $\dot p = -\partial H/\partial q$; energy conservation is then a consequence of the parameterisation rather than a loss term that competes with fitting accuracy.
Recap
- Noether: a continuous symmetry of the action gives a conserved \(Q = \frac{\partial L}{\partial \dot q}K\), derived by substituting Euler–Lagrange into \(\delta L = 0\).
- Time translation gives energy, space translation gives momentum, rotation about \(\hat n\) gives \(\hat n\cdot(\mathbf r\times\mathbf p)\) — one component, not the whole vector.
- Discrete symmetries are outside the theorem; so are symmetries of the equations that are not symmetries of the action.
- Gauge symmetry is a local labelling redundancy; demanding local invariance generates the interaction field.
- Equivariance, \(f(\rho_{\text{in}}(g)x)=\rho_{\text{out}}(g)f(x)\), is a hard constraint on the function class — exact everywhere, unlike augmentation.
References
- Noether, E. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235–257, 1918. English translation: arXiv:physics/0503066.
- Cohen, T., & Welling, M. Group Equivariant Convolutional Networks. ICML 2016.
- Elesedy, B., & Zaidi, S. Provably Strict Generalisation Benefit for Equivariant Models. ICML 2021.
- Greydanus, S., Dzamba, M., & Yosinski, J. Hamiltonian Neural Networks. NeurIPS 2019.
- Satorras, V. G., Hoogeboom, E., & Welling, M. E(n) Equivariant Graph Neural Networks. ICML 2021.