• Emmy Noether Noether’s Theorem pixel-art physics sticker, 3" × 3" / White

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Emmy Noether Sticker

Symmetry In. Conservation Out. | Physics

\[Q = \sum_i \frac{\partial L}{\partial \dot q_i}\,\delta q_i\]
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Symmetry in, conservation out: one of the deepest results in physics. Emmy Noether’s 1918 theorem: every continuous symmetry of physical law gives a conserved quantity, like energy or momentum. A pixel-art portrait on a 3" × 3" sticker; the physics panel below has the equation, every symbol and the sources.

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3″ × 3″ kiss-cut sticker.

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The physics

Noether's theorem says that every continuous symmetry of a system's laws comes with a conserved quantity. Symmetry under shifts in time gives conservation of energy, shifts in space give momentum, and rotations give angular momentum.

The sticker says Symmetry In. Conservation Out. That is Emmy Noether's theorem, drawn as a vending machine.

In 1918, in Göttingen, Noether proved that every continuous symmetry of a system's laws comes with a conserved quantity. If the laws don't change when you shift the clock, energy is conserved. If they don't change when you slide everything sideways, momentum is conserved. Turn the whole set-up through an angle and angular momentum is conserved. The three cans in the machine are exactly those three trades.

Before Noether, physicists found conservation laws one at a time and trusted them because experiments kept agreeing. After her, you go looking for the symmetry, and the conservation law falls out of the mathematics. The formulas above are the mechanical version: if nudging the coordinates leaves the Lagrangian unchanged, the quantity Q stays fixed as the system moves.

Her 1918 paper is far more general than the table on the machine. It covers fields as well as particles, and a second theorem deals with symmetries that can vary from place to place, the kind found in general relativity and in modern gauge theories. Charge conservation is now traced to a symmetry of the quantum phase. That is a later application of her result, not something she wrote down.

She did this work while the University of Göttingen would not let a woman habilitate, the qualification needed to lecture in her own name. For years her courses were advertised under David Hilbert's name. She was finally allowed to habilitate in 1919.

Press θ. Collect L.

Equations

\[Q = \sum_i \frac{\partial L}{\partial \dot q_i}\,\delta q_i\]
\[\delta L = 0 \;\Rightarrow\; \frac{dQ}{dt} = 0\]
\[\begin{gathered} E = \sum_i \dot q_i\,\frac{\partial L}{\partial \dot q_i} - L \\ \frac{\partial L}{\partial t} = 0 \;\Rightarrow\; \frac{dE}{dt} = 0 \end{gathered}\]
\[\begin{aligned} t \to t+\epsilon \;&\Rightarrow\; E \text{ conserved} \\ \vec x \to \vec x+\vec\epsilon \;&\Rightarrow\; \vec p \\ \theta \to \theta+\epsilon \;&\Rightarrow\; \vec L \end{aligned}\]
SymbolMeaningUnit
\(Q\) conserved Noether charge (momentum for a shift, angular momentum for a rotation) \(\mathrm{kg·m/s, J·s}\)
\(L\) Lagrangian \(\mathrm{J}\)
\(q_i\) generalised coordinates (lengths or angles) \(\mathrm{m, rad}\)
\(\dot q_i\) generalised velocities \(\mathrm{m/s, rad/s}\)
\(\epsilon\) small symmetry parameter (a length for a shift, an angle for a rotation, a time for a clock shift) \(\mathrm{m, rad, s}\)
\(\delta q_i\) direction of the symmetry shift (its size is carried by ε) \(\mathrm{1}\)
\(\delta L\) change in the Lagrangian under the shift \(\mathrm{J}\)
\(t\) time \(\mathrm{s}\)
\(E\) energy \(\mathrm{J}\)
\(\vec x\) position \(\mathrm{m}\)
\(\vec p\) linear momentum \(\mathrm{kg·m/s}\)
\(\theta\) rotation angle \(\mathrm{rad}\)
\(\vec L\) angular momentum \(\mathrm{J·s}\)

Sources

  1. Noether (1918) Invariante Variationsprobleme, Nachr. Ges. Wiss. Göttingen 235–257; English translation by M. A. Tavel
  2. MacTutor — Emmy Noether biography (habilitation refused until 1919; courses listed under Hilbert's name)
  3. Wikipedia — Noether's theorem

Source Noether, E. (1918). Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 235–257.

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