Noether's theorem

Submitted by Kostas Maistrelis on · last edited
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Noether’s theorem states that every continuous symmetry1 of the action2 of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems published in 1918 by the mathematician Emmy Noether (German: Amalie Emmy Noether).

The classic examples

SymmetryWhat it meansConserved quantity
Translation in timethe laws do not change if we perform the same experiment tomorrowEnergy
Translation in spacethe laws do not change if we move the system somewhere elseLinear momentum
Rotation in spacethe laws do not change if we rotate the entire systemAngular momentum

Here lies the turning point: before the theorem, these three conservation laws appeared as separate laws, three independent cases of “that is just what we find.” Noether showed that they are not three things but one, and that they arise from something more fundamental: symmetry.

A more conceptual reading of Noether’s theorem

Let us look at the theorem through the concept of energy:

  • Noether’s theorem is formulated within Lagrangian mechanics.3
  • If Lagrangians do indeed describe the world (and all the evidence suggests that they do), and there is symmetry with respect to time, then the Lagrangian of the system yields a quantity called energy, which must be conserved (energy is the conserved quantity corresponding to the symmetry of the action under translations in time). In this way, we both characterize what energy is and derive the principle of conservation of energy.

Corresponding formulations can also be made for momentum and angular momentum.

  1. Here, symmetry does not mean the shape of an object, but a change that leaves the laws themselves unchanged: you do something to the system and its description does not change at all. “Continuous” means that the change can be made arbitrarily small: you can shift by one minute or by one thousandth of a second, or rotate through any angle. Noether’s proof requires this kind of symmetry because it starts from an infinitesimal change and reaches a conclusion that holds throughout the motion. ↩︎

  2. Action is a concept from Lagrangian mechanics3: a quantity obtained by summing/integrating the Lagrangian over the system’s whole path. The actual path is the one that makes it stationary (or, less precisely, the one that minimizes it). ↩︎

  3. Lagrangian mechanics is an alternative perspective on classical physics: it makes the same predictions as Newtonian mechanics, but begins with energy and paths rather than forces, and generalizes much more readily. The cost is that, beyond differential calculus, it also requires the calculus of variations. ↩︎ ↩︎

Topics: PhysicsEmmy NoetherMathematics

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