Noether's theorem

Noether's theorem is a mathematical result which relates continuous symmetries and conserved quantities in mechanical systems. For instance, physical laws are independent of location; this symmetry implies that the momentum is constant. Noether's theorem was published by Emmy Noether in 1918.

Informal description

There are several formulations of Noether's theorem, but the basic principle is always the same: continuous symmetries correspond to conserved quantities in variational systems. We see that three concepts have to be explained before we can understand the result. What are continuous symmetries, conserved quantities and variational systems?

Of these, conserved quantity is the easiest to understand. They are numbers associated to the system under study which do not change in time, even though the components of the system may move around. In our example, the conserved quantity is momentum. The momentum of an object is the product of its mass and its velocity, while the momentum of a system is the sum of the momenta of all objects that comprise the system.

Let us now have a look at continuous symmetry. In science, a symmetry of a system means that it behaves the same under some transformation. For instance, it does not matter if a marble rolls from left to right or from right to left on a flat table; that is why the world still looks natural if you look in a mirror. This is an example of a discrete symmetry, because there are only two possibilities: you exchange left or right, or you do not. An example of a continuous symmetry is that the marble behaves the same, no matter what position it starts from. This is called translational invariance, and it covers infinitely many transformations: shifting the system one millimeter to the left, or an inch to the right, and so on. Noether's law only works for continuous symmetries.

The final concept is that of a variational system. This is the most abstract concept, but the important point is that most mechanical systems fall within the class of variational systems. These systems satisfy a principle of least action, which means that the system evolves in such a way that its action is minimal (or, more precisely, stationary). For instance, Fermat's principle states that light takes that path which it can transverse in the least time.

Roughly, Noether's theorem says that if a variational system allows a continuous symmetry, then the corresponding quantity is conserved. A precise explanation of Noether's theorem requires the precise definitions of the key concepts of conserved quantity, continuous symmetry, and variational system.

References

• V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, Berlin, 1989. ISBN 978-0-387-96890-2.
• E. Noether, "Invariante Variationsprobleme". Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, pp. 235–257, 1918 (in German).
• P. J. Olver, Applications of Lie groups to differential equations. Springer-Verlag, Berlin, 1986. ISBN 978-0-387-96250-4.