In the spirit of previous work in abstract algebra, I have, erm, adapted another nursery rhyme.
After all, the songs are so common and commonly known; why not update them with some definite content?
To the tune of “Mary had a little lamb” (with no disrespect to the original, which seems to be an endearing story of an actual lamb), a discussion of Noether’s theorem.
If you haven’t heard of Noether’s theorem, it is very nice. (It should be distinguished from several other theorems of Emmy Noether, and indeed other mathematical Noethers.)
Roughly speaking, Noether’s theorem states that whenever a physical system has a nice symmetry, there is always some numerical quantity which is conserved along with it.
For instance, if a physical system is invariant under translation, then there is a conserved quantity associated to it, known as momentum. (And there are translations in three independent directions in space, so there are three components of momentum which are conserved. In other words, momentum as a vector quantity is conserved.) Similarly, if it’s invariant under rotations, then there is a conserved quantity known as angular momentum. Invariant under moving forward and backward in time — a conserved quantity known as energy. And so on.
This is not very precise, and there are different ways of formulating it, and of course physicists and mathematicians have different perspectives about it — as well as the level of mathematical precision and rigour with which it should be stated and understood.
The wikipedia page, at least at the time of writing, has a very physics-oriented discussion, which would offend many mathematicians’ sensibilities — certainly including my own. The nicest mathematical formulation uses symplectic geometry, and hence some fairly serious prerequisite knowledge, well beyond the Australian undergraduate curriculum. (Unless you take an undergraduate research unit with me at Monash, perhaps!)
A good discussion may be found in the lecture notes of Ana Cannas da Silva, available online here. Once enough machinery is developed to state the principle cleanly (um, in section 24 on page 147…), the theorem is proved in a leisurely half a dozen lines.
Anyway, less talk, more nursery rhymes!
Emmy had a theorem,
Emmy had a theorem
Its proof was clear as day.
Everywhere a symmetry,
Everywhere a symmetry
A conserved quantity.