Today I thought I’d write a blog post about an interesting theorem I learnt whilst studying my Variational Principles module – Noether’s Theorem.
To understand Noether’s Theorem, we must first understand what is meant by a symmetry of a functional.
suppose we change the variables by the transformation t –> t*(t) and x –> x*(t) to obtain a new independent variable and a new function. This gives
where α* = t*(α) and β* = t*(β).
If F*[x*] = F[x] for all x, α and β, then this transformation * is called a symmetry.
What is a continuous symmetry?
Intuitively, a continuous symmetry is a symmetry that we can do a bit of. For example, a rotation is a continuous symmetry, but a reflection is not.
Noether’s Theorem – proven by mathematician Emmy Noether in 1915 and published in 1918 – states that every continuous symmetry of F[x] the solutions (i.e. the stationary points of F[x]) will have a corresponding conserved quantity.
Consider symmetries that involve only the x variable. Then, up to first order, the symmetry can be written as:
t –> t, x(t) –> x(t) + εh(t)
where h(t) represents the symmetry transformation. As the transformation is a symmetry, we can pick ε to be any small constant number and F[x] does not change, i.e. δF = 0. Also, since x(t) is a stationary point of F[x], we know that if ε is any non-constant, but vanishes at the end-points, then we have δF = 0 again. Combining these two pieces of information, we can show that there is a conserved quantity in the system.
For now, do not make any assumptions about ε. Under the transformation, the change in F[x] is given by
Firstly, consider the case where ε is constant. Then the second integral vanishes and we obtain
So we know that
Now, consider a variable ε that is not constant, but vanishes at the endpoints. Then, as x is a solution, we must have that δF = 0. Therefore,
If we integrate the above expression by parts, we get that
Hence the conserved quantity is:
Not all symmetries involve just the x variable, for example we may have a time translation, but we can encode this as a transformation of the x variable only.