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.*

Given

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

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.

### Why?

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.

M x