A simplified, but not yet simple, statement of Noether’s theorem is that for every global symmetry there is an associated conserved quantity.
It is this property that I will aim to demonstrate by the use of the Lagrangian formalism of mechanics.
Starting with Lagrangian mechanics and the definition of the Lagrangian,
where is the kinetic energy of the system and is the potential energy of the system.
The action of the system is then defined to be
and the principle of least action then tells us that the equations of motion of a system are determined by minimising the action, by finding a point where ,
Using the chain rule, can be represented by
Note that and are small changes in functions rather than small changes in a variable.
Both are functions of time.
can be written as
where and are functions that have the same start and end points but take a different path, ie .
Similarly can be written as
All of these expressions can be used in equation 1,
At this point cannot be factorised out due to having a time derivative taken of it, so the second term in the integral is reordered using integration by parts,
But recall the definition of and , they have the same start and end points. The first term therefore is zero.
Substituting this back in gives
where we have factorised out the .
Since can be any arbitrary function, the only way for this to be zero in general is for the section in brackets to be zero
This is known as the Euler-Lagrange equation.
Newton’s Second Law
I will now take a moment to use the Euler-Lagrange equation and obtain the Newtonian equation of motion.
Firstly using the definition of the Lagrangian,
using the expression for kinetic energy of a single particle at non relativistic speeds.
Using this in the Euler-Lagrange equation gives
Resolving the left hand side requires knowing that , and the right hand side is following through the differentiation,
The well known result of Newton’s Second Law.
Now on to the main attraction, a demonstration of Noether’s theorem.
Using the definition given at the start, “for every global symmetry there is an associated conserved quantity”, we will first look at what a symmetry is.
A symmetry is a transformation of the path, , that does not change the action, .
It is of the general form
Unlike previously, this does not need to be zero at the start and end points.
is the time derivative of ,
Firstly we consider the variation of the action of the system to be zero,
Then, like before, we rewrite the second term in the integral using integration by parts,
Which is then substituted back in giving
Since the system follows a path described by the Euler-Lagrange equation, the first term must be zero as previously shown,
We can now define a value to be
Since, by equation 2, with and being arbitrary, must be independent of time.
Therefore for every symmetry there must be a conserved quantity.