Noether's Theorem, As I Understand It

Noether's Theorem, As I Understand It 6 July 2020

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.

Lagrangian Mechanics

Starting with Lagrangian mechanics and the definition of the Lagrangian,

\mathcal{L} = T - V

where $T$ is the kinetic energy of the system and $V$ is the potential energy of the system. The action of the system is then defined to be

S = \int_{t_1}^{t_2} \mathcal{L}(x, \dot{x}, t) dt

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 $\delta S = 0$ ,

\delta S = \int_{t_1}^{t_2} \delta \mathcal{L}(x, \dot{x}, t) dt = 0. \qquad \text{(1)}

Using the chain rule, $\delta \mathcal{L}$ can be represented by

\delta \mathcal{L} = \frac{\partial \mathcal{L}}{\partial x} \delta x + \frac{\partial \mathcal{L}}{\partial \dot{x}} \delta \dot{x}.

Note that $\delta x$ and $\delta \dot{x}$ are small changes in functions rather than small changes in a variable. Both are functions of time. $\delta x$ can be written as

\delta x(t) = x(t) - x'(t)

where $x(t)$ and $x'(t)$ are functions that have the same start and end points but take a different path, ie $\delta x(t_1) = \delta x (t_2) = 0$ . Similarly $\dot{x}$ can be written as

\begin{aligned} \delta \dot{x}(t) &= \delta \frac{dx(t)}{dt} \\ &= \frac{dx(t)}{dt} - \frac{dx'(t)}{dt} \\ &= \frac{d}{dt} \left( x(t) - x'(t) \right) \\ &= \frac{d}{dt} \delta x(t). \end{aligned}

All of these expressions can be used in equation 1,

\delta S = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial x} \delta x + \frac{\partial \mathcal{L}}{\partial \dot{x}} \frac{d}{dt} \delta x \right) dt = 0.

At this point $\delta x$ 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,

\int_{t_1}^{t_2} \frac{\partial \mathcal{L}}{\partial \dot{x}} \frac{d}{dt} \delta x dt = \left[ \frac{\partial \mathcal{L}}{\partial \dot{x}} \delta x \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}} \right) \delta x dt.

But recall the definition of $x(t)$ and $x'(t)$ , they have the same start and end points. The first term therefore is zero. Substituting this back in gives

\delta S = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt} \left(\frac{\partial \mathcal{L}}{\partial \dot{x}}\right) \right) \delta x dt = 0

where we have factorised out the $\delta x$ . Since $\delta x$ can be any arbitrary function, the only way for this to be zero in general is for the section in brackets to be zero

\frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{x}}\right) = 0.

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,

\mathcal{L} = T - V = \frac{1}{2} m \dot{x}^2 - V

using the expression for kinetic energy of a single particle at non relativistic speeds. Using this in the Euler-Lagrange equation gives

\frac{\partial \left(-V\right)}{\partial x} = \frac{d}{dt}\left(\frac{\partial \left(\frac{1}{2}m \dot{x}^2 \right)}{d \dot{x}}\right).

Resolving the left hand side requires knowing that $\frac{\partial V}{\partial x} = -F$ , and the right hand side is following through the differentiation,

F = m \ddot{x} = ma.

The well known result of Newton’s Second Law.

Noether’s Theorem

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, $x(t)$ , that does not change the action, $\delta S = 0$ . It is of the general form

\delta x = x(x) - x'(x) = g(x, \dot{x}).

Unlike previously, this $\delta x$ does not need to be zero at the start and end points. $\delta \dot{x}$ is the time derivative of $\delta x$ ,

\delta \dot{x} = \frac{d}{dt} \delta x.

Firstly we consider the variation of the action of the system to be zero,

\delta S = \int_{t_1}^{t_2} \delta \mathcal{L} dt = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial x} \delta x + \frac{\partial \mathcal{L}}{\partial \dot{x}} \frac{d}{dt} \delta x \right) dt = 0.

Then, like before, we rewrite the second term in the integral using integration by parts,

\int_{t_1}^{t_2} \frac{\partial \mathcal{L}}{\partial \dot{x}} \frac{d}{dt} \delta x dt = \left[ \frac{\partial \mathcal{L}}{\partial \dot{x}} \delta x \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}} \right) \delta x dt.

Which is then substituted back in giving

\delta S = \int_{t_1}^{t_2} \left( \frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt} \left(\frac{\partial \mathcal{L}}{\partial \dot{x}}\right) \right) \delta x dt + \left[ \frac{\partial \mathcal{L}}{\partial \dot{x}} \delta x \right]_{t_1}^{t_2} = 0.

Since the system follows a path described by the Euler-Lagrange equation, the first term must be zero as previously shown,

\delta S = \left[ \frac{\partial \mathcal{L}}{\partial \dot{x}} \delta x \right]_{t_1}^{t_2} = \left[ \frac{\partial \mathcal{L}}{\partial \dot{x}} g(x, \dot{x}) \right]_{t_1}^{t_2} = 0. \qquad \text{(2)}

We can now define a value $Q$ to be

Q = \frac{\partial \mathcal{L}}{\partial \dot{x}} g(x, \dot{x}).

Since, by equation 2, $Q(t_1) - Q(t_2) = 0$ with $t_1$ and $t_2$ being arbitrary, $Q$ must be independent of time. Therefore for every symmetry there must be a conserved quantity.