Noether’s Theorem

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

Screen Shot 2017-08-08 at 11.45.48 AM.png

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 givesScreen Shot 2017-08-08 at 11.45.51 AM.png

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 ofFor example, a rotation is a continuous symmetry, but a reflection is not. 

Noether’s Theorem

Noether.jpg

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

Screen Shot 2017-08-08 at 11.57.37 AM.png

Firstly, consider the case where ε is constant. Then the second integral vanishes and we obtain

Screen Shot 2017-08-08 at 11.58.21 AM.png

Screen Shot 2017-08-08 at 11.58.38 AM.png

So we know that

Screen Shot 2017-08-08 at 11.58.41 AM.png

Now, consider a variable ε that is not constant, but vanishes at the endpoints. Then, as is a solution, we must have that δF = 0. Therefore,

Screen Shot 2017-08-08 at 12.00.00 PM.png

If we integrate the above expression by parts, we get that

Screen Shot 2017-08-08 at 12.00.03 PM.png

Hence the conserved quantity is:

Screen Shot 2017-08-08 at 12.01.56 PM.png

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

MATHS BITE: Shoelace Theorem

The Shoelace theorem is a useful formula for finding the area of a polygon when we know the coordinates of its vertices. The formula was described by Meister in 1769, and then by Gauss in 1795.

Formula

Let’s suppose that a polygon P has vertices (a1, b1), (a2, b2), …, (an, bn), in clockwise order. Then the area of P is given by

\[\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|\]

The name of this theorem comes from the fact that if you were to list the coordinates in a column and mark the pairs to be multiplied, then the image looks like laced-up shoes.

Screen Shot 2017-08-04 at 11.59.29 AM.png

Proof

(Note: this proof is taken from artofproblemsolving.)

Let $\Omega$ be the set of points that belong to the polygon. Then

\[A=\int_{\Omega}\alpha,\]

where $\alpha=dx\wedge dy$.

Note that the volume form $\alpha$ is an exact form since $d\omega=\alpha$, where

\[\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}\]

Substitute this in to give us

\[\int_{\Omega}\alpha=\int_{\Omega}d\omega.\]

and then use Stokes’ theorem (a key theorem in vector calculus) to obtain

\[\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.\]

where

$\partial \Omega=\bigcup A(i)$

and $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$, i.e. Screen Shot 2017-08-04 at 12.05.20 PM.png is the boundary of the polygon.

Next we substitute for $\omega$:

\[\sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}.\]

Parameterising this expression gives us

\[\frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}.\]

Then, by integrating this we obtain

\[\frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- (y_{i}+y_{i+1})(x_{i+1}-x_i)].\]

This then yields, after further manipulation, the shoelace formula:

\[\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).\]

M x

Proof Without Words #3

Proof of the identity

Screen Shot 2017-07-30 at 5.34.22 PM.png

HCfGOYp.gif


dkoCZ.png

The figure for general n is similar, with n right pyramids, one with an (n-1)-cube of side length xk as its base and height xk for each k=1,….,n.


The derivative of sin is cosine.

g4n8SZX.jpg



From ‘Proof without words‘ by Roger Nelsen

Screen Shot 2017-07-30 at 5.47.56 PM.png

By Sidney H. Kung

Screen Shot 2017-07-30 at 5.48.12 PM.png

By Shirley Wakin


Previous ‘Proof Without Words‘: Part 1 | Part 2

M x

Maths Bite: Impossible cube

The impossible cube was invented by M.C. Escher for his 1958 print Belvedere. It is based on the Necker cube, and seems to defy the rules of geometry; on the surface resembles a perspective drawing of a 3D cube, however its features are drawn inconsistently from the way they would be in an actual cube.

The impossible cube draws upon the ambiguity present in a Necker cube illustration, in which a cube is drawn with its edges as line segments, and can be interpreted as being in either of two different three-dimensional orientations. – Wikipedia

impossiblecube.jpg

Source: kidsmathgamesonline

How would this cube look like in real life? The below video attempts to demonstrate that.

M x

New Podcast!

Today is a quick post to let you know that the writer of one of my favourite blogs Roots of Unity, Evelyn Lamb, and the mathematician Kevin Knudson from the University of Florida have created a new podcast called ‘My Favourite Theorem‘!

Lamb says:

In each episode, logically enough, we invite a mathematician on to tell us about their favourite theorem. Because the best things in life are better together, we also ask our guests to pair their theorem with, well, anything: wine, beer, coffee, tea, ice cream flavours, cheese, favourite pieces of music, you name it. We hope you’ll enjoy learning the perfect pairings for some beautiful pieces of math.

Click here to listen to the first episode, which features Lamb and Knudson telling us about their favourite theorems!

M x

MATHS BITE: The Kolakoski Sequence

The Kolakoski sequence is an infinite sequence of symbols {1,2} that is its own “run-length encoding“. It is named after mathematician Willian Kolakoski who described it in 1965, but further research shows that it was first discussed by Rufus Oldenburger in 1939.

This self-describing sequence consists of blocks of single and double 1s and 2s. Each block contains digits that are different from the digit in the preceding block.

To construct the sequence, start with 1. This means that the next block is of length 1. So we require that the next block is 2, giving the sequence 1, 2. Continuing this infinitely gives us the Kolakoski sequence: 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, etc.

M x

2017 LMS Winners

The London Mathematics Society (LMS) has announced the winners for their various prizes and medals this year. In this blogpost I will give a breakdown of who won.

  • Polya Prize: This was awarded to Professor Alex Wilkie FRS from the University of Oxford due to his contributions to model theory and its connections to real analytic geometry.
  • Senior Whitehead Prize: Professor Peter Cameron, from the University of St Andrews, was awarded this prize for his research on combinatorics and group theory.
  • Senior Anne Bennett Prize: Awarded to Professor Alison Etheridge FRS from the University of Oxford for her “research on measure-valued stochastic processes and applications to population biology” as well as outstanding leadership.
  • Naylor Prize and Lectureship: This was given to Professor John Robert King from the University of Nottingham due to profound contributions to the theory of non-linear PDEs and applied mathematical modelling.
  • Berwick Prize: This was awarded to Kevin Costello of the Perimeter Institute in Canada for his paper entitled The partition function of a topological field theory (published in the Journal of Topology in 2009). In this paper Costello “characterises the function as the unique solution of a master equation in a Fock space.”
  • Whitehead Prize:
    • Dr Julia Gog (University of Cambridge) for her research on the mathematical understanding of disease dynamics, in particular influenza.
    • Dr András Máthé (Univeristy of Warwick) due to his insights into problems in the fields of geometric measure theory, combinatorics and real analysis.
    • Ashley Montanaro (University of Bristol) for her contributions to quantum computation and quantum information theory.
    • Dr Oscar Randal-Williams (University of Cambridge) due to his contributions to algebraic topology, in particular the study of moduli spaces of manifolds.
    • Dr Jack Thorne (University of Cambridge) for research in number theory, in particular the Langlands program.
    • Professor Michael Wemyss (University of Glasgow) for his “applications of algebraic and homological techniques to algebraic geometry.”

M x