theorem

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

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

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So we know that

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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,

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

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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.

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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.

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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:

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

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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!

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Goodstein Theorem

On reading a magazine on Gödel’s Incompleteness Theorems, I came across a family of sequences of non-negative integers called Goodstein sequences and the Goodstein Theorem involving these sequences.

Goodstein’s thoerem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence converges to 0.

What is a Goodstein Sequence?

To understand what a Goodstein sequence, first we must understand what hereditary base-n notation is. Whilst this notation is very similar to the usual base-n notation, base-n notation is not sufficient for Goodstein’s theorem.

To convert a base-n representation to a hereditary base-n notation, first rewrite all of the exponents in base-n notation. Then rewrite any exponents inside the exponents, and continue this way until every number in the expression has been converted to base-n notation.

For example, 35 = 25 + 2 + 1 in ordinary base-2 notation but Screen Shot 2017-07-11 at 5.37.45 PM.png in hereditary base-2 notation.

Now we can define the Goodstein sequence G(m) of a natural number m. In general the (n+1)-st term G(m)(n+1) of the Goodstein sequence of m is given as follows (taken from Wikipedia):

  • Take the hereditary base-n + 1 representation of G(m)(n).
  • Replace each occurrence of the base-n + 1 with n + 2.
  • Subtract one. (Note that the next term depends both on the previous term and on the index n.)
  • Continue until the result is zero, at which point the sequence terminates.

In spite of the rapid growth of the terms in the sequence, Goodstein’s theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.

Mathematicians Laurie Kirby and Jeff Paris proved in 1982 that Goodstein’s theorem is not provable in ordinary Peano arithmetic. In other words, this is the type of theorem described in 1931 by Gödel’s incompleteness theorem.

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VIDEO: Banach–Tarski Paradox

The Banach-Tarski Paradox is a theorem in geometry which states that:

“It is possible to decompose a ball into five pieces which can be reassembled by rigid motions to form two balls of the same size as the original.”

It was first stated in 1924, and is called a paradox as it contradicts basic geometric intuition.

An alternate version of this theorem tells us that:

“It is possible to take a solid ball the size of a pea, and by cutting it into a finite number of pieces, reassemble it to form a solid ball the size of the sun.”

Below is an awesome video explaining how this paradox works:

 

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Fermat’s Little Theorem

Statement:

Let p be a prime then ap ≡ a (mod p), for any natural number a.

Proof using Modular Arithmetic:

Firstly, we need to discuss Wilson’s theorem. This states:

(p-1)! ≡ -1 (mod p) is p is prime.

We must first prove this theorem:

If p is prime, then 1, 2, …, p-1 are invertible mod p. Now we can pair each of these numbers with its inverse (for example 3 with 4 in mod 11). The only elements that cannot be paired with a different number are 1 and -1, which are self-inverses, as shown below:

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Now (p-1)! is a product of (p-3)/2 inverse pairs together with -1 and 1, whose product is -1.

So (p-1)! ≡ -1 (mod p).

Back to the proof of Fermat’s Little Theorem.

The statement of Fermat’s Little Theorem is equivalent to ap-1 ≡ 1 (mod p) if a ≢ 0 (mod p).

Consider the numbers a, 2a, …., (p-1)a. These are each distinct mod p and so they are congruent to 1, 2, …., (p-1) (mod p) in some order.

 Hence a·2a···(p-1)a ≡ 1·2···(p-1) (mod p).

So ap-1(p-1)! ≡ (p-1)!.

And therefore ap-1 ≡ 1 (mod p).

We can extend this to a≡ a (mod p) as shown below:

When a ≡ 0 (mod p): 0≡ 0 (mod p). So a≡ a (mod p).

When a ≢ 0 (mod p): We can multiply through by a, as a and p are coprime. Then we get a≡ a (mod p), as required.

Hence, we have proved Fermat’s Little Theorem, a very important result in number theory.

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NEWS: 2017 Breakthrough Prize in Mathematics

The Breakthrough Prizes is awarded in three categories: Life Sciences, Fundamental Physics and Mathematics, in recognition of great scientific advance.

This year, the Breakthrough Prize in Mathematics was awarded to Belgian mathematician Jean Bourgain, for his “multiple transformative contributions to analysis, combinatorics, partial differential equations, high-dimensional geometry and number theory”.

He has published, on average, 10 papers a year tackling some of the most challenging problems in a range of mathematical fields. For example, his most recent work is on the decoupling theorem: whilst Pythagoras showed that the length of the two shorter sides of a right-hand triangle are related to the hypotenuse, the decoupling theorem, proven by Bourgain and Ciprian Demeter, shows a similar relationship in the superposition of waves.

“It is of course an immense honour for me to be awarded the Breakthrough Prize and also an occasion to thank all those who helped me along my career. Over the years I have been fortunate to interact with several truly exceptional individuals who introduced me to different subjects and from whom I learned a lot. Collaborations on all levels played an important part in my work and are greatly valued. Appointments at research institutions such as the Institut des Hautes Études Scientifiques in Bures/Yvette (France) and the Institute for Advanced Study in Princeton provided ideal conditions for a full dedication to mathematics. I am most grateful for their trust. Last but not least, thanks to my family for their love and continuous support over the years.” – Jean Bourgain

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