Symmetry for Dummies: Noether’s Theorem

Einstein referred to her as the most important woman in the history of mathematics. Her theorem has been recognized as “one of the most important mathematical theorems ever proved in guiding the development of modern physics.” Yet many people haven’t the slightest clue of who this woman was, or what she did that was so significant to our understanding of how our world works. If you count yourself as one of those who have never heard of Emmy Noether and wish to enlighten yourself, please read on. I can only hope I do her memory justice. Not just by telling you who she was, but by also giving you an understanding of how her insight led to the coming together of symmetry and quantum theory, pointing academia’s arrow toward quantum electrodynamics.

N_01
Emmy Noether

Being a female in Germany in the late 1800s was not easy. She wasn’t allowed to register for math classes. Fortunately, her father happened to be a math professor, which allowed her to sit in on many of his classes. She took one of his final exams in 1904 and did so well that she was granted a bachelors degree. This allowed her to “officially” register in a math graduate program. Three years later, she earned one of the first PhD’s given to a woman in Germany. She was just 25 years old.

1907 was a very exciting time in theoretical physics, as scientists were hot on the heels of figuring out how light and atoms interact with each other. Emmy wanted in on the fun, but being a woman made this difficult. She wasn’t allowed to hold a teaching position, so she worked as an unpaid assistant, surviving on a small inheritance and under-the-table money that she earned sitting in for male professors when they were unable to teach. She was still able to do what professors are supposed to do, however – write papers. In 1916, she would pen the theorem that would have her rubbing shoulders with the other physics and mathematical giants of the era.

Noether’s Theorem – The Basics

Emmy Noether’s Theorem seems simple on the onset, but holds a fundamental truth that explains the fabric of our reality. It goes something like this:

For every symmetry, there is a corresponding conservation law.

We all have heard of laws such as Newton’s first law of motion, which is about the conservation of momentum. And the first law of thermodynamics, which is about the conservation of energy. Noether’s theorem tells us that there must be some type of symmetry that is related to these conservation laws. Before we get into the meaning, we must first understand a little known subject called The Principle of Least Action.

The Universe is Lazy

N_02I would wager a few Raspberry Pi Zeros that many of you already have an intuitive grasp of this principle, even if you’ve never heard of it before now. The principle of least action basically says that the universe has figured out the easiest way possible to get something done. Mathematically, it’s the sum over time of kinetic energy minus potential energy as the action occurs. Let us imagine that you’re trying to program an STM32 Discovery eval board in GCC. After about the 6,000th try, you toss the POS across the room and grab your trusty Uno. The graph depicts the STM32 moving through time and space.

 

The green points represent particular points of how how high the STM32 is at a given point in time. Note that there are no values for height and time – this example is meant to explain a principle. We can say that at these points (and all points along the curve), the SMT32 has both kinetic and potential energies. Let us call the kinetic energy (kt) and the potential energy (pt). The ‘t‘ subscript is for time, as both the energies are functions of time. The action for each point will be called s, and can be calculated as:

s = k_t-p_t

However, action is the total sum of the difference of energies at each point between t1 and t2. If you’ve read my integral post, you will know that we need to integrate in order to calculate the total action.

S = \int_{t_1}^{t_2} (k - p) dt

Now before you get your jumper wires in a bunch, all that is saying is that we’re taking the difference in potential (p) and kinetic (k) energies at each point along the curve between t1 and t2, and we’re adding them together. The elongated S symbol means a sum, and the (dt) means as it changes over time. The path that the STM32 will take will be the path where the action S is at its minimum value. Check out the video in the source section below if you’re confused. It’s only 10 minutes and goes into this concept in easy to follow details.

Noether’s Theorem – The Details

N_03Noether’s theorem is based upon a mathematical proof. It’s not a theory. Her proof can be applied to physics to develop theories, however. Now that we know what the principle of least action is, we can do just this.

Any law of nature can be traced back to a symmetry and the least action principle. Let’s consider two very simple examples – Newton’s first law of motion and the first law of thermodynamics.

Conservation of Momentum

Space has what is known as translational symmetry. That’s just fancy-pants talk for saying that what you do in one point in space is the same as what you do in another point in space. It doesn’t matter what hacker space you throw your STM32, it will act the same at all hacker spaces on earth. Space itself provides the symmetry. And because the principle of least action applies, you have a natural law – the first law of motion.

Conservation of Energy

Time has the same translational symmetry as space does. If I toss the STM32 now, and toss it tomorrow, it will act the same. It doesn’t matter what point in time I toss it, the results will always be the same. Thus energy is conserved between different points in time. Time is our symmetry, and the 1st law of thermodynamics is the result.

Now, I realize these examples might seem a bit useless. But when you dig a bit deeper, things get interesting. Electrical charge is also conserved. Noether says there must then be some type of symmetry involved. What do you suppose that symmetry might be? Keep following that rabbit hole, and you’ll end up face to face with QED. We’ll get there in a future article, so for now just keep Noether’s Theorem in mind.

Sources

Physics Helps, The principle of least action, video link.

Ransom Stephens, Ph.D., Emmy Noether and The Fabric of Reality, video link

Prime Numbers are Stranger than You Thought

If you’ve spent any time around prime numbers, you know they’re a pretty odd bunch. (Get it?) But it turns out that they’re even stranger than we knew — until recently. According to this very readable writeup of brand-new research by [Kannan Soundararajan] and [Robert Lemkein], the final digits of prime numbers repel each other.

More straightforwardly stated, if you pick any given prime number, the last digit of the next-largest prime number is disproportionately unlikely to match the final digit of your prime. Even stranger, they seem to have preferences. For instance, if your prime ends in 3, it’s more likely that the next prime will end in 9 than in 1 or 7. Whoah!

Even spookier? The finding holds up in many different bases. It was actually first noticed in base-three. The original paper is up on Arxiv, so go check it out.

This is a brand-new finding that’s been hiding under people’s noses essentially forever. The going assumption was that primes were distributed essentially randomly, and now we have empirical evidence that it’s not true. What this means for cryptology or mathematics? Nobody knows, yet. Anyone up for wild speculation? That’s what the comments section is for.

(Headline photo of researchers Kannan Soundararajan and Robert Lemke: Waheeda Khalfan)

Bone Up on Your Multiplication Skills

John Napier was a Scottish physicist, mathematician, and astronomer who usually gets the credit for inventing logarithms. But his contributions to simplifying mathematics and building shorthand solutions didn’t end there. In the course of performing the many calculations he needed to practice these subjects in the 1500s, Napier invented a kind of computing mechanism for multiplication. It’s a physical manifestation of an old system known as lattice multiplication or gelosia.

Lattice multiplication makes use of the multiplication table in order to multiply huge numbers together quickly and easily. It is thought to have originated in India and moved west into Europe. When the lattice method reached Italy, the Italians named it gelosia after the trellised window covering it resembled, which was commonly used to keep prying eyes away from one’s possessions and wife.

Continue reading “Bone Up on Your Multiplication Skills”

How to Find a Lost Drone with the Integral

If I asked you to find the area of a square, you would have no problem doing so. It would be the same if I asked you to find the volume of a cone or rectangle or any other regular shape. You might have to turn to Google to get the proper formula, but it would be a trivial process nonetheless. But what if I asked you to find the volume of some random vase sitting on a kitchen counter? How does one go about finding the volume of irregular shapes?

One way would be to fill the vase with much smaller objects of a known volume. Then you could add up the smaller volumes to get an estimate of the total volume of the vase. For instance, imagine we fill the vase with marbles. A marble is a sphere, and we can calculate the volume of each marble with the formula zeno014/3πr3. We count all of our marbles and multiply the total by the volume of a single marble and arrive at our answer. It is not perfect, however. There is a lot of empty space that exists between the marbles as they fill the vase. We are forced to conclude that our estimated volume will be lower that the actual volume.

It would be about this time when our good friend Isaac Newton would ask the question “What if you made the marbles smaller?” Reducing the size of each marble would reduce the empty space that exists between them as they pile up in the vase, giving us a more accurate total volume. But how small? Is there a limit to how small we can make them? “Do not trouble yourself with the limit.” says [Newton]. “You will find that as you make the marbles smaller and smaller, you will begin to converge on a single number – and that number will be the exact volume of your vase.”

Reducing the size of the marble to get a more exact volume demonstrates the idea of the integral – one of the two fundamental principles of The Calculus. The other principle is known as the derivative, which we explained in our previous article by taking a very careful and tedious examination of an arrow in flight. In this article, we shall take the same approach toward the integral. By the end, you will have a fundamental understanding of what the integral is, and more importantly, how it works. Our vase example gives you a good mental image of what the integral is all about, but it is hardly a fundamental understanding of it. Just how do you make those marbles smaller? To answer this question, let us look again at one of Zeno’s moving arrows.

Continue reading “How to Find a Lost Drone with the Integral”

Calculus is not Hard – The Derivative

The Calculus is made up of a few basic principles that anyone can understand. If looked at in the right way, it’s easy to apply these principles to the world around you and to see how the real world works in their terms. Of the two main ideas of The Calculus — the derivative and the integral — today we’ll focus on the derivative.

You can enjoy this article by itself, but it is also worth looking back at the previous installment in this series. We went over the history of The Calculus and saw how it arose from two paradoxes put forth by a 4th century philosopher named Zeno of Elea. These paradoxes lead to the derivative/integral ideas that revolutionized mankind’s understanding of motion.

Continue reading “Calculus is not Hard – The Derivative”

Calculus is Not as Hard as you Think

“Everything should be made as simple as possible, but not simpler.”

                                                                                                Albert Einstein

Our journey begins with a fictitious character whom we shall call [John Doe]. He represents the average professional worker who can be found in cities and towns across the world. Most everyday, [John] wakes up to his alarm clock and drives his car to work. He takes an elevator to his office and logs on to his computer. And he does these things without the slightest clue of how any of them work. While he may be interested in learning about the inner workings of the machines and devices he uses on a daily basis, [John] does not have the time and energy to invest in doing so. To him cars, elevators, computers and alarm clocks are completely different and complicated machines with hardly any similarities. It is simply not possible to understand how each of them work without years of study.

The regular readers of Hackaday might see things a bit differently than our [John Doe]. They would know that the electric motor that moves the elevator is very similar to the alternator in his car. They would know that the PLC that controls the electric motor that moves the elevator is very similar to the computer he logs in to. They would know that on a fundamental level, the PLC, alarm clock and computer are all based on relatively simple transistor theory. What is a vast complicated mess to [John Doe] and the average person is nothing but the use of simple mechanical and electrical principles to the hacker. The complication resides in how those principles are applied. Abstracting the fundamental principles from complicated ideas allows us to simplify and understand them in a way that pays homage to Einstein’s off-the-cuff advice, quoted above.

Zeno of Elea 430 - 490BC
Zeno of Elea 490 – 430BC

Many of you look at The Calculus the same way [John Doe] looks at machines. You see the same vast, complicated mess that would require a great deal of time and effort to understand. But what if I told you that calculus shares a commonality in much the same way many different machines do. That there are a few basic principles that anyone can understand, and once you do, it will unlock a new way of looking at the world and how it works.

The average calculus course book is a thousand pages long. The [John Does] of the world will see a thousand difficult things to learn. The hacker, however, will see two basic principles and 998 examples of those principles. In this series of articles, I’m going to show you what these two principles – the derivative and the integral – are.  Based on work done by Professor [Michael Starbird] of The University of Texas at Austin for The Teaching Company, we’ll use everyday examples that anyone can understand. The Calculus reveals a particular beauty of our world — a beauty that arises when you’re able to view it dynamically as opposed to statically. It is my hope to give you this view.

Before we get started, it pays to understand a little of the history of how The Calculus came about, and how its roots lie in the very careful analysis of change and motion.

Zeno’s Paradox

Zeno of Elea was a philosopher in the fourth century BC. He posed several subtle but profound paradoxes, two of which would eventually give rise to The Calculus. It would take over 2,000 years for man’s ingenuity to solve the paradoxes. As you can imagine, it wasn’t easy. The difficulties largely revolved around the idea of infinity. How do you deal with infinity from a mathematical perspective? Sir Isaac Newton and Gottfried Leibniz would go on to independently invent The Calculus in the mid 17th century, finally putting the paradoxes to rest. Let us take a close look at them and see what the fuss was all about.

The Arrow

zeno_03Consider the arrow flying through the air. We can say with reasonable and competent assurance that the arrow is in motion. Now consider the arrow at any given instant in time. The arrow is no longer in motion. It is at rest. But we know the arrow is in motion, how can it be at rest! This is the paradox. It might seem silly, but it’s a very challenging concept to deal with it from a mathematical point of view.

We’ll find out later that what we’re really dealing with is the concept of an instantaneous rate of change, which we will elaborate on with the idea of one of the two principles of calculus – the derivative. It will allow us to calculate the velocity of the arrow at an instant in time – a monumental feat that took over two millennia for mankind to reach.

The Dichotomy

zeno_02Let us consider the same arrow again. This time let’s say the arrow is coming at us. Zeno says we don’t have to move, because it can never hit us. Imagine that as the arrow is in flight, it has to cover half the distance between the bow and the target. Once it reaches the half way point, it has to do this again – move half the distance between it and the target. Imagine that we keep doing this. The arrow is constantly moving halfway between its origin and target. By doing this, the arrow can never hit us! In real life, the arrow does eventually hit the target, leaving us with the paradox.

As with the first paradox, we’ll see how to resolve this issue with one of the two principles of calculus – the integral. The integral allows us to deal with the concept of infinity as a mathematical function. It is an extremely powerful tool to scientists and engineers.

The Two Principles of Calculus

The two main ideas of The Calculus will be demonstrated by using them to solve Zeno’s paradoxes.

  • The Derivative – The derivative is a technique that will allow us to calculate the velocity of the arrow in “The Arrow” paradox. We will do this by looking at positions of the arrow through incrementally smaller amounts of time, such that the precise velocity will be known when the time between measurements is infinitely small.
  • The Integral – The integral is a technique that will allow us to calculate the position of the arrow in the Dichotomy paradox.  We will do this by looking at velocities of the arrow through incrementally smaller amounts of time, such that the precise position will be known when the time between measurements is infinitely small.

It’s not difficult to notice some similarity between the derivative and integral. Both values are calculated by examining the arrow with increasingly finer time intervals. We will learn later that the integral and derivative are in fact two sides of the same ceramic capacitor.

Why Should I Learn Calculus?

We are all familiar with Ohm’s Law, which relates current, voltage and resistance in a simple equation. However, let us consider “Ohm’s Law” for a capacitor. A current flow through a capacitor is dependent on the voltage across it and time. Time is the critical variable here, and must be taken into account in any dynamic event. Calculus lets us understand and measure how things change over time. In the case of a capacitor, the current through it is equal to the capacitance multiplied by volts per second, or: i = C(dv/dt) where:

  • i  = current (instantaneous)
  • C = Capacitance in Farads
  • dv = change in voltage
  • dt = change in time

zeno_04In this circuit, there is no current flow through the capacitor. The volt meter will read the battery voltage and the ammeter will read zero amps. So long as the potentiometer is not moved, the voltage on the meter will be steady. Our equation would say that i = C(0/dt) = 0 amps. But what happens when we adjust the potentiometer? Our equation says there will be a resulting current flow in the capacitor. This current flow will be dependent on the rate the voltage changes, which is tied to how fast we move the potentiometer.

zeno_05

These graphs show the casual relationships between the voltage across the capacitor, the current through the capacitor and the speed we turn the potentiometer. It starts with the potentiometer turning slowly. An increase in speed results in a faster changing voltage which in turn results in a dramatic increase in current. At all points, the current through the capacitor is proportional to the rate of change of the voltage across it.

Calculus, or more specifically the derivative,  gives us the ability to quantify this rate of change, so that we can know the exact value of current running through the capacitor at any given instant in time. The same way we can know the instantaneous velocity of Zeno’s arrow. It is an incredibly powerful tool to have in your hacking arsenal.

In the next article, we will go into deep detail of how we calculate the derivative using a modern but still simple representation of Zeno’s “The Arrow” paradox and some basic algebra. A following article will do the same for the integral using the Dichotomy paradox. Then we will tie things up by showing how the two are related, something known as The Fundamental Theorem of Calculus.

Slide Rules were the Original Personal Computers

Unless you are above a certain age, the only time you may have seen a slide rule (or a slip stick, as we sometimes called them) is in the movies. You might have missed it, but slide rules show up in Titanic, This Island Earth, and Apollo 13. If you are a fan of the original Star Trek, Mr. Spock was seen using Jeppesen CSG-1 and B-1 slide rules in several episodes. But there was a time that it was common to see an engineer with a stick hanging from his belt, instead of a calculator or a cell phone. A Pickett brand slide rule flew to the moon with the astronauts and a K&E made the atomic bomb possible.

Slide rules are a neat piece of math and history. They aren’t prone to destruction by EMP in the upcoming apocalypse (which may or may not include zombies). Like a lot of things in life, when it comes to slide rules bigger is definitely better, but before I tell you about the 5 foot slide rule in my collection, let’s talk about slide rules in general.
Continue reading “Slide Rules were the Original Personal Computers”