# 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.

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

I 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

Noether’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

# OneSolver Does What Wolfram Can’t

Wolfram Alpha has been “helping” students get through higher math and science classes for years. It can do almost everything from solving Laplace transforms to various differential equations. It’s a little lacking when it comes to solving circuits, though, which is where [Grant] steps in. He’s come up with a tool called OneSolver which can help anyone work out a number of electrical circuits (and a few common physics problems, too).

[Grant] has been slowly building an online database of circuit designs that has gotten up to around a hundred unique solvers. The interesting thing is that the site implements a unique algorithm where all input fields of a circuits design can also become output fields. This is unique to most other online calculators because it lets you do things that circuit simulators and commercial math packages can’t. The framework defines one system of equations, and will solve all possible combinations, and lets one quickly home in on a desired design solution.

If you’re a student or someone who constantly builds regulators or other tiny circuits (probably most of us) then give this tool a shot. [Grant] is still adding to it, so it will only get better over time. This may be the first time we’ve seen something like this here, too, but there have been other more specific pieces of software to help out with your circuit design.

# A Look Into the Future of Slicing

I’ve had a few conversations over the years with people about the future of 3D printing. One of the topics that arises frequently is the slicer, the software that turns a 3D model into paths for a 3D printer. I thought it would be a good idea to visualize what slicing, and by extension 3D printing, could be. I’ve always been a proponent of just building something, but sometimes it’s very easy to keep polishing the solution we have now rather than looking for and imagining the solutions that could be. Many of the things I’ll mention have been worked on or solved in one context or another, but not blended into a cohesive package.

I believe that fused deposition modelling (FDM), which is the cheapest and most common technology, can produce parts superior to other production techniques if treated properly. It should be possible to produce parts that handle forces in unique ways such  that machining, molding, sintering, and other commonly implemented methods will have a hard time competing with in many applications.

Re-envisioning the slicer is no small task, so I’m going to tackle it in three articles. Part One, here, will cover the improvements yet to be had with the 2D and layer height model of slicing. It is the first and most accessible avenue for improvement in slicing technologies. It will require new software to be written but does not dramatically affect the current construction of 3D printers today. It should translate to every printer currently operating without even a firmware change.

Part Two will involve making mechanical changes to the printer: multiple materials, temperatures, and nozzle sizes at least. The slicer will need to work with the printer’s new capabilities to take full advantage of them.

Finally, in Part Three, we’ll consider adding more axes. A five axis 3D printer with advanced software, differing nozzle geometries, and multi material capabilities will be able to produce parts of significantly reduced weight while incorporating internal features exceeding our current composites in many ways. Five axis paths begin to allow for weaving techniques and advanced “grain” in the layers put down by the 3D printer.

It was the year of 1687 when Isaac Newton published “The Principia“, which revealed the first mathematical description of gravity. Newton’s laws of motion along with his description of gravity laid before the world a revolutionary concept that could be used to describe everything from the motions of heavenly bodies to a falling apple. Newton would remain the unequivocal king of gravity for the next several hundred years. But that would all change at the dawn of the 20th century when a young man working at a Swiss patent office began to ask some profound questions. Einstein had come to the conclusion that Newtonian physics was not adequate to describe the findings of the emerging electromagnetic field theories.  In 1905, he published a paper entitled “On the Electrodynamics of Moving Bodies” which corrects Newton’s laws so they work when describing the motions of objects near the speed of light. This new description became known as Special Relativity.

It was ‘Special’ because it didn’t deal with gravity or acceleration. It would take Einstein another 10 years to work these two concepts into his relativity theory. He called it General Relativity – an understanding of which is necessary to fully grasp the significance of gravitational waves.

# Hacking Candle Extinguishing

Anyone can put out a candle by blowing on it. According to [Physics Girl], that method is old hat. She made an educational video that shows five different ways to put out a candle using–what else–physics.

You might not need alternate ways to put out a candle, but if you are looking to engage students in STEM (Science, Technology, Engineering, and Math), this video along with others from [Physics Girl] might spark interest.

# Suspension Bridges of Disbelief

Suspension bridges are far and away the target of choice in America’s action blockbusters. In just the past three years, the Golden Gate Bridge has been destroyed by a Kaiju, Godzilla, a Skynet-initiated nuclear blast, and a tsunami. Americans don’t build real bridges anymore, or maintain the ones that we have, but we sure love to blow them up in movies.

There is logic here: A disaster scene involving a famous bridge serves both to root the film in the real world and to demonstrate the enormity and the immediacy of the threat. The unmaking of these huge structures shocks us because many bridges have gained an aura of permanence in our collective consciousness. Although we know when the Brooklyn Bridge was built and who built it, we feel like it has always been there and always will be. The destruction of our familiar human topography is even more disturbing than the deaths of the CGI victims, and I’m not just saying that as a misanthrope who loves bridges.

However, in all of the planning, storyboarding, rendering, and compositing of these special effects shots, nobody pauses to consider how suspension bridges actually behave. I can accept messianic alien orphan superheroes and skyscraper-sized battle robots, but I will not stand for inaccurate portrayals of structural mechanics. It’s fine to bend the laws of physics if the plot warrants it, but most suspension bridge mistakes are so needless and stupid that their only function seems to be irritating engineers.

# Don’t Look Now, Nothing Will Happen –Zeno of Elea

The Greek philosopher [Zeno of Elea] proposed that an arrow in flight was in fact not in motion and its visible movement is only an illusion. A simple example of this is to glance at an arrow in flight, doing this causes our mind to store a snapshot of a motionless arrow. [Zeno] further defended this argument by stating that if an object has to travel a finite distance to reach a destination then the finite distance can be divided in half and the object must first reach this halfway point before arriving at the destination. This process can be repeated an infinite number of times, creating an infinite number of points that the object must occupy before reaching the destination thus it can never arrive at the destination.

Whoa, that’s a bit heavy. Let’s take a second here to think about this and never arrive at the conclusion, shall we?

So what does a fancy mathematics parlor trick have to do with the fact that we have all seen an arrow arrive at its destination? Recent experiments conducted at Cornell University have in fact verified the Zeno Effect. Researchers were able to achieve this by having atoms suspended between lasers in temperatures ~1 nano degree above absolute zero so that the atoms arrange themselves in a lattice formation. As per usual in quantum mechanics when observed, the atoms had an equal possibility of being anywhere within the space of the lattice. However, when they were observed at high enough frequencies the atoms remain motionless, bringing the quantum evolution to a halt.