The Unabashed Academic

07 September 2016

Work and energy in introductory physics

Some of my Physics Education Research Facebook friends have been questioning the value of some forms of the work-energy theorem in teaching introductory physics for life sciences (IPLS). Since over the past few years I have made this theorem an important component of my teaching of mechanics, I thought I'd take the opportunity to describe how I do it.
I've been teaching a – what? – reformed? reinvented? renewed? – IPLS course.[1] (link to NEXUS paper) I struggle with the adjective since our reformation process went beyond usual course reform. We spent a lot of time communicating and discussing (and arguing) content and approach with biologists and chemists, and a lot of time researching student responses and what they brought to the table. This produced a deep philosophical change in the way we designed and that I teach the class.
We learned that it wasn't just that many life science students didn't know the required math (perhaps because they hadn't used it in previous courses) or that they weren't familiar with physics concepts (perhaps because they hadn't taken high school physics, or if they had, hadn't taken it seriously). Rather, there were some serious barriers in the way many students were thinking about the nature of the scientific knowledge they were learning – epistemological barriers, if you will.
Here are some of the issues that we found:
·      Life science students often saw scientific knowledge as bits and pieces of memorized knowledge, failing to build a coherent picture. Although they had learned some heuristics (usually in chemistry), they had little or no experience with the use of deep and powerful principles such as those that drive even introductory physics.
·      Life science students often were deeply skeptical of highly simplified "toy" models. Since life depends in a critical way on complexity, simplification was seen as "playing irrelevant games." Few had any experience with the concept of modeling and few understood the insight that could be derived from studying simplified systems.
·      Even when life science students knew and were comfortable with the required math, almost all saw math as a way to calculate, rather than as a way to think about physical relationships. They were missing not only the skill of estimation and intuitions of scale, they missed being able to read qualitative implications from equations.
Our course is designed to address these epistemological issues as well as the issues of choosing content relevant for life science students (like doing more fluids and including diffusion and random walk). We try to stress coherence, modeling, and the value of using equations to build understanding and insight. The work-energy theorem plays a pivotal role in this structure.
Newton's three laws form the framework for building understanding of mechanics and building models of physical motion. I treat the three laws [2] as the basic structure. Any analysis of a particular motion requires a model – a choice of what we are going to treat as "objects" and how we are going to model their interactions. We use the method of System Schema [3] as a tool for analyzing systems and building models. This is a pre-requisite to drawing free-body diagrams. Interactions are two-way connecting a pair of objects. When the focus is on one object the interaction is realized as a force. By Newton's third law, the forces on either end of an interaction are equal and opposite. This is the tool that focuses student attention on the modeling character of each system considered.
Newton's second law tells how an object responds to the forces it feels. If the forces are not balanced (cancel out), the object accelerates – changes its velocity according to the rule:
Acceleration = (Sum of the forces)/(mass of the object)
This is a vector law, so forces are requires to change either the object's speed or the direction of its motion.
This naturally leads to the question:
If I only care about the change in an object's speed and not its direction,
what does Newton's second law tell me?
It's pretty easy to figure out how to do this, at least in principle. Forces that are in the same direction as the motion tend to speed it up, forces that are in the opposite direction of the motion tend to slow it down, and forces that are perpendicular to the motion tend to change its direction. So to consider only the speed, we multiply Newton's second law by a small displacement along (or against) the direction of motion. After a little simple algebra (no calculus needed), we get the one-body work-kinetic energy theorem:[4]
The change in an object's kinetic energy = work done on it by the sum of all the forces it feels
Or as an equation, this is written
Δ(1/2 mv2) = Fnet.Δr
(Bold here indicates a vector.)This law is not particularly useful by itself unless it is used in connection with the System Schema so one can see that it helps to provide clear and simple answers to two rather subtle questions:
·      Why is there such a thing as "potential energy" but no such thing as "potential momentum"? The impulse-momentum theorem and the work-energy theorem look very similar.
·      Why do we sometimes treat potential energy as belonging to a single object (e.g., the gravitational PE, mgh) but sometimes treat it as belonging to a pair of objects (e.g., the PE between two electric charges, kqQ/r).
To answer the first, let's consider the impulse-momentum theorem in contrast to the work-energy theorem written above:
Δ(mv) = FnetΔt
If we consider a system with two objects interacting, since they interact for the same amount of time, and since the forces they exert on each other are equal and opposite (by Newton 3), they change each other's momenta in equal and opposite ways. This means that if we add together the impulse-momentum theorems for the two objects, their momentum changes will cancel. We can then see easily see what the conditions are for momentum conservation to hold. (All the other forces acting on the two objects have to cancel.)
For the work-energy theorem, things are a bit difference. If we consider a small time interval when the two objects are interacting, their time intervals are the same, but the distances that they move do not have to be. Therefore, if we add together the work-energy theorems for two interacting objects, even if there are no other forces acting on them, the work terms for the two objects do not have to cancel. And we can easily see that the extract term is the force dotted into the change in the relative separation of the two objects.
This extra term is why we introduce a potential energy (but not a potential momentum). And it makes clear that the potential energy belongs to the interaction between the two objects.
It also helps us understand when we can treat the PE as belonging to a single object rather than to a pair of objects. Since the momentum changes of the two objects are the same, its easy to find that the KE change of each object is Δ(p2/2m). If one object is much larger than the other, the KE change (and therefore the PE) can be totally assigned to the lighter object. This is why the gravitational PE of an object on the earth's surface can be assigned to the object, and why, in an atom or molecule, the electric PE if the interaction of an electron and a nucleus can be assigned to the electron and we can talk about "the PE of the electron."
These are nice results, if abstract. But I like the work-energy theorem for more reasons. Here are three:
·      When we have a situation where there one of the interacting objects is much larger than the other, there are a lot of nice examples where one can write energy conservation and create equations relating position and velocity. This gives the students good practice with using manipulating symbolic equations and interpreting the result.
·      It can be used to generate other relations and show the relation between other principles that are often treated as independent.
·      If provides the link between the fundamental concepts of force and energy, building another powerful coherence.
The first doesn't need much elaboration, but I was a bit surprised at the second. I knew in principle the power of the work-energy theorem, but it wasn't until I included a substantial discussion of fluids in my class that I realized how cool it was. The work-energy theorem, when applied to a bit of fluid in a pipe easily reduces to:[5]
·      The dependence of pressure on depth and the related Archimedes' principle (by assuming no motion and only gravitational and pressure forces)
·      Bernoulli's principle (by assuming no resistive forces)
·      The Hagen-Poiseuille equation (by assuming resistive force but no gravitational change)
It can also be used to generate new equations, such as a modified H-P equation for fluids flowing vertically in a tree.
Of course, each of these can be derived from forces as well, but tying everything to work-energy and thereby back to forces and Newton two emphasizes the coherence of the whole structure and the reliance on powerful overarching principles.
I've seen this work with my students. They all come in knowing that "energy is the ability to do work," but for most, these are just words. Once we've gone through the work-energy theorem they begin to be able to translate forces into work.
My favorite specific example of this occurred in an interview done with Carol, a student in the class's second term. We had completed a discussion of free energy and done a recitation analyzing the separation of oil and water and the formation of lipid cell membranes. The result is somewhat counterintuitive, since it is actually pretty easy for students who have taken chemistry to see that the interaction (electric attraction) between water and oil molecules is stronger than the interaction between two oil molecules. So why does oil and water separate? Why do lipid membranes form?
In the interview, Carol answered the question by referring to the equation for the Gibb's free energy:
ΔG = ΔHTΔS
As all biology and chemistry students know, Gibb's free energy is what drives chemical reactions. Spontaneous reactions go to a lower free energy. (Here, H is the enthalpy, which, for this discussion, is equivalent to the internal energy.)
She said (paraphrasing), "The force between the molecules goes into the work which creates potential energy. That goes into the H term since it's energy. Since it's attractive, that tends to make the H lower for the separated oil molecules. But the other term competes. It comes from the losing of the opportunities for the water molecules to interact. In this case, that term wins."
I've seen many students reason like this and it makes me happy. They are using equations to reason with qualitatively and bringing together the idea of forces and energy, building an overall coherence and reasoning from principle.
The single-particle work-energy theorem is easy to think about and reason qualitatively and quantitatively with. This is why I like it and why I make it a central element of my IPLS class.


[1] NEXUS/Physics: An interdisciplinary repurposing of physics for biologists, E. F. Redish, C. Bauer, K. L. Carleton, T. J. Cooke, M. Cooper, C. H. Crouch, B. W. Dreyfus, B. Geller, J. Giannini, J. Svoboda Gouvea, M. W. Klymkowsky, W. Losert, K. Moore, J. Presson, V. Sawtelle, K. V. Thompson, C. Turpen, and R. Zia, Am. J. Phys. 82:5 (2014) 368-377.

[2] I actually introduce a "zeroth law" – that every object responds only to forces it feels and only at the instant it feels them. While this might seem trivial to an experienced physicist, a significant fraction of the errors that introductory students make are a violation of this law.
[3] V. Sawtelle & E. Brewe, System Schema Introduction, NEXUS/Physics; L. Turner, System Schemas, Am. J. Phys. 41:9 (2003) 404.

Could dark matter be super cold neutrinos?

Probably the greatest physics problems of the current generation are the cosmological questions. Thanks to the development of powerful new telescopes (many of them in space) in the last years of the twentieth century, startling new and unexpected results have pointed the way to new physics. These currently go under the names of "dark matter" and "dark energy", but those aren't real descriptions; rather they are suggestions for what might provide theoretical solutions to experimental anomalies. And, as naming often does, they guide our thinking into explorations of how to come up with new physics.

The problem that "dark matter" is supposed to resolve began in the 1970s with the observations of Vera Rubin. By making a careful analysis of the motion of stars in galaxies, she found an unexpected anomaly. As any first year physics student can tell you, Newton's law of gravitation tells you how planets orbit around the sun. The mass of the sun draws the planets towards it, bending their velocities ever inward in (nearly) circular orbits. The mathematical form of the law produces a connection between the distance the planets are from the sun and the speed (and therefore the period) of the planets.

That connection was known empirically before Newton to Kepler (Kepler'sthird law of planetary motion: the cube of the distance from the sun is proportional to the square of the planet's period). The fact that Newton's laws of motion together with his law of gravity explained that result was considered a convincing proof of Newton's theories.

A galaxy has a structure somewhat like that of a solar system. There is a heavy object in the center – a massive black hole – that is responsible for most of the motion of the stars in the galaxy. Rubin found that the speed of the stars around the center didn't follow Kepler's law. The far out stars were going too fast. This suggested that there was an unseen distributed mass that we didn't know about (or that Newton's law of gravity perhaps failed at long distances; In my opinion this option has not received enough attention, though that's for another post.).

Observations in the past thirty years have increasingly supported the idea that there is some extra matter that we can't see – and a lot of it. More than the matter that we do see. As a result, a growing number of physicists are exploring what might be causing this.

I saw a lovely colloquium yesterday about one such search. Carter Hall, one of my colleagues in the University of Maryland Physics Department, spoke about the LUX experiment. This explores the possibility that there is a weakly interactive massive particle (a "WIMP") that we don't know about – one that doesn't interact with other particles electromagnetically so it doesn't give off or absorb light, and it doesn't interact strongly (with the nuclear force) so it doesn't create pions or other particles that would be easily detectable in one of our accelerators. This would make it very difficult to detect. The experiment was a tour de force, looking for possible interactions of a WIMP with a heavy nucleus – Xenon. (The interaction probability goes up like the square of the nuclear mass so a heavy nucleus is much more likely to show a result.) The experiment was incredibly careful, ruling out all possible known signals. It found no results but was able to rule out many possible theories and a broad swath of the parameter space – eliminating many possible masses and interaction strengths. An excellent experiment.

But as I listened to this beautiful lecture, I wondered whether the whole community exploring this problem hadn't made the mistake of looking under the lamppost for our lost car keys. It's sort of wishful thinking to assume that the solution to our problem might be exactly the kind of particle that would be detectable with the incredibly large, powerful, and expensive tools that we have built – particle accelerators. These are designed to allow us to find new physics – in the paradigm we have been exploring for nearly a century: finding new sub-nuclear particles and determining their interactions in the framework of quantum field theory.

This reflects a discussion my friend Royce Zia and I have been having for five decades. Royce an I met in undergraduate school (at Princeton) and then became fast friends in grad school (at MIT). We spent many hours there (and since) arguing about deep issues in physics. We both started out assuming we wanted to be elementary particle theorists. That, after all, was where the action was. Quarks had just been proposed and there was lots of interest in the nuclear force and how to make sense of all the particles that were being produced in accelerators. But we were both transformed by a class in Many Body Quantum Theory given by Petros Argyres, a condensed matter theorist. In this class we saw many (non-relativistic) examples of emergent phenomena – places where you knew the basic laws and particles, but couldn't easily see important results and structures from those basic laws. It took deep theoretical creativity and insight to find a new way of looking at and rearranging those laws so that the phenomena emerged in a natural way.

There are many such examples. The basic laws and particles of atomic and molecular physics were well known at the time. Atoms and molecules are made up of electrons and nuclei (the structure of the nuclei is irrelevant for this physics – only their charge and mass matters) and they are well described by the non-relativistic Schrödinger equation. But once you had many particles – like in a large atom, or a crystal of a metal – there were far too many equations to do anything useful with. Some insight was needed as to how to rearrange those equations so that there was a much simpler starting point.

Three examples of this are the shell model of the atom (the basis of all of chemistry), plasmon oscillations in a metal (coherent vibrations of all the valence electrons in a metal together), and superconductivity (the vanishing of electrical resistance in metals at very low temperatures). Each of these were well described by little pieces of the known theory arranged in clever and insightful ways – ways that the original equations gave no obvious hint of in their structure.
I was deeply impressed by this insight and decided that this extracting or explaining phenomena from new treatments of known physics was just as important – as just as fundamental – as the discovery of new particles or new physical laws. Royce and I argued this for many hours and finally decided to grant both approaches the title of "fundamental physics" – but we decided they were different enough to separate them. So we called the particle physics approach "fundamental-sub-one" and the many-body physics approach "fundamental-sub-two". (Interestingly, both Royce and I went on to pursue physics careers in the f2 area, he in statistical physics, me in nuclear reaction theory.) In the decades since we had these arguments, physics has made huge progress in f2 physics – from phase transition theory to the understanding and creation of exotic (and commercially important) excitations of many body systems.

So yesterday, I brought my f2 perspective to listening to Carter talk about dark matter and I wondered: He was talking all about f1 type solutions. Interesting and important, but could there also be an f2 type solution? We already know about weakly interacting massive particles: neutrinos. They only interact via gravity and the weak nuclear force, not electromagnetically or strongly. 

Could dark matter simply be a lot of cold neutrinos? They would have to be very cold – travelling at a slow speed – or else they would evaporate. When we make them in nuclear reactions in accelerators they are typically highly relativistic – travelling at essentially the speed of light. The gravity of the galaxy wouldn't be strong enough to hold them.

That leads to a potential problem for this model. Whatever dark matter is, it has to have been made fairly soon after the big bang – when the universe was very dense, very uniform, and very hot -- hot enough to generate lots of particles (mass) from energy. (Why we believe this is too long a story to go into here.) So you would expect that any neutrinos that were made then would be hot – going too fast to become cold dark matter.

But suppose there were some unknown emergent mechanism in that hot dense universe -- a phase transition -- that squeezed out a cold cloud of neutrinos. Neutrinos interact with matter very weakly – and their interaction strength is proportional to their energy so cold neutrinos interact even more weakly than fast neutrinos. If there were a mechanism that spewed out lots of cold neutrinos, I expect they would interact too weakly with the rest of the matter to come to thermal equilibrium. If the equilibration time were, say, a trillion years, they would stay cold and, if their density were right, could serve as our "dark matter".

Most of the experimental dark matter searches wouldn't find these cold neutrinos. Searching for them at this point would have to be a theoretical exploration: Can we find a mechanism in hot baryonic matter that will produce a phase transition that spews out lots of cold neutrinos? I don't know of any such mechanism or where to start, but wouldn't it be fun to consider?

19 May 2016

Still a physicist! Thanks, Emmy Noether

Recently while browsing my FaceBook feed, I was tempted to take one of the BuzzFeed quizzes that regularly pop up. Usually, I'm immune to this kind of clickbait, not really being interested in "Which American Idol judge are you?" or "Which Game of Thrones character are you like?" (Though as a frequent traveler, I do often do the ones that ask, "How many states have you visited?" or "How many of the top 150 world travel sites have you seen?") This one asked, "Are you more of a physicist, biologist, or chemist?" This was clearly a quiz for scientists and, though I'm a lifelong physicist (practicing for 50 years), I've always been a "biology appreciator", collecting Wildlife Stamps as a boy, and reading Stephen J. Gould, E. O. Wilson, Konrad Lorenz, and lots of other as an adult. And for the past half dozen years or so, I've been holding many conversations with multiple biologists and learning some serious bio in the service of carrying out a deep reform on algebra-based physics to create an IPLS (Introductory Physics for Life Scientists) class – NEXUS/Physics. I wondered whether I had been sufficiently infected with biology memes to have gone over to the dark side.

I needn't have worried. As expected, I came out "Physicist". Their description of a physicist was one I liked and that describes my favorite physicists (and I hope me too): "You’re a thinker who loves nothing more than getting stuck into a good intellectual challenge. You love to read, and you’ve got so much information (useless and otherwise) stored in your brain that everyone wants to have you on their pub quiz team. Physics suits you because it lets you spend your time contemplating some of the smallest and biggest things in the universe, and tackle some really huge questions while you’re at it."

But I particularly found one item in the quiz interesting: "Select a real scientist." They offered three female scientists: Emmy Noether, Jane Goodall, and Rosalind Franklin. Although I assume that they matched Emmy to Physics, Jane to Biology, and Rosalind to Chemistry, I think of both Goodall and Franklin as biologists. I have read some of both of their work – one of Jane Goodall's books on chimpanzees (and I regularly contribute to her save the chimps foundation), and Rosalind Franklin's paper on X-ray diffraction from DNA crystals. I've never read any of Emmy Noether's original writings, but her work was introduced into my physics classes in junior year and had a powerful impact on my thinking about the world and about physics. That's what I want to talk about here.

[But first, I'm inspired to make one of my typical academic digressions about a topic I've been thinking about: the structure of biological research. Reading E. O. Wilson's memoir, Naturalist, clarified for me a lot of what I have been seeing in my recent conversations with multiple biologists. I refer to this as "the Wilson/Watson abyss". About 1960, E. O. Wilson and J. D. Watson were both new Assistant Professors in the Harvard Biology Department. Over the next few years they engaged in a fierce battle for the soul of biology. What were the key issues for biology research for the next few decades? E. O., a field biologist rapidly becoming the world's greatest expert on ants, argued vigorously for a holistic approach: looking at whole animals, their behavior, how they interacted with others and their environments. J. D., fresh off his success in deciphering the structure of DNA and offering a molecular model for evolution, argued vigorously for a reductionist approach: studying the molecular mechanism of biology and the genome. The result was a split into two departments, and, essentially, a victory for Watson. Although there is excellent research in both areas, for the past half century, the strongest focus has been on microbiology and molecular models. Premier biology research institutes are often entirely focused on molecular and cellular biology and far more funding goes into that area. I personally think this is a problem and that the critical biological problems for the next half century are going to be that we HAVE to understand the systemic aspects of ecology – both for our interaction with the planet and even for medicine (through consideration of the human as an ecosystem by including our microbiome and the implications of social and environmental interactions on it).
Of course this digression is inspired by the choices of Jane Goodall – a premier field biologist in the Wilson model (though she came through anthropology as a student of Louis Leakey), and of Rosalind Franklin – a premier biochemist in the Watson model (and her work was critical in allowing the Watson-Crick breakthrough).

An interesting point for another post, is to note that evolution is the bridge that spans the Wilson/Watson abyss. Evolution is not a hypothesis or even really a theory, but rather a conclusion that grows out of a number of fundamental principles based strongly in observation and experiment: heredity (through DNA and its copying mechanism), variation, morphogenesis (the building of a phenotype – the individual organism – from the genomic info), and natural selection. (One might choose a different set, but this is one I like so far.) The first lies firmly on the Watson side, the last on the Wilson side. You can't make sense of evolution unless you are willing to consider both ends.]

We now return to our main program. Why did I pick Emmy over Jane and Rosalind, both of whose work I have actually read and I think are immensely important?

The reason is because for me as a physicist, Emmy Noether's result was a total game changer for me in the way I think about physics, the epistemology of physics, and how the world works. To state her result crudely in a way that the non-mathematician might understand, Noether's theorem says:

If you have a system of interacting objects whose behavior in time is governed by a set of equations that have a symmetry, then you can find a conserved quantity.

By a "symmetry", she means that you can change something about your description that doesn't change the math. By a "conserved quantity" she means something you can calculate that doesn't change as the system changes through time. (Of course Noether's theorem is a mathematical statement and there are conditions and a process to find the conserved quantity, but that requires a lot of math to elaborate. I refer you to the Wikipedia article on Noether's theorem for those who want the details. Warning: It requires knowledge of Lagrangians and Hamiltonian – junior level physics.)

This is a little dense. Let's take an example or three to see just what it means.
Suppose I have a set of interacting objects – something like the planets in the solar system interacting via gravity, or a set of atoms and molecules interacting via electric forces. We can describe these interactions either using forces or energy. (These approaches can be shown to be mathematically equivalent, though each tends to foreground different ways of thinking about the system.) The key is that the interactions of the objects only depend on the distances between them. This means that I can choose any coordinate system to describe the system: I can put my reference point – the 0 of my coordinates or origin – anywhere I want. Whatever origin I choose, the distance between two objects is the difference of the positions of those two objects and when you subtract their positions to get their relative distance, the position of the origin cancels.

This is a symmetry. The equations that describe the motion of the system do not change depending on the position of the origin of the coordinate system. You can choose it as you like – and we typically pick an origin that will make the calculation simpler. This symmetry is called translation invariance. It means you can shift (translate) the origin freely without anything changing.

But what Noether's theorem shows is the symmetry doesn't just mean we are allowed to choose the coordinate system that makes the calculation simpler, it says there is a conserved quantity and it allows you to find and calculate it.
In the case of translation invariance, Noether's conserved quantity is momentum – in most cases, the product of the mass and velocity for each object. You calculate the momentum of each object in the system, add them up at one time, and for any later time you will always get the same answer, no matter how the objects have moved, even though the motions may be amazingly complicated – and may involve billions of particles!

This is immensely important and has powerful practical implications. One technical example is, "How can you figure out how protons move inside a nucleus or electrons move inside an atom?" In the case of protons, you don't actually know exactly what the force law between two protons is (though there are lots of models), but we are pretty sure that they only depend on the distance between them.* But we can shoot very fast protons at a nucleus. Sometimes they will strike a proton moving in the nucleus and knock it out. If we measure the momenta of the two outgoing protons, and since we know the momentum of the incoming proton, we can infer the initial momentum of the struck proton inside the nucleus using momentum conservation. We then do a lot of these scatterings and get a probability distribution for the velocities of protons inside the nucleus. 

Since we do know the force between electrons and the nucleus (the electric force), this technique is extremely powerful for studying the structure of atoms and molecules. While this seems rather technical, we'll see that there are even more important implications that providing a measurement tool for difficult to observe quantum systems.

Two other fairly obvious symmetries in our description of systems are:

  • ·            Time translation invariance
  • ·            Rotational invariance

The first, time translation, means that it doesn't matter when you start your clock (what time you take as 0 of time). This is true for most dynamic models in physics. Gravitational forces don't depend on time and neither do electrical ones. Since these are the two forces that dominate everything bigger than a nucleus, this symmetry holds for everything from atoms up to galaxies (where there are some as yet unsolved anomalies). Emmy's theorem says that due to the time translation symmetry there is a conserved quantity – in this case energy.

The second, rotational invariance, means that it doesn't matter in which direction you point your axes. You can take the positive x direction as being towards the north star or towards the middle star of Orion's belt. (You want your coordinates to be fixed in space, not rotating with the earth or you introduce fake forces like centrifugal force and Coriolis forces.) The conserved quantity that goes with this is angular momentum, another useful principle (though more complicated to use because of more vectors).

OK. That tells us what Noether's theorem tells us – about important conservation laws like (linear) momentum, energy, and angular momentum. But we learn about these in introductory physics classes without needing a sophisticated theorem. What does it add?

For me, it adds something deeply epistemological – something fundamental about what we know in physics and how we know it. It shows that two very different things are tightly related: how we are allowed to describe the system at a given instant of time without changing anything (where we can choose our space and time coordinates) – a purely static statement about what kinds of forces or energies we have – and how the system moves in time – a dynamic statement about how things change.

This is immensely powerful. This means that if I have created a mathematical model of a system and I find that energy is NOT conserved, I know that either I have made a mistake, or I have assumed interactions that change with time. If I find that momentum is NOT conserved, I know that I must have tied something to a fixed origin rather than to a relative coordinate between two objects.

Now this isn't always wrong or bad. If I have a particle moving through a vibrating fluid I might want to treat the fluid like a fixed time dependent potential energy field. What this will mean is that the energy of my particle will not be conserved and where the energy goes (into the fluid) will not be correctly represented in this model. 

A more common example is projectiles or falling bodies. Since the earth is so much larger than our projectiles we take the origin of our coordinates as a fixed point on the earth instead of taking the force as depending (as it actually does) on the distance between the center of the earth and the projectile. This means we won't see momentum conserved since we have fixed the earth. Momentum transfer to it will not be correctly represented. This might not matter depending on what we want to focus on.

But what Noether's theorem shows us is that there are powerful – and absolute – links between two distinct ways of thinking about complex systems: the structure of the mathematical models we set up to describe the evolution of systems and characteristics of how those systems evolve in time. And that the result can be something as powerful and useful as a conservation law blew me away. More, that we now know exactly what characteristics of a mathematical model leads to a conservation law! There is nothing analogous to this in biology or chemistry – except as it is inherited from Noether's theorem in mathematical models biologists or chemists build or as they use energy or charge conservation. But as far as I can tell they rarely pay attention to conservation laws – even when they might do them some good.

It also showed me that when you build mathematical models you occasionally hit the jackpot: you get out more than you thought you put in. Extensions of Noether's theorem to other symmetries have become a powerful tool in constructing new models of dynamics. Instead of trying to invent new force laws, we look experimentally for conservation laws, find symmetries that can give those conservation laws, and construct new dynamical models by putting together variables that fit the symmetry. This is the way much of particle physics has functioned for the past 50 years.


So that question on the quiz is probably the best selector of the "physicist" category. Goodall and Franklin both did essential and pivotal work in their fields; but Noether's was a core pillar for all of 20th century physics and for me, won hands down. Thanks Emmy!