The coming revolution in physics education

[Dale]

why not teach the computational methods?

You have yet to produce actual evidence that they are not taught. Personally, they were taught to me sophomore year, and I gave several example of different universities that do the same.

You seem to not recognize that a course is more than a textbook and that a degree program is more than a course.

 

[bhobba]

One good thing about the US system is there is much greater diversity in teaching subjects at HS. This means different approaches can and are used at different schools. At Wilberforce School all students study calculus (including multivariable calculus and differential equations) and MATLAB so they can forge ahead in physics, and presumably other areas like data science, at a faster pace.
https://www.wilberforceschool.org/academics/high-school/curriculum/matlab

Here in Australia IMHO we should do the same, but, for most students, are locked in a rather old fashioned system. It is possible to break free of it by simply doing 2 university subjects with a credit average and you are automatically accepted into many universities eg
https://www.mq.edu.au/study/other-study-options/open-universities-australia

Most do not know about it which is a pity. Anyway as far as this thread goes MATLAB and similar packages that solve differential equations with ease help students forge ahead in physics much quicker. Later of course they can study analytical methods of solutions, and the limitations of computer packages, but not having taken such courses is no longer an impediment for doing calculus based physics properly.

Thanks
Bill

 

[bhobba]

You have yet to produce actual evidence that they are not taught. Personally, they were taught sophomore year, and I gave several example of different universities that do the same.

I have given an example where they are taught at HS. In the degree I did they are now taught from first year:
https://www.qut.edu.au/courses/bachelor-of-mathematics-applied-and-computational-mathematics

Back when I did it MATLAB etc was not around so you had to wait until you had done Fortran and Pascal before undertaking computational work - that was done second year. But these days it has all changed and is used right from the start.

However IMHO it should be done in HS and more advanced work started at university.

Thanks
Bill

 

[Will Flannery]

I have given an example where they are taught at HS. In the degree I did they are now taught from first year:
https://www.qut.edu.au/courses/bachelor-of-mathematics-applied-and-computational-mathematics

I checked out the curriculum at Wilberforce and read ...

One of the hallmarks of the Upper School is an advanced math and science curriculum that enables students to enter into a two-year calculus, physics, and programming course in their junior and senior years.

and a description of the MATLAB course:

* The course integrates the study of calculus, physics, and programming.
* Students develop the mental disciplines of programming - careful, logical, and concrete problem-solving.
* Students learn to create working mathematical models of physical phenomena so that they can better understand physics.

Perfect. The school is integrating calculus, physics, and the computer right from the start. This seems to me to be obviously the way to go, and it will inevitably happen, at least in the universities. And in fact I taught a pilot course at my kid's high school that did exactly that, and it was from my perspective 100% successful. Some of the projects we did are in the paper referenced earlier. There were meetings with the school's calculus teacher and also the physics teachers, and they were uniformly opposed to the idea. Subsequently the school 'curriculum committee' decided it was not to be continued.

 

[Dale]

Anyway as far as this thread goes MATLAB and similar packages that solve differential equations with ease help students forge ahead in physics much quicker.

That is also my preference. I would prefer to teach them to use standard ODE solvers rather than hand programming Euler’s method.

 

[mpresic3]

I read both papers by Will Flannery provided. The 28 page paper, and the 6-7 page Physics Teacher paper. There is good material in the papers and some I agree with up to a point, and some I disagree with. First off, in your paper to the physics teacher, the ,point in the fourth paragraph refers to the "process model". As soon as you used that term, I strongly suspected, you have a background in control theory, and aerospace engineering or electrical engineering. I read the rest, but fast-forward to the end, and we learn that you have worked in control systems engineering and avionics.

(Fear not, I have a strong background and worked in control theory so, your paper has found me as a receptive audience). I will go through the "good stuff" first. I really liked the simulations with MATLAB included, that showed planetary flyby's and lunar intercepts. These are fun (to me). I know the University of Colorado has PHET simulations though, that are fun to play with. Also, in my experience, the simulations that are fun to me are considered less fun to younger students and kids that grew up with video games. I like arduino and robotics, and yet I cannot get my (much) younger relatives interested in them. Computer games have won them over however.

When I consider your (our) background in control theory, I understand why you place great importance to differential equation "paradigms" for physics. I have many opinions on your two papers, some supportive, and some less supportive, but I do find the idea of a coming revolution in physics as being, too grandiose. By the way, why stop with physics instruction. Almost all engineering instruction at the universities I am familiar with also have computer software as part of their syllabi.

Some of the revolution is already happening. I know Colorado has a number of PHET simulations in JAVA, that you can download from the net. Ohanian, Classical Electrodynamics, now 20-30 years old, has boundary value problems with finite differencing. I can think of many others if given some time.

I can tell you from my experience with work interns in their upper undergraduate years, that although they may not have experience in the numerical solutions to differential equations, they do not have any problems picking up numerical analysis techniques (and especially MATLAB software) at their rather advanced ages > 20 years old!

The part I am having a hard time is the coming revolution... To what extent? Are physics educators supposed to give a "control theory state space" slant to all their physics courses. I can hear the objections from the workplace now.

Employer: My interns can solve differential equations by computer, but they don't understand why a figure skater spins faster, when she pulls in her arms. My interns don't understand the path of the orbit is an ellipse, although they can use the computer to show the path as a function of time. My interns don't understand the pressure on a submarine increases as the depth submerged.

Perhaps you can tell us, in your experience, are new-hires in the workspace, showing up with deficiencies in their education involving differential equations and numerical analysis in physics?

I expect as a educator, you are familiar with the "force-concept-inventory" The promoters of the FCI seem to be of the opinion that the problem with our physics education system is that the students are completing the sources, without even the basics: such as:

For a planet in orbit around the Sun, if the Sun's gravity is removed... Many of the students believe the planet will remain along the path and not "fly off" in a straight line. Or they do not understand the force in the elliptic orbit is always towards the Sun, not along the path etc.

It seems the promoters of the FCI would downplay the role of differential equations in favor of concentrating on the "basics".

I think the best idea is to incorporate (customize) elements of both and more points of view. It should not be "all or nothing"

This is a lengthy note and I come back to a question in case you skimmed it, (sorry for the length)

Perhaps you can tell us, in your experience, are new-hires in the workspace, showing up with deficiencies in their education involving differential equations and numerical analysis in physics? Do you feel introducing the computer earlier will ameliorate the problem?

 

[Will Flannery]

I read both papers by Will Flannery provided.

Thanks, you are a brave man.

The part I am having a hard time is the coming revolution...

Basically it's like this - the revolution has already happened in the real world, about 40-50 years ago, and it revolutionized science and engineering. The revolution is like a freight train, not coming toward us, it's already passed right through town. But, it missed the university. It is inevitable that it will eventually make it to the U.

The revolution in science and engineering is that now everything is analyzed using the computer.

So, how does a computer analyze a process, i.e. something that changes ?

From my experience everything is analyzed by simulation.

I'll give three examples - an EE designs a new circuit, what he does then is he enters the circuit description into a simulator like SPICE (simulation program with integrated circuit emphasis), there really is no other way to analyze a complex circuit. I have worked with EEs and that is how they do it.

The other example is more dramatic - consider the analysis that proceeded the Apollo mission. The flight of the Apollo represents a solution to a three-body problem, the earth, the rocket, and the moon. The three body problem is completely intractable analytically, that is you can't do anything at all with it. So, what was the design tool NASA used to design the Apollo mission? Simulation.

I even have several example from my own experience. Here is one: the first stage of Space Station Freedom was unpowered. Variation in Earth's gravitational field is sufficient to induce instability in the unpowered station to the extent that it would eventually cause it to tumble. This phenomenon has a name but I've forgotten it. The solution was to install dampers in the form of magnetic balls suspended in a viscous liquid, the balls would align with Earth's magnetic field, and the torque the balls exerted on the station, acting through the viscous fluid, would stabilize it. The stabilizers were to be built by Honeywell, my employer. How to analyze this ? I was given the job. The 'tool' was a flexible space station simulation already developed elsewhere and used for space station performance analysis. Then all that was necessary was to model the stabilizers, the gravity and magnetic field models were in the space station simulator, and fly a number of prescribed orbits. I did that. I still have the report ! Unfortunately Honeywell lost the contract for the stabilizers so I didn't see how it played out.

Now, what is the core of process simulation? Hint: In has three steps ... :).
Ans: the paradigm: physics, differential equation model, analysis using computational calculus, i.e. simulation.

So, computers model physical processes by simulating them, and the at the core of the simulation is computational calculus.

Computational calculus, together with the ability of the computer to combine the models of any number of interconnected components, literally thousands, without difficulty, is the basis of the computer revolution in science and engineering.

That is, the computer and computational calculus is not just a 'numerical method' that can supply numbers when analysis comes up short, it is now the fundamental method of analysis for complex systems. Analytic methods are in no way applicable.

The university is still teaching physics as it did 50 years ago, with, as an earlier post documents, occasional asides to the fact that if you want to analyze real processes you need to do it 'numerically'.

There are other points in your post, but I'd like to concentrate on this first.

 

[Dale]

But, it missed the university.

No, it didn’t. It was at my university last century. Maybe you just need to expand your horizon beyond USF.

 

[Will Flannery]

No, it didn’t. It was at my university last century. Maybe you just need to expand your horizon beyond USF.

USF is nearby and it has a huge physics department with 34 faculty members. Plus its degree programs are nearly identical, same course names and numbers, same degree requirements, to those of the Univ. of Florida physics department, which is ...

The Department of Physics at the University of Florida is making strides toward becoming one of the premier physics departments in the United States. We have active groups in astrophysics, biological physics, condensed matter/materials physics, and elementary particle physics. Our faculty are internationally renowned in their areas of expertise at the various frontiers of physics. Our undergraduate and graduate students participate in cutting-edge research that prepares them for successful careers in a wide variety of fields, many in of them pure or applied sciences but others drawing on the broader problem-solving and communication skills fostered by an education in physics.

So, I think it is representative.

 

[Dale]

So, I think it is representative.

And yet you portray its curriculum as being sadly deficient in an important skill that is demonstrably taught at many other universities.

Again YOU claim that this is not taught at “university”, and it is in fact taught at many universities. So be specific and fact-based in your claims. If you don’t have general evidence then don’t make general claims. If you have specific evidence then make only specific claims for which you actually have evidence.

 

[mpresic3]

Not to hijack the thread but here is something I think you should really be howling about:

In the study of differential equations as it is currently taught in math and physics at the university, you may be given a second order differential equation, say the motion of an ideal Hooke's law spring mass system for simplicity. Intermediate physics and (I suspect) math books illustrate examples where the position and velocity at some time taken to be zero are set as initial conditions, and the differential equation is solved. With state-space and control theory, you find out the observability of the system indicates the motion can be solved if you have the position at one prescribed time and the position at another prescribed time. Or you can have the velocity at two prescribed times, or the position at one time and the velocity at another time. I have never seen these problems treated outside of a course in control theory.

State space techniques are not emphasized very much in physics, yet my knowledge of them has improved my knowledge of mechanics. To some degree, I agree with you in this However, my experience with new hires suggest that lack of simulation with regards to physics is not a problem. I find quite the opposite. I find many new-hires and interns have a over-reliance on computers and software, and are likely to believe results which are clearly incorrect. (Sometimes it is not just new-hires either)

Case in point with an intern, I had. The intern had a background in physics at a rather prestigious university. The intern was given software which would visualize the Earth in space from a perspective of a rocket given an initial velocity and set off. The initial velocity in each component was 5 km/sec. The intern experimented with the program and increased the initial velocity to 6 km/sec in each component. The software showed the Earth at 5 km /sec as quite large. At 6 km /sec the Earth was shown almost as a point. The intern was sure something was wrong with the software. When he showed me this result, he was alarmed and promised to "fix" it.
I poured myself some coffee, and before I finished about a third of it, it occurred to me to do a quick calculation. It turned out the "escape" velocity of the Earth was around 6.32 (square root 3) km /sec. The software was working correctly. (Any "fix", the intern could have made to the trajectory simulation to make the Earth bigger, would have been wrong). What was disquieting to me was that I had hoped the intern with a background in physics would play to his strength and analyze the system and its inputs first, before trying to indulge in software fixes of the simulation.

Just so I do not pick on interns and new-hires. I had a colleague doctorate in aerospace engineering give me an assignment to simulate the differential equation x dot = 1 / (1 + x squared), using the runge-kutta. He was pleased when I showed him the results. Then I said to him, do you want me to compare these results to the true value?
He asked how could I know the true value?
I told him you need to solve the differential equation.
He told me, you cannot solve the differential equation, that it was non-linear.

Apparently, he believed NO nonlinear differential equation had a elementary solution and only linear differential equations could be solved. He was shocked when I demonstrated to him x = inverse tangent t was the solution.
The point I am making that computers can enhance our field, but many of the errors I see are from a over-reliance on computers and am underappreciation for approximation and elementary results.

Just to do a mea culpa, a Sr colleague of mine 20 years older, said, you were the same way. Many times I told you you would get further along with a sharp pencil and paper and putting the computer aside. He used to joke, the computer went down for a week around here and productivity went up 50%.

 

[mpresic3]

Correction to my earlier thread, x dot = (1 + x squared),

Edit by mentor with the above written using LaTeX:
$$\dot x = 1 + x^2$$

Or the version in the previous thread:
$$\dot x = \frac 1 {1 + x^2}$$
I believe the latter was the one meant.

 

[physicsponderer]

Why? They already use lots of black boxes. I simply disagree with avoiding black boxes as a desideratum. Will you also teach them how compilers work, or how computer chips calculate floating point arithmetic, or how transistors work, or how CPUs are manufactured, or ...? Why is one black box “to be completely avoided” and not others?

You cannot avoid black boxes, so their avoidance is not a good desideratum. Euler’s method is a poor method, so it seems like a total waste of time to me. Doing something badly simply to avoid one out of a thousand black boxes seems detrimental to me.

Do you want to teach numerical methods or do you want to teach physics? If you want to teach numerical methods that is fine, but don’t bill it as a revolution in teaching physics.

I think that if students can learn numerical methods, and computer programming as a bonus with no extra investment in time while learning the same amount physics, that is good for them.

 

[physicsponderer]

I read the beginning of the paper. Yes, it may really be a revolution. Every student should do some coding, arguably. All the more true of a physics student.

A small nitpick, if I may? The acceleration due to gravity is 9.8 m/s and so it is distracting and confusing to see 9.9 m/s being used. I know it was calculated, but surely it would have been worth having a more accurate calculation so that students see the familiar, correct value ?

 

[physicsponderer]

A slightly bigger nitpick/question: wouldn't it have been better to have the MATLAB variables labelled the same way as in the equations? As it is, the by hand calculations have v(1) = -9.9 does which does not match v(1) = 0; % Initial velocity in the code. Wouldn't it reduce the cognitive load if the labeling was the same? Or am I missing something?

 

[physicsponderer]

I kind of like Euler's method as a teaching tool and have used it to teach high school students to compute rocket trajectories as well as to compute forces from videos of experiments. For systems where the forces are changing slowly, it is not bad, and it can be implemented in a spreadsheet. Students can learn it even before they have a clue about calculus with the idea that for sufficiently short times, the assumption of constant acceleration is "good enough" for sufficiently short time steps. Then students can shorten the time step and see if their calculations change much. Sure, the spreadsheets might have 1000-10000 rows, but they are all cut and pasted from the first few, and this allows students to easily see how the motion is progressing.

I see use of Euler's method in a spreadsheet as a stepping stone for students to using numerical methods in black boxes. It gives some insight into what is going on inside those black boxes and allows solving a bunch more systems than the assumption of constant acceleration. There will always be a more efficient numerical method just around the corner. But computers today are pretty powerful - it may be a fine approach to sacrifice numerical efficiency for conceptual simplicity for certain parts of introductory training.

But having taught introductory college physics for lots and lots of years, making scientific programming part of the course or a prerequisite for it is going to be a hard sell for most intro courses (except majors courses). It will be a great way to make students hate the teacher. Students in introductory classes are open to spreadsheets, not so much to learning to program.

I agree with your use of spreadsheets. I think it is a step that should not be omitted between the calculations and the graph when studying. Learning to make a table is a very useful thing. It can even be used outside of physics, and there is no need to go the whole way and plot a graph. I have used a table to work out how much electricity I was using doing different things by entering times and electricity meter reading into a table on paper (and how much it cost in money) which is sort of physics but also sort of business or just life.
Hard to believe physics students would hate to learn a tiny easy bit of coding. They should love the teacher, not hate him or her.

 

[physicsponderer]

Without math, one isn't really teaching physics. The question is how to approach the challenges to spend less time with the math and more time with the physics. Black boxes are one approach. Euler's method is another. It may well prove to be a revolution in teaching physics, at least for courses that adopt it. I doubt I'll ever teach another physics (or other quantitative science class) without using spreadsheets on a weekly basis. A spreadsheet takes repetitive calculations (and the calculator) out of the student's hands and also leaves a documented trail students can refer back to for what was done. Work on calculators disappears every time the student presses C.

When I was first exposed to numerical methods as an undergraduate, it was a real eye opening experience. The challenges of tough integrals, derivatives, and differential equations melted away, and I realized that the real challenge of physics was reducing the problem to an equation that could then be solved on the computer. Before then - I saw reducing the problem to an equation as about the first 30% of the problem and then beating that equation into submission with the traditional analytical (pencil and paper) techniques as the last 70% of the task at hand. Physics was mostly math, and I was not good at math, and I did not like math. About the same time, I also felt cheated by realizing that all the prior problems had been cherry picked to be solvable with traditional pencil and paper methods - most problems cannot be solved with pencil and paper. These revelations should really occur before a physics major is a junior in college.

So, I do think an approach to empower physics majors to solve more general problems in their first year would be a revolution in physics education, at least for majors. Some of the students I mentor are beginning to see similar things in their introductory undergraduate courses.

"Without math, one isn't really teaching physics." I disagree. You seem to be implying that physics is a branch of mathematics. I take the view that maths is an indispensable part of physics, but physics without maths is still physics. It's possible understand parts of physics without maths, and very deeply, too. In fact, too great an emphasis on maths, in the ordinary sense of equations and calculations, can result in losing a certain intuitive understanding, I think. So many physicists who are great at maths, will flunk simple, fun, physics puzzles that are put in the form of words. Relativity Visualized by Lewis Carroll Epstein is light on maths indeed and yet I think gives a better understanding than most degree courses at the time did.

 

[physicsponderer]

"When I was first exposed to numerical methods as an undergraduate, it was a real eye opening experience. The challenges of tough integrals, derivatives, and differential equations melted away, and I realized that the real challenge of physics was reducing the problem to an equation that could then be solved on the computer. Before then - I saw reducing the problem to an equation as about the first 30% of the problem and then beating that equation into submission with the traditional analytical (pencil and paper) techniques as the last 70% of the task at hand. Physics was mostly math, and I was not good at math, and I did not like math. About the same time, I also felt cheated by realizing that all the prior problems had been cherry picked to be solvable with traditional pencil and paper methods - most problems cannot be solved with pencil and paper. These revelations should really occur before a physics major is a junior in college."

I totally agree and I am blown away by this. I think many students of physics think the same way. I know I did. Solving differential equations numerically is so much easier and so much fun, using a computer, as shown by 3Blue1Brown in a Youtube video:[/QUOTE]

 

[physicsponderer]

Have you seen this? It has a bit where a Python program is used to solve numerically the ODE of a free pendulum. Did he really write that code that fast or was it sped up or rehearsed I wonder (lol).

Maybe your students would like it.

 

[physicsponderer]

I taught high school physics for 15 years with some success. The difficulties were not in solving problems once the underlying physics was understood but in undoing the years of “false physics” that most students arrived having internalized. In the 90s there was a movement to measure the basic physics beliefs of incoming students and then to do a second measurement after taking basic physics. Most students in traditional courses never improved their basic understanding, so in response a course was developed with much success at Dicinson College and elsewhere. Unfortunately, thought the course had a high rate of success in improving basic understanding, it required a lot of technology to implement and never caught on widely.

I'd like to know more about the "years of false physics" and the technology needed to counter their effects.

 

[Will Flannery]

A slightly bigger nitpick/question: wouldn't it have been better to have the MATLAB variables labelled the same way as in the equations? As it is, the by hand calculations have v(1) = -9.9 does which does not match v(1) = 0; % Initial velocity in the code. Wouldn't it reduce the cognitive load if the labeling was the same? Or am I missing something?

You're missing something. v(1) is not the velocity after 1 second, it's the velocity at the beginning of the first subinterval, that is, when t = 0, then v(2) is the velocity at the start of the 2nd subinterval, and so on.

 

[physicsponderer]

You're missing something. v(1) is not the velocity after 1 second, it's the velocity at the beginning of the first subinterval, that is, when t = 0, then v(2) is the velocity at the start of the 2nd subinterval, and so on.

I didn't say it was the velocity after 1 second, I said there seemed to be an unnecessary cognitive load caused by the mismatch.

 

[Will Flannery]

I didn't say it was the velocity after 1 second, I said there seemed to be an unnecessary cognitive load caused by the mismatch.

Great Scott, on going back and checking, you're right. Actually, the variable does have the same name, v in the hand calculation and v in the program, it's the change in the nature of the argument that creates the cognitive dissonance, v(1) = -9.9 and v(1) = 0. In the hand calculation v(1) = -9.9 the argument is time, in the program statement v(1) = 0 the argument is an index into an array, with v(i) being the velocity at the start of the ith subinterval. As we sometimes say on the bandstand, my bad.

 

[Frabjous]

Sherwin’s book Basic Concepts of Physics (1961) had students solving difference equations. There might be some ideas in it that resonate with you.

 

[Dr. Courtney]

"Without math, one isn't really teaching physics." I disagree. You seem to be implying that physics is a branch of mathematics.

Not at all. What if I had said, "Without reading, one cannot really teach law" or "Without reading, one cannot really teach history"?

At the high school and college levels, one is not really teaching law or history with the appropriate level of rigor if one does not require the students to _READ_.

So sure, there are some elementary physical principles that can be taught to students without requiring them to do math. But with the exception of physics courses with "Conceptual" in the name, one is not being honest about the rigor if one is teaching high school or college physics without requiring students to do the math.

What would you think of a law school that did not require their law students to read? This is how I regard physics teachers who do not require their students to solve quantitative problems.

 

[Will Flannery]

So sure, there are some elementary physical principles that can be taught to students without requiring them to do math. But with the exception of physics courses with "Conceptual" in the name, one is not being honest about the rigor if one is teaching high school or college physics without requiring students to do the math.

Well, how about being more specific? What college physics? What math?

My premise is that physics since the time of Newton has been based on differential equations. Physical laws governing processes are written as differential equations. Physical systems are modeled using differential equations. A real world system is analyzed by deriving a differential equation model of the system and analyzing the model.

So, I claim that there is very little physics without differential equations.

Trying to be more specific, what physics requires what kind of math? First, what kinds of math are there? Arithmetic, algebra, calculus, and differential equations**. What physics can be taught with arithmetic only? Algebra only? Calculus minus differential equations only? Calculus with differential equations?

**and I've just become aware of another type of math that used to be very prominent in physics but seems to be less so now(?), real analysis as is typified by A First Course in Modern Analysis - Whitaker - pdf. There is a section on 'The Equations of Mathematical Physics' that seems to be various functions useful for solving PDEs, e.g. Legendre functions, Bessel functions.

 

[Dr. Courtney]

So, I claim that there is very little physics without differential equations.

No need to chase all those rabbits. I would agree that one can only learn a small fraction of ALL physics without differential equations, but a small fraction of a very huge amount can still be a large amount.

I can recall learning a large amount of physics (a semester's worth) in my 9th grade physical science class with little more than 3 letter formulas (algebra) and a tiny bit of trig (SOHCAHTOA). I felt it was an excellent use of the 90 class days, it included the fundamentals of experimental physics (comparing theory with experiment), and laid a foundation for the vector analysis I'd see in 12th grade physics. The simplifications needed (no air drag, constant forces and accelerations) and limitations were clear. More importantly, the class inspired me to further study of physics (eventually majoring in it, earning a PhD, and a career in physics). Trying to do much with differential equations would have been confusing and likely turned me off - I had neither the math prowess nor the computer skills for it. I took the same approach in teaching my two sons' physical science - it has been wildly successful, as they are now both physics majors at a big state school doing well in their coursework and pleasing their research advisers as they plan for grad school.

My 12th grade physics class was based on algebra and trig. The lack of calculus and diff eq did not prevent it from being a very productive year for me. I learned a lot of physics - both key experimental things as well as just about all of the mechanics, E&M, and thermodynamics one could learn with the math skills I had. I was still far off from learning calculus and had zero computer skills. I took the same approach to my sons' high school physics class. No need for calculus or diff eq, but a strong experimental focus - accurate experiments testing whether the experiment agrees with predictions of theory. One son did have a research course in which I taught him to numerically integrate diff eqs to understand the motion of a complicated case of changing forces, but the other son did not. If all the students have the math and computer skills, the 12th grade physics class is a reasonable place to introduce some of these things, but I wouldn't build a whole course around it unless all the students had calculus.

I've often taught algebra and trig based college physics courses. There is plenty of physics one can teach to make good use of the two semesters without calculus or diff eqs. I like a strong experimental focus, lots of stress on the limitations and simplifying assumptions, and full vector-based quantitative problem solving. If students have the computer skills, I do like to introduce spreadsheet solutions to problems with changing forces. But I don't write down the differential equation or talk about calculus - that would freak them out. I simply show them how to apply the rules for constant forces point by point over smaller time scales and explain that it can give a good approximation, because over a short enough time scale, most varying forces are approximately constant. This paper gives the gist:
https://arxiv.org/pdf/0903.1555.pdf

 

[Frabjous]

Legendre and Bessel functions are still taught in physics. For example, see TOC of Arfken, a common mathematical physics book.
https://www.amazon.com/dp/0123846544/?tag=pfamazon01-20

Real analysis provides many of the theoretical underpinnings of calculus and differential equations. With this effort being successful, there is less direct overlap of physics and modern real analysis. Whitaker is a century old and was written when these topics were at the forefront of real analysis.

Update: There is a great book by Duren ”invitation to classical analysis” which has all sorts of neat analysis topics at the undergraduate level

 

[Will Flannery]

No need to chase all those rabbits.

Well ... on the other hand ... I remember when I was a freshman at UF taking an honors physics course, and thinking to myself ... here I am 18 years old in an honors class and learning Newtonian physics, when I should be learning Einsteinian physics, 100 years (not really!) after the fact. It was only as a post doc working at Honeywell that I discovered I had not learned Newtonian physics in the freshman class, what I had learned was pre-Newtonian physics, and that Newtonian physics was much more difficult than I'd thought. It is so difficult that I think it's rare to see a solution to the Kepler problem, which marks the beginning of modern science and technology, in college physics. To my mind physics begins with Newton (I was never an experimentalist).

And, I think you're wildly underestimating the capabilities of high school students. Of course I base my opinion on just two students, the best in my kid's high school, whom I taught in a pilot class. They went through the stuff in the OP paper like fish through water. I would like to have a bigger class but the 'curriculum committee', i.e. the math and science teachers at the school, thought the class was nonsense. I think the subject as per my class, which totally replaces analytic calculus with Euler's method, is simple enough that any motivated ! high school science student would not have a problem. I can't imagine even trying to teach it to unmotivated students.

And, I remember from high school that the best thing about high school physics was trajectories of thrown objects. So, differential equations occur right at the start of the study of physics. And in your linked pdf you inject a little calculus lite with differentials and equations containing differentials. So it's no great leap to write down the differential equation for a falling object, and that's usually done in high school physics, A=GM/RR, even though it's not called a differential equation. So all you have to do is introduce Euler's method and you can generate orbits, comet trajectories, rocket trajectories, etc., just like I did at Honeywell. Euler's method is basically an application of the formula distance = velocity*time, and that's an easy rabbit to catch !

 

[Dr. Courtney]

I can't imagine even trying to teach it to unmotivated students.

I can't imagine ever having a class of any size without a significant number of unmotivated students. It's a fantasy to think one will revolutionize anything without succeeding with unmotivated students.

 

[physicsponderer]

Great Scott, on going back and checking, you're right. Actually, the variable does have the same name, v in the hand calculation and v in the program, it's the change in the nature of the argument that creates the cognitive dissonance, v(1) = -9.9 and v(1) = 0. In the hand calculation v(1) = -9.9 the argument is time, in the program statement v(1) = 0 the argument is an index into an array, with v(i) being the velocity at the start of the ith subinterval. As we sometimes say on the bandstand, my bad.

Thanks for saying I was right, but I did not say 'cognitive dissonance', but rather 'cognitive load.'
Can the lesson in your paper be changed to get rid of the mismatch? If so, how? Not sure what you mean by 'As we sometimes say on the bandstand'. Could you please clarify that for me?

 

[Will Flannery]

Thanks for saying I was right, but I did not say 'cognitive dissonance', but rather 'cognitive load.'
Can the lesson in your paper be changed to get rid of the mismatch? If so, how? Not sure what you mean by 'As we sometimes say on the bandstand'. Could you please clarify that for me?

Well, it's a sticky wicket ... that I recognized from the beginning but I didn't have an easy solution so I just let it slide and forgot about it. I still don't have an easy solution.

Everything is solid in the paper thru the line - 'End of first lecture'.

It's the program that lacks sufficient explication.

Now, looking at the problem, I see I should have included in the paper -
"The time interval of interest 0-T is divided into N evenly spaced subintervals - 0=t1,t2, ... tN+1=T".
This line is necessary and I'm surprised I didn't include it, I've written it many times.

Note: I start with t1 instead of t0 because MATLAB array indices start with 1.

And I now see that I did not explain that the program stores the computed positions and velocities in arrays, so they can be graphed. That line should be included and there should have been program statements declaring the arrays, for clarity (they're not required in MATLAB).

And the program comment should be changed to
v(1) = 0; % velocity at start of first subinterval, v(t1)=0

I think that fixes it. My hobby is playing music and the only place I've heard the expression 'my bad' is on the bandstand, following aural dissonance.

 

[Dale]

I base my opinion on just two students, the best in my kid's high school, whom I taught in a pilot class. They went through the stuff in the OP paper like fish through water.

This is a bad basis for a pedagogical opinion. The two brightest students with a two to one student teacher ratio is essentially a guaranteed success regardless of how good or bad the pedagogical approach itself is.

 

[Buzz Bloom]

The debate in this tread seems to ignore an important general truth. A great many problems have more than one right answer.

 

[jasonRF]

Well ... on the other hand ... I remember when I was a freshman at UF taking an honors physics course, and thinking to myself ... here I am 18 years old in an honors class and learning Newtonian physics, when I should be learning Einsteinian physics, 100 years (not really!) after the fact. It was only as a post doc working at Honeywell that I discovered I had not learned Newtonian physics in the freshman class, what I had learned was pre-Newtonian physics, and that Newtonian physics was much more difficult than I'd thought. It is so difficult that I think it's rare to see a solution to the Kepler problem, which marks the beginning of modern science and technology, in college physics.

As an 18 year-old I was a lot like you were - I thought a lot of things that weren't true!

More seriously, in my opinion a physics or engineering program is failing their students if they graduate without being exposed to some set of tools they can use to compute rocket trajectories or solutions of Kepler's equation (perhaps with some guidance from standard resources available in the library or online). Whether or not they solved that exact problem in their curriculum is much less important. I've worked with people holding physics and engineering degrees from a number universities. Anecdotally, it has been much more common for new-hires to have conceptual shortfalls or weak analytical skills than to have trouble throwing the computer at a problem. This is of course a non-scientific observation with a small sample size (although >> 2!).

**and I've just become aware of another type of math that used to be very prominent in physics but seems to be less so now(?), real analysis as is typified by A First Course in Modern Analysis - Whitaker - pdf. There is a section on 'The Equations of Mathematical Physics' that seems to be various functions useful for solving PDEs, e.g. Legendre functions, Bessel functions.

I've had a copy for years and agree that Whittaker and Watson has a lot of gems that aren't easy to find other places. While it does cover some real analysis, the majority of the text is on functions of a complex variable. As caz wrote, many of the topics in that book are standard fare in mathematical physics texts/classes. At least where I was in school, every physics major I knew took at least one (and most took 2 or more) upper division courses in mathematical physics or applied math that included some combination of complex analysis, method of Frobenius, various special functions such as Bessel and Legendre, integral-transform and series solutions of PDEs, etc. I took a couple of those classes even as an engineering major.

Of course, in some sense this approach is the opposite of your "revolution", especially when you look at some uses of the more obscure special functions that allow for more problems to be "cooked" to get a closed-form solution. For example, Whittaker and Watson has an entire chapter on Mathieu functions, which (among other things) exactly solve an ODE for a simple parametric oscillator. Fortunately, instead of just quoting the exact solution, the professor who taught my junior-level mechanics course introduced us to perturbation theory which we then used to find approximate solutions of that same ODE. This provided us with more physical insight, and gave us a tool that is useful in many (most?) branches of physics and engineering.

If you are ever bored, try working the exercises in the book. They should keep you occupied for a long time!

jason