The coming revolution in physics education

[Will Flannery]

Is this true? https://www.amazon.sg/University-Physics-Modern-Hugh-Young/dp/0135159555 says it's for courses in calculus-based physics.

The text for PHY 2048-2049 is 'calculus based', calculus is a prerequisite for the courses at USF. Yet, the text is almost completely differential equation free ... you can download a free pdf, for example at https://www.academia.edu/41736532/University_Physics_With_Modern_Physics_14th_Edition_by_Hugh_D_Young_Roger_A_Freedman

Checking the curriculum guide for the University of South Florida (USF) [1], a physics major takes one calculus course, MAC 2311 – Calculus I, and no physics courses in the freshman year and two math courses, MAC 2312 – Calculus II and MAC 23113 – Calculus III, and two courses, PHY 2048 – General Physics I and PHY 2049 – General Physics II, in the sophomore year. Differential equations are not covered in the three math courses [3]. Differential equations are covered in the third-year course PHZ 3133 – Mathematical Method for Physics.
Note: I've got links for all this.

The first mention of differential equations in Young and Freedman (use the search function on the pdf) is on page 276, they are mentioned in passing. The fourth mention, on page 415 in the section on planetary motion, reads

These results can be derived by a straightforward application of Newton’s laws and the law of gravitation, together with a lot more differential equations than we’re ready for.

Thus planetary motion is covered without the DE model, i.e. A=GM/RR, electric circuit analysis without the DE models of capacitor and inductor, heat transfer is covered without Fourier's law, fluid mechanics without the Navier-Stokes equations, electrodynamics without Maxwell's equations in differential equation form, etc.

 

[hutchphd]

So USF would not be my first choice for a physics major.
This is not evidence for your thesis however.

 

[Vanadium 50]

If your ideas have merit, once is enough. If they don't, even a thousand won't help.

This was not a suggestion you work your way up past a thousand.

 

[Dale]

university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations

This may be the case at USF, but I dispute your claim that it is “typical”. Do you have a scientific reference supporting your claim?

I gave several counter-examples earlier in the thread, so I am highly skeptical about the correctness of your claim here.

 

[Will Flannery]

This may be the case at USF, but I dispute your claim that it is “typical”. Do you have a scientific reference supporting your claim?

I gave several counter-examples earlier in the thread, so I am highly skeptical about the correctness of your claim here.

I base it on the following: the Young and Freedman text is one of the world's most popular introductory physics texts according to wiki.

I've just checked the Resnick and Halliday text used at UF (a free pdf download is available) which is similar but does use differential equations in the section on electric circuit analysis.

Also the OpenStax text is similar (a free download is available).

All of the texts cover planetary motion without Newton using Kepler's laws, heat transfer without Fourier's law, fluid mechanics without the Navier-Stokes equations, and electrodynamics without Maxwell's equations in differential equation form.

Also from a post above ... https://web.mit.edu/physics/prospective/undergrad/transfer.html ... the way I read it MIT has allows a student to transfer credit for their introductory physics courses from another college by taking a test ...

The Physics Validation Exams are three-hour, closed book exams covering Classical Mechanics (8.01) or Classical Electromagnetism (8.02) at a level of calculus-based introductory physics texts for science and engineering students such as: University Physics by Young and Freedman; Physics by Halliday, Resnick and Krane; ...

So, I don't claim the avoidance of DEs is universal (the UC Berkeley text does give an analytic derivation of Kepler's laws I think, but that's atypical for an introductory course, the junior level course in classical mechanics at USF doesn't, for example), but I think it is typical.

 

[Dale]

None of which constitutes adequate support for your claim on this forum.

 

[gmax137]

just as a single meaningless data point, when I was in school the intro classes for physics majors was mechanics by Kleppner & Kolenkow followed by EM by Purcell. if they offered a "Halliday..." intro physics course, it was for the pre-meds, not physics majors.

maybe this wasn't all to the good - the physics majors all knew our path in high school or even earlier. Maybe the root of the diversity problem often discussed.

 

[Will Flannery]

just as a single meaningless data point, when I was in school the intro classes for physics majors was mechanics by Kleppner & Kolenkow followed by EM by Purcell. if they offered a "Halliday..." intro physics course, it was for the pre-meds, not physics majors.

maybe this wasn't all to the good - the physics majors all knew our path in high school or even earlier. Maybe the root of the diversity problem often discussed.

Kleppner is the UC Berkeley text (I downloaded a free pdf), and ... I'm glad I didn't take that class, and I was math major. The chapter on central force motion is way tough ... yet I note that it does not solve Kepler's problem, i.e. position as a function of time ... instead we have ...

Equation (10.11) formally gives us r as a function of t, although the integral may have to be done numerically in some cases.

where 10.11 is a nasty looking integral from r0 to r involving expressions for total mechanical energy and effective potential energy.

 

[hutchphd]

There is a considerable difference between showing analytically that certain functions solve differential equations of motion and developing the skillset to solve ab initio those equations. For the most part that latter skillset is learned in the junior and senior level courses. But one certainly should not delay providing the differential equations and analytic solutions.
When I arrived at Cornell 50 yrs ago I literally did not know what an integral sign was and took "noncalculus" freshman physics from Sears and Zemansky (no Young). Even in that context I was exposed to the details of the harmonic oscillator first semester much to my edification. The solutions to uniform constant accelerated motion were presented and conjoined with initial conditions to solve the differential equations. By the beginning of sophomore year I was much the better for it. Incidentally the calculus based course was taught from Halliday and Resnick.
If remedial work is required by some students then by all means we need to be creative in supplying it efficaciously. But to abjure the teaching of analytic methods central to the subject is ridiculous, and the notion that existing programs postpone these subjects as you describe is absolutely not true for any institution that I know.

 

[Will Flannery]

Here is what happens - (as I recall) - in high school you derive the equation A = GM/RR, but that is deemed to difficult to solve so it is simplified to A = -9.8 and it's easy even in high school to solve this heuristically, so that is done, and you calculate the trajectories of falling and thrown objects near the surface of the earth.

Here's what they don't tell you. You will never see the equation A = GM/RR solved, it does not have a closed form solution and I'll challenge anyone to find an analytic series solution in any text. I don't know of one. (I do know of a Texas A&M website where Kepler's laws are derived from the DE, and a solution is given for Kepler's problem (I think.), i.e. computing position as a function of time from Kepler's equation)

Let's see what Kleppner does ... and I've seen this elsewhere ... he alludes to a solution, noted in my previous post, but doesn't provide it. Instead he writes ...

Often we are interested in the path of the particle, which means knowingr as a function of θ rather than as a function of time.

and that problem is solved.

However, I worked for years on space projects, Star Wars !, the first use of laser gyros in space, Space Station Freedom stability control, and the Mars Observer. and in every project position as a function of time was basic, and position as a function of angle was never a consideration. That is, incidentally, when I discovered that computational calculus is how problems are solved. Not in the U. It was a revelation to me.

The reality is that the two-body problem doesn't have a closed form solution, and the three body problem, e.g. rocket trajectories between to Earth and moon, are completely intractable analytically.

They never tell you this at the university. So ...

There is a considerable difference between showing analytically that certain functions solve differential equations of motion and developing the skillset to solve ab initio those equations. For the most part that latter skillset is learned in the junior and senior level courses.

No, the differential equation models of real systems are almost always completely intractable. The skillset to solve them doesn't exist.

But to abjure the teaching of analytic methods central to the subject is ridiculous, and the notion that existing programs postpone these subjects as you describe is absolutely not true for any institution that I know.

I'm not suggesting to delay teaching analytic calculus, and I'm not saying that current programs do. I'm saying they delay teaching differential equations. What I am suggesting is teaching modeling physical processes with differential equations at the very beginning, in and of themselves they are not difficult at all, they don't even involve calculus. It's when you try to solve them analytically that things get difficult, and I would delay that just as it is done now.

However, computational calculus is trivially easy, it computes solutions to all differential equations, even analytically unsolvable ones, no problem, and it can be taught to high school students in a single one hour lecture and immediately be used to analyze any number of real complex physical systems, e.g. the Apollo trajectory, electric circuits, etc. This will completely transform physics and STEM education.

And computational calculus is not a 'black box', in its simplest form, Euler's method, it is a trivially easy formula, distance = velocity * time, or generally change = (rate of change)* time. That's it ! It is intuitively obvious and transparent, it completely demysitifies differential equations.

 

[Dale]

They never tell you this at the university.

This is certainly false. The claim that “they never tell you this at the university” means literally that no university teacher at any university has ever told this fact to any single university student at any time in history.

You really oversell this thing so much that you completely destroy your credibility.

 

[hutchphd]

They never tell you this at the university. So ..

They certainly told me at university. Why else would I have learned (at university) variational methods and sequential approximation methods and perturbation expansion methods and constants of the motion and Lagrange multipliers and, yes, numerical methods. So...give me a break...

 

[gmax137]

My high school math teacher told us all the integrals we were working on had been cooked up to be solvable, and "in the real world" integrals were solved numerically, or by plotting and counting the squares, or by plotting and cutting them out with scissors and weighing the paper.

 

[Will Flannery]

Well, I can think of several examples myself ... this is from the text for the USF physics department classical mechanics course - Classical Mechanics by Thornton and Marion:

“The actual motion of a rocket attempting to leave earth’s gravitational field is quite complicated. For analytical purposes, we begin by making several assumptions. The rocket will have only vertical motion, with no horizontal component. We neglect air resistance and assume that the acceleration of gravity is constant with height. We also assume that the burn rate of fuel is constant. All these factors that are neglected can reasonably be included with a numerical analysis by computer”

The above is on page 374 in a section on Rocket Motion, and the text continues analyzing 'Vertical Ascent Under Gravity', with lots of fairly complex math including integral equations, to analyze a rocket going straight up with constant gravity. They vary this and that parameter to give themselves a problem to solve. And, that's it for rocket motion.

That's the only thing a USF physics major will learn about rocket motion.

Come to think of it, when I studied the upper division courses at USF in the physics, ME and EE departments, several of the texts noted that the systems they were analyzing were artificially simplified to make them solvable using analytic methods, and that computational methods were used for realistic systems ... and I took notes ...

From the text for the ME course on heat transfer ... Heat and Mass Transfer by Y.Cengel, A. Ghajar:

So far we have mostly considered relatively simple heat conduction problems involving simple geometries with simple boundary conditions because only such simple problems can be solved analytically. But many problems encountered in practice involve complicated geometries with complex boundary conditions or variable properties, and cannot be solved analytically. In such cases, sufficiently accurate approximate solutions can be obtained by computers using a numerical method.

From the text for the ME course in vibration ... Engineering Vibration by D. Inman

So far we have mostly considered relatively simple heat conduction problems involving simple geometries with simple boundary conditions because only such simple problems can be solved analytically. But many problems encountered in practice involve complicated geometries with complex boundary conditions or variable properties, and cannot be solved analytically. In such cases, sufficiently accurate approximate solutions can be obtained by computers using a numerical method.

”

From the text for the ME course in Fluid Dynamics -Fundamentals of Fluid Mechanics by P. Gerhart, A. Gerhart, J. Hochstein,

Unfortunately, because of the general complexity of the Navier–Stokes equations (they are nonlinear, second-order, partial differential equations), they are not amenable to exact mathematical solutions except in a few instances
...
With the advent of high-speed digital computers it has become possible to obtain numerical solutions to these (and other fluid mechanics) equations for many different types of problems, including both inviscid flows and boundary layer flows.

So the texts are admitting that realistic problems are solved using the computer, even while they are teaching the analytic methods that cannot be applied to real problems.

Which begs the question of course, since the computational methods are very easy and very powerful, and the analytic methods are very difficult and can't be used to analyze realistic problems, then ... why not teach the computational methods?

So I guess I got carried away for a minute ... but also ... maybe a year ago I did look at how five or six upper division classical mechanics books treated central force motion - and none gave a method of solving Kepler's problem, i.e. of computing position as a function of time, and that's the problem that usually needs to be solved*, and none explicitly admitted that fact, and this includes Thornton and Marion. So, that's where I got the notion.

* and note that Newton's solution to Kepler's problem marks the start of modern math and physics, so this is probably the most important problem in the history of science and math.

 

[hutchphd]

Well, I can think of an example myself ... I quote an example in the appendix to a paper I'm working on now .. this is from the text for the mechanics course - Classical Mechanics by Thornton and Marion:

The fact that one can use computers to solve real world problems does not mean they should be used pedagogically the way you propose. I happen to know Steve Thornton I can guarantee you he would think this a bad idea. But that doesn't mean NASA shouldn't have used computers on the space shuttle when his wife Kathy was being launched to fix Hubble.

Over and out. I'm done.

 

[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?