The coming revolution in physics education

[Will Flannery]

Is this really before calculus? At the bottom of page 1, you say:

"Newton’s model for a falling object consists of state variables for position, r, and velocity, v, and the rate equation
r' = v(t)
v' = -Gm/r^2

How is that to be understood without calculus - aren't r' and v' derivatives?

r'(t) = v(t) is an equation for the velocity of the object at time t, the name of the function is r', so, it's just the name of a function. When you take your calculus course you'll find out that the velocity of a function r(t) is usually denoted r'(t) and it's called a derivative.

Ditto for v'(t) = -Gm/r(t)^2

That is, a process model is a set of variables, var1, var2, ... varn, and an equation for the rate of change, i.e. velocity, of each. So I need a name for the function that is the rate of change of var1, (and var2, etc.) and I called it var1' just as is done in calculus. I could have called it rate_of_change_var1 or any other name.

Try this - read thru the paper attached in the OP, up to 'end of first lecture'. You should be able to understand every step. If not, it's on me - post a question.

 

[Tom Hammer]

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 hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

 

[Dale]

I hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

I agree wholeheartedly with this. I had asked a similar question earlier. I don’t know if there is solid evidence that the use of numerical methods improves conceptual understanding.

https://www.physicsforums.com/threa...-physics-education.954664/page-3#post-6055004

@Will Flannery is clearly of the opinion that it does, and @bob012345 is clearly of the opinion that it is detrimental. I would like to see scientific evidence of its efficacy one way or the other.

 

[atyy]

r'(t) = v(t) is an equation for the velocity of the object at time t, the name of the function is r', so, it's just the name of a function. When you take your calculus course you'll find out that the velocity of a function r(t) is usually denoted r'(t) and it's called a derivative.

Ditto for v'(t) = -Gm/r(t)^2

That is, a process model is a set of variables, var1, var2, ... varn, and an equation for the rate of change, i.e. velocity, of each. So I need a name for the function that is the rate of change of var1, (and var2, etc.) and I called it var1' just as is done in calculus. I could have called it rate_of_change_var1 or any other name.

Try this - read thru the paper attached in the OP, up to 'end of first lecture'. You should be able to understand every step. If not, it's on me - post a question.

If v' is just the name of a function, then it has no relation to v?

 

[Will Flannery]

*

If v' is just the name of a function, then it has no relation to v?

I'm using this convention, which is standard: v' is the name of the function that returns the velocity of v, but it's just a convention. I could have used a different one: v# is the name of the function that computes the velocity of v (maybe I should have).

 

[atyy]

*

I'm using this convention, which is standard: v' is the name of the function that returns the velocity of v, but it's just a convention. I could have used a different one: v# is the name of the function that computes the velocity of v (maybe I should have).

So what is "velocity"?

 

[Dr. Courtney]

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.

My experience has been much different. Pretty good Physical Science courses early in high school have done a good job with my students undoing the "false physics." The challenges I've had were more on the math side: very poor algebra skills, no trig to speak of, complete inability to solve word problems involving multiple steps, etc.

I hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

My view is that the focus of the lab side of the course should be the scientific method. How many high school classes spend 25-40% of the class time on real labs (that test hypotheses, require data analysis, and written discussions)? How many high school classes complete 15-20 real physics experiments?

It is very hard to teach and learn the scientific method in a meaningful way when the lab portion of the courses consistently get short shrift.

 

[Zeynel]

https://mail.aol.com/webmail/getPar...ope=STANDARD&saveAs=PDE+chart+small.jpg[/IMG]

* what Newton did: see http://farside.ph.utexas.edu/teaching/336k/lectures/node32.html

Sorry, a nitpick. This language does not reflect reality. Newton did none of the things shown in the link. Newton did not work with equations but with proportions. Newtonian dynamics as presented in the link was developed years after Newton's death. Newton did not even know about "Newton's constant G". Why is this important? I think it is important to conform to historical developments.

 

[Will Flannery]

Sorry, a nitpick. This language does not reflect reality. Newton did none of the things shown in the link. Newton did not work with equations but with proportions. Newtonian dynamics as presented in the link was developed years after Newton's death. Newton did not even know about "Newton's constant G". Why is this important? I think it is important to conform to historical developments.

You're right - although I wouldn't worry about the gravitational constant which just adjusts for the units of measurement - the reference should have read something like ...
* There is no analytically defined r(t) such that r'' = G∙m2 / (r∙r). Newton derived Kepler’s laws, from which Newton (and Kepler) derived numerical procedures to approximate r(t). For a modern treatment see ... link.

 

[grandbeauch]

The answer to that question seems obviously to be the black boxes. It will take slightly less time to teach the inputs and outputs of the black box than to teach Euler’s method, and it will take far less time to use it.

When they are done they will have the same understanding of physics in a shorter amount of time and they will know how to use a computational tool that they can use professionally for the rest of their life even to solve problems where Euler’s method flat out fails.

 

[Dr. Courtney]

https://aapt.scitation.org/doi/10.1119/1.5055324

Nice article in this month's TPT on using Euler's Method (in cognito) in a spreadsheet to solve for motion with a varying force.

My classroom approach is very similar and accessible to high school students who don't even know what Calculus really is yet.

 

[Will Flannery]

Since Newton, the basic paradigm for the analysis of physical systems in classical physics has been:
1. State the physical laws governing the system
2. Derive a differential equation model of the system
3. Solve or otherwise analyze the model from step 2.

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible. At USF (U of South Florida) a physics major takes a course in DEs in his/her junior year!

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

The remedy is to teach the paradigm, along with computational calculus in the form of Euler's method for ODEs and the finite difference method for PDEs, at the very beginning of physics/STEM education, in a course dedicated to that purpose and called Scientific Programming. I've written a paper that describes the course titled 'A New Curriculum for Classical Physics', that fills in many of the details, you can see it here ...
http://www.berkeleyscience.com/ANewCurriculum.pdf

 

[mpresic3]

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible.

This is NOT the reason the "basic paradigm" , as you call it, is not taught in high-school and university. One important reason simple harmonic motion for a pendulum is taught with simplifying assumptions, is to show how much mileage can be obtained from solving a (elementary and straightforward) second order differential equation. Solving the same problem with elliptic functions, or a numerical method will lead to a better solution, but at the expense of time and effort. The time and effort may be manageable for students in physics, but what about the pre-med, or social science student, does he or she need the differential equations?
The simple pendulum is only one example. I am not a professional educator but my time as a recitation instructor, and time in post graduate education (probably around 10-15 years in total), suggests that physics educators should stress how approximations are useful in getting to the heart of physics problems, and how (as another example), the central force problem for the path of a body under the inverse-square law can be (cleverly) solved using conservation laws, changes of variable, and other techniques.

This is not to say that we should dismiss your idea of including scientific programming, and numerical solutions to differential and partial differential equations. I make my living doing just this. A course (or maybe even two) should probably added to the physics curricula, but the curricula is pretty tight these days with quantum mechanics, electricity and magnetism, laboratory, statistical physics and so on.

I just take issue with trying to justify the effort to introduce this course by creating a new "paradigm". I also do not buy into the idea that the educational system deemphasizes differential equations for as long as possible. Lately, there have been other posters to this forum, to deemphasize calculus in high-schools
I think it would be better to call to mind that companies that hire and graduate schools for research are interested in solving "practical problems" not "textbook" problems, and a course in numerical methods is important to these ends

 

[hutchphd]

I would like to be a little harsher in my criticism of your plan.
The introduction of numerical solutions and canned programs as the initial exposure to physics education is a terrible idea. It promotes what I like to call the Oracle Approach to physics which I consider anathema:
Student uses computer program to show it takes a ball 1s to fall 16 ft.
Ask student how far it will fall in 2s. Student says "Let me plug it into the program"
Anathema.Excuse me but "paradigm" is one of my trigger words
'

 

[Will Flannery]

I realized that I had a hard sell on my hands, and I thought my first post was a winner ! But ... clearly my initial post was unconvincing, so I've added an illustrative example that will hopefully improve it, in the last paragraph below. (Note: part of my reason for posting is to develop a concise intriguing, if not convincing, argument.)

Since Newton, the basic paradigm for the analysis of physical systems in classical physics has been:
1. State the physical laws governing the system
2. Derive a differential equation model of the system
3. Solve or otherwise analyze the model from step 2.

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible. At USF (U of South Florida) a physics major takes a course in DEs in his/her junior year!

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

The remedy is to teach the paradigm, along with computational calculus in the form of Euler's method for ODEs and the finite difference method for PDEs, at the very beginning of physics/STEM education, in a course dedicated to that purpose and called Scientific Programming. I've written a paper that describes the course titled 'A New Curriculum for Classical Physics', that fills in many of the details, you can see it here ...
http://www.berkeleyscience.com/ANewCurriculum.pdf

To illustrate using planetary motion: in the USF text 'University Physics', and in high school physics, the section on central force motion begins with Newton's law of gravity and Newton's 2nd law of motion. So, the first step of the paradigm is accomplished. It is a trivial matter to derive the differential equation model for a falling object, A = GM/RR, and as I recall that's done in high school physics. However, 'University Physics' doesn't derive this equation, instead it introduces, not derives, Kepler's Laws, and uses them for the section on planetary motion. So, for planetary motion the student studies pre-Newtonian physics. The alternative is to use Euler's method to compute solutions to the differential equation as is shown in the paper. Euler's method is simple, intuitive, and can be taught to high school science students and used to compute 1-D trajectories of falling objects in a single one-hour lecture. Orbits can be calculated in a second lecture. In terms of analyzing physical systems, we've replaced two years of study of analytic calculus that is ineffective for analyzing complex systems with a one-hour lecture on computational calculus that is not only effective in analyzing real complex systems, it is the only way that real complex systems can be analyzed and it has revolutionized science and engineering outside the university ! The paper demonstrates how computational calculus is applied in most areas of classical physics.

 

[Will Flannery]

Note: I published a paper on the subject of the thread, i.e. using computers in physics education, in 'The Physics Teacher' for 10/19, you can see it here ... http://www.berkeleyscience.com/TheComingRevolution.pdf.

 

[pbuk]

I think this is a terrible idea. Plugging equations into black box doesn't give any understanding and creates a reliance on that black box to solve problems. And Euler's method is a terrible black box anyway.

Then that isn’t a physics course. So sell it as what it is: a numerical methods course. It is not a revolution in teaching physics. One of the problems I think that you are having is that you are mislabeling the course and people rightly object.

If it is a numerical methods course then that is even worse - presenting Euler's method as some kind of universal solution generator without an understanding of its limitations and how they arise from Taylor's theorem (with also an understanding of round-off error) is about as useful as teaching multiplication by repeated addition.

 

[Vanadium 50]

Spamming the forum with links to your paper is not an effective way to get your ideas across.

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

 

[atyy]

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

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

That University Physics by Young and Freedman is for calculus-based physics is also mentioned at https://web.mit.edu/physics/prospective/undergrad/transfer.html
"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; Physics for Scientists and Engineers by Serway; Physics for Scientist and Engineers by Fishbane, Gasiorowicz, and Thornton. The exams will be similar to the final exams given in 8.01 and 8.02, with problems based on a selection of the topics listed below. Neither calculators nor formula sheets may be used during Validation Exams."

 

[kith]

Is this true?

I don't know about the US but it is far from true in Europe. Physics students at university not only encounter calculus right from the start but usually take a course in real analysis in the first semester.

 

[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.