For those that don’t want to read any more, here’s a laser bouncing around inside a cow:
I was hoping that there would be an easy function like “normal to surface” or something but I couldn’t find anything close enough. There is a nice “RegionDistance” function that tells you how far from a region (a cow) you are. Unfortunately I couldn’t use that as my potential energy because to do the differential equation solving I need to be able to take a derivative of that (that’s what gives you the forces and hence the acceleration of the ball) and Mathematica didn’t really like that (I don’t blame it, it’s not really a differentiable function).
So then I started playing around with “RegionNearest” which tells you what part of a region is closest to your current position (ie the position of the ball). With that, here’s what I do:
What’s really handy is the “WhenEvent” feature of Mathematica’s differential equation solver “NDSolve.” WhenEvent basically does part 4 above and triggers the function for 5 and 6. It’s so useful that I use it for this situation that isn’t really a differential equation. Basically I just say the force is zero everywhere and add the WhenEvent feature to do the reflections. It works great!
I should note that I started this project wondering what happens in a sphere. But it turns out that it’s super boring:
Then I went on to an ellipsoid, but that was pretty boring too. So, I went with the cow (my sons helped pick that). When my son saw the one above, he was really interested in what happens when it goes down a leg. Here you go:
Cool, huh?
Thoughts? Here are some starters for you:
On estimation days the students work in groups to determine things like “how many piano tuners work in the twin cities?” or “how many gallons of gas are used in the US every year?” My grading scale is determined by:
which is the same as saying a letter grade off for every factor of 3 you are off from the actual answer.
We’ve only had one of these days so far (the gas question) and I was really pleased with the result. Out of the 11 teams, only 2 were more than a factor of 3 off and both of those were due to a simple mistake where they multiplied by average gas mileage instead of dividing. It took the slowest team an hour so next time (tomorrow) I’m going to do multiple ones so that the groups can divide and conquer.
The connection to science is just developing a good number sense and to realize that they know a lot more than they think they know. By the way, I put a box at the front of the room so that they couldn’t just google all the answers:
The day before debate day (actually on Friday to prep for Monday) they are told to bring in 2 pro and 2 con sources for debate day. On our first debate day we did “The US should use nuclear power for it’s dominant source of energy.”
In class they broke into the pro and con teams (randomly) and were tasked with finding 5 main topics and to break into groups to develop good quotes from their sources about their topic. Each group in the main debate was responsible for presenting one new idea and rebutting one idea from the other side. The debate ran: pro new idea, con rebuttal, con new idea, pro rebuttal repeat.
I mostly enforced the rule that when they presented they had to basically just read from their sources. This really worked well! My only complaint is that they didn’t really attack each others’ sources’ authenticity.
I graded the efforts with 50% of the points for turning in their annotated sources (making clear which quotes they used) and 50% based on a reflection paragraph on who won and how the authenticity of the sources affected the debate (the consensus was that it didn’t).
I told them to familiarize themselves with Randall Monroe’s What If blog and to prepare a blog post of their choosing. I’ve graded the first drafts so far and they’re all doing some pretty interesting things, ranging from picking on Trump through ice in place of sidewalks to having all cats disappear. I’m encouraging them to really take things to as many new levels as they can. They’ve got some work to do on future drafts, including drawing some funny/ironic cartoons, but I’m really excited about where this is going.
Borrowing again from Randall Monroe, I’m asking them to submit an explanation of a scientific or technical things using the 10 100 most used words in the English language. That’s what they’re doing peer editing on as I type this post.
We talked about the analogy between a scientist needing to change her vocabulary when talking with the general public and them having to do this assignment. We talked about possibly allowing a glossary where they could introduce new words that must be defined using the 10 100 words.
Lateral thinking puzzles are those party games where you lay out a situation and the audience has to ask yes/no questions to determine the whole story. Here’s an easy one: 5 pieces of coal, a carrot, and a scarf and found on a lawn. There’s a logical reason why they’re there, what is it? The answer is that a snowman melted.
I’m using them sparingly (ok, I did 3 of them today) because I think they are good practice for the questions they need to ask themselves during their mythbusting explorations. To do well at lateral thinking puzzles you have to constantly question your assumptions and double check what you think you know. Sound familiar? I think it’s quite similar to the traits of good scientists!
Ok, that’s all I’ve got for now. It’s a fun class, though the grading is killing me. Do you have any thoughts? Here are some starters for you:
Typically students pick something like a projectile problem and they present it kind of like a mix between an oral exam and a mock teaching situation. The former is weird because it’s usually a really easy problem (for juniors) and the latter is problematic because that kind of presentation tends not to be how we’d choose to teach the material (no interactivity, for example). Normally we end with some token physics criticism/question but them concentrate on suggestions for how they could present better.
My biggest problem with this setup is how fake it is. It’s not the type of presentation they’ll likely be giving in the future. Then they’d much more likely be talking about a data-driven decision or conclusion they’ve reached. They likely wouldn’t be defending their understanding of a simple concept and they certainly shouldn’t be teaching like that (mostly dark room, only one person talking, etc).
So my current idea is to have them still have a pretty safe and easy thing to prepare but to have it be much more in the spirit of the types of presentations we’d like to see from them. I thought it might be cool if they could present their data from a lab they’ve already done. It might be a simple lab from general physics or something, so likely none of us would be surprised by the outcome, but they could be asked to approach it as trying to convince someone of the conclusions they’ve drawn. We could then really help them focus on what aspects of the presentation need focus (well done data plots, clear explanation of any theory necessary, etc). We still might make minor physics criticisms or questions but we could focus on the presentation skills we really care about.
I figure we could ask students in their first one or two physics classes to pick the lab write up they are most proud of and we could keep it for them to hand back when their juniors. We could do that early in the semester so they could have time to get their presentations ready.
Your thoughts? Here are some starters for you:
Option one is just do nothing and hope that they learn their lesson eventually. Perhaps enough 1’s and 2’s will get them to realize they really should work harder (or perhaps differently) on these problems.
Because I use a Standards-Based Grading approach they know they can repair their grade by turning something in later so there’s not a lot of pressure to perform in class. Of course many of these students pile up a lot of 1’s and 2’s and tend to fall far enough behind that my two week rule gets them in trouble.
I guess what I’m feeling right now (facing the prospect of (once again) turning in lots of F’s for midterm grades) is that the lessons aren’t being learned.
I could remind them that since the quiz grade doesn’t set in stone that standard’s score, if they feel like they’re likely to get a 1 or a 2 they could try to explain on their quiz what they’re struggling with. When this happens now, admittedly rarely, I tend to either give that person some feedback on their quiz paper, send them a video, or send the whole class a video with some explanation. I think in the smaller classes that I could handle this scaling up a little and I think I’d understand better where the students are at than grading a crappy quiz performance.
I think if I didn’t stay on top of it and continue to give personalized feedback, students would likely stop writing a whole lot and instead continue to use their 10 minutes in class to do other things (with a defeated look on their face).
I was brainstorming with a student who’s in class with me right now about what would work better for him on days when he doesn’t really know how to start the quiz. I asked him about the notion laid out above and he thought that seemed reasonable. Then we started to brainstorm this new notion. Maybe I could give students a choice: either take the quiz or come with me to another room where I’d do some additional instruction on the topic. Maybe that could just be me trying to articulate every thought in my head as I try to do the quiz.
I’m definitely attracted to this if only to avoid looking out and seeing defeated faces. On the down side I feel like this sort of direct instruction 1) makes the students feel like they’re learning, but 2) doesn’t really help their learning. It reminds me of the students who say “I own the solution manual but I don’t use it to do my homework. Instead I love to study from it much more than struggling over any suggested exercises.” I’ve never actually believed students when they say that (not the first part, the second part). I think they look at well solved problems and dutifully nod their heads saying “oh yeah . . . of course . . . yep that’s right” instead of what we want our students to be thinking: “hmmm … not sure… oh wait … let me try … ooh, now I know!”
I’m also not sure if students would tend to make the right choice (for them) when offered to either take the quiz or come with me. I guess I’m not sure what to think.
Your thoughts? Here are some starters for you:
First an update on the database and our prototype. I haven’t added much to the database. Really only a dashboard list for the admin (me) showing the total number of measurements for each machine and when the last measurement was. It turns out that even though I fixed the pretty common wifi freeze with the watchdog approach, it still goes down about twice a week. I’m not really sure what’s up with that. The most recent time it happened I didn’t notice for about a day so when I went to look at it I noticed it was pretty hot. Unplugging it for 10 seconds or so and then replugging it seems to have fixed it for now. Any thoughts on how I can figure out what causes those very intermittent problems would be greatly appreciated. The class has devoted one lab to Arduinos so far and they all got to the point where they were measuring and calibrating temperature and controlling an LED. Pretty good for 90 minutes. We meet again about it this coming week when I’m hoping they’ve all found a “client” that they can program the arduino for.
The first new idea is to deal with client requests that are time-based. Something like “every night at midnight check the XXXX and report” as opposed to what we already have: “every Y minutes do something.” The arduino has a millisecond timer on it, but you can’t trust it to know in absolute terms what time it is. So at first I thought we couldn’t handle those requests. But then I remembered that we have a working wifi card on all of these! I figure we can just have it check the absolute time on some time server somewhere and at most be off by Y minutes. That’ll likely satisfy those sorts of clients (at least I hope so).
Already a few potential clients have asked for things like “every time the door opens measure XXX and report.” If they can handle a resolution of Y minutes, then we’re all set. Otherwise we’ll have to use the built-in interrupt stuff for arduinos. Then if the door opens an interrupt will be thrown and we’ll execute the whole “measure XXX and hook up to the internet” code.
This is the one that I would really like some advice on. Some students and I were brainstorming how we might communicate to the arduino to make a measurement. Really this would be an extension of the “using a clock” idea above. We’re thinking of updating the database web site so that the owner of the machine (which is currently defined as the person who most recently updated the code for that machine) could update a page with something like “yes” or “no” returned so that the machine would check that page first (every Y minutes, say) before deciding whether to do anything else. We could, of course, get more complicated with various messages triggering lots of different responses from the machine. The limit would be the storage on the arduino (20 totally different codes stored might not be possible, for example) but I think we can still do some cool stuff.
I guess my question is: is this the right way to go about it? I read a lot about making the arduino basically a web page server so that you can go to it and ask for things. That’s the bulk of the pages you get if you search for “arduino internet of things” so I know it’s pretty popular. My problem is that it seems like it’s asking the arduino to do quite a lot, and essentially be fully on (as opposed to a boring “pause” loop) all the time. I figure this approach saves a lot of that, but maybe I’m missing something.
So, we’d love your help. If you have any ideas/hare-brained-schemes/concerns/anecdotes/love-letters-to-mathematica let us know in the comments below. Here are some starters for you:
This article helped me clarify my thoughts about why web and not mobile apps. Honestly, for me, it’s a learning curve thing, since I know how to do web stuff (I’ve written my own LMS, a course schedule web site for my institution, a site for concept tests/quizzes, a jeopardy quiz site, and an Arduino Internet of Things site (which I wrote about in my last post)). Also it seems that you don’t have to make a choice between apple and android if you go with web apps. However, I think we’ll concentrate on things that’ll work well on mobile browsers just to keep that audience in mind. It helps that Twitter Bootstrap is mobile-first.
The hope would be that we could develop something that people need. I wrote about something that I’d like a while ago so maybe we’d just start with that one (it’s a site where students could upload photos that would be added to an ongoing slideshow that would be projected). I’d love it if we could respond to the needs of the community to make things that are useful (and teach students some useful coding skills).
We would use github and trello to keep organized. Github would allow us to has a core of working code along with branches where new features are developed. Github would also let new people join in pretty easily, though a working project might be intimidating to some. At first at least we’d be coding in the Laravel PHP framework. Participants would need a development machine that could run Laravel/php along with mysql. Installing xampp would do the trick, along with composer to easily handle various code dependencies. Once they have those working (really it only takes about 10 minutes to get all that working on a windows machine) they would just pull down the githup repository they’d like to work on and they’re in business. Really the notion of handling code with others using git is a useful skill.
Trello is an awesome way for a group to keep track of what there is to do, who’s doing what, and what’s done. I used it to great effect this summer with my research students (here’s our public list).
Once students have got a cool feature working on their local machines I would just have to merge the repository and pull it down to the development machine (for now it’ll be physics.hamline .edu which is where all the examples above are hosted – eventually I would hope to get something like webapps.hamline.edu or something). That way I’d be the only one dealing with the production machine and any headaches that come with getting it working right. I really like how github would help with that (and how students can see the working code on their development machines without having to wait for me to get it up for the whole world).
I think it’ll be fun to work with students who are developing these types of skills. I think I’d be a little hands-off with dictating what we work on and how, but I’d probably stick to Laravel, Test Driven-Development, and mysql. If someone comes along with skills to get other approaches working, then sure! I got all excited about Meteor a few months ago but getting it to work on a server with ease of community-based updates was a big hassle (for me). So Laravel/PHP/MYSQL for us for now. I think that students who really put some effort into this will be able to easily sell their scripting and collaboration skills with some production code to show off.
So what do you think. This is really just a 2-day-old brainstorm for the moment. I’d love some help moving this project forward. Here are some starters for you:
With some more internal funds we now have enough Arduino Uno’s, CC3000 wifi boards, and DHT 22 temperature/humidity probes for my whole class this fall (13 students). The total cost was something like $1200.
I based the code on two examples that come once you download the cc3000 library from Adafruit. The “build test” example checks the functionality of the board, and, importantly for how we wanted use it, sends the MAC address of the board back along the serial line to the computer. Then the “web client” example is what I made slight adjustments to in order to get it to work with our database. Really you’re just having it connect to a simple web page (with a GET request) and then having it read the result and then shut down. There’s a lot of code to help with connecting to the wifi but it’s pretty straightforward. At my institution the main wifi signal uses WPA2-EAP (I think) for security (the kind where you put in a username and password in the wifi configuration on your computer as opposed to the kind where you put your username and password into a browser page) and the cc3000 can’t handle that. I thought we were out of luck but one forum I read suggested seeing if my ITS department would be willing to just store the MAC address of the boards and allow them on the network without all the certification stuff. It turns out they were willing to do that! It’s on the “guest” wifi signal as opposed to the main one but it seems to work great. (Note the comment above about being able to have the arduino print out the MAC address with the initial test).
So you load the cc3000 library, run the “build test,” grab the MAC address, send it to ITS, then run a doctored version of “web client” with the appropriate wifi settings and the correct url for the database (and collecting the appropriate probe data).
One major headache that happened was the arduino would hang every few hours or so. It gets stuck in a loop and won’t do anything. Hitting the reset button fixes it, but that’s a pain for a machine placed somewhere on campus. Luckily there’s a very cool “watchdog” library for arduino that allows you to program the machine to reset if it goes into an infinite loop. Unfortunately the longest you can have it do something before it’s decided that it’s stuck is 8 seconds using the default watchdog approach. That sucks because the whole “connect to the wifi, send the info, parse the result” takes something like 24 seconds. Luckily, I found a very cool work around that allows you to multiply that 8 seconds by any integer you’d like. It uses interrupts on the watchdog and is very slick. Check it out. So now if it hangs in the wifi hookup process, it just resets itself and continues right along. If I look at my data I see usually one or two gaps of 6 minutes instead of 5 minutes between my data points every day. Not too bad.
I wanted to build a pretty flexible database for all this data. I wanted authorized uses to be able to add new machines (register the MAC address), attach probes to it (so the system could make sure that the type of data submitted was of the right sort), and describe the location it’s in. I also wanted to have a place to store the arduino code on the machine so that people could check it to understand any calibration issues. I’ve got nearly everything done:
This was the first time I programmed an application using Test Driven Development and I really liked it. I now have a suite of tests (over 50) that test every aspect of the application. That way if I make a change I’ll know if I’ve screwed something else up. Basically it’s a way to test all your web pages and user-clickable events to make sure they’re all still working. In the old days I’d just make a change, make sure the new thing I was working on was working and only find out days later that I screwed something else up. Very cool.
You can see the working database here.
I’m pretty excited about what we can do with that. We can have machines all over campus submitting data that we can really put to use. I’m also excited because I think I could easily handle other people’s data as well. If you’ve got a wifi-enabled arduino, just register on the site (and maybe send me a message), I can approve you and decide if you should be an admin or what and then you’d have access to your data. Alternatively people could feel free to fork my application and just host it on their own server.
By the way, there’s also a Plotly approach having arduino stream data to a graph if that’s all you’re interested in.
In the old days I’d just make up some dumb thing for my students to work on when learning how to work with an Arduino (make a working stoplight circuit, move a motor at a speed determined by the temperature, etc). What I’m excited about with this project is that they’ll be learning many of the same skills while adding to a larger university-wide project that other people care about. I hope that helps motivate them.
Working with my math colleague has been fun because he has his students deal with data sets in his statistics courses. Now he’s excited to use this data instead with really very little change to the learning outcomes. Again motivating the students to come up with hypotheses that could be tested using our data should be easier with this.
The other thing I’m excited to do in this upcoming class is to encourage the students to engage with community members. My hope is to identify several potential “clients” that they can go talk to and figure out what kind of data they want logged and why. It’s a math methods class for physics majors so maybe they can do some modeling of the system being measured. Not sure about that (though simple heat flow should be pretty easy) but it’s certainly possible.
So I’m pretty excited. Your thoughts? Here are some starters for you:
My partner and I talk a lot about how our son is learning. We think he does a great job with some things and just an ok job at others. We’ve been with him in stressful situations and we’ve all made it through (even the car!). But as we reflect on what we would have done in those situations we start to realize just how much better we are as drivers than he is, or that he will be even after another few months of intensive training. We’ve had (cough cough) 30 years of practice, and now we’d say we’re pretty good at it.
What I’m depressed about is the realization that years of experience (or 10,000 hours, if you prefer) can’t be taught. I’m pretty sure that my son will become a great driver, but I don’t think there’s anything I could do to help that along very fast. I’m depressed because my profession is teaching physics, and all I ever get is four years with a student. For most of my students all I ever get is one semester. Trying to teach “physics maturity” (to borrow and slightly change a phrase from my mathematics buddies) in a semester is really hard. Maybe impossible. If I knew my students were going to go off and continue to think about physics and practice physics and model things like crazy throughout their life, I suppose I could take solace that they’d eventually become the experts I want them to be. But the students that do that are in a minority so small that it’s probably not worth it to count them.
I’m realizing that all I can do is set the table for them. I can try to make a course experience that gives them some tools and gives them glimpses of others. Just as I can’t make my son a great driver in just a few months, I can’t make an expert in physics in one course.
So I’m depressed, but super excited to be heading off to the AAPT conference today so I can get the usual pick-me-up that I get from all my friends there. Who knows, maybe when I get back I’ll have a post with a title like “Teaching physics is the greatest thing you can do” or something like that.
Your thoughts? Here are some starters for you:
That sounds cool and all, but the details are proving to be tough. I’d like to brag a little about what we’ve been up to in this post (mostly so there’s a good record of it somewhere), but if you’re wondering about the title of this post, just go here where there’s a little explanation for what we need for you. Read on for more details.
This is actually the easy part. If you know the shape of the drum head you’re interested in (and can describe it mathematically — see below for that hassle) you just need a single command in Mathematica:
{frequencies, functions}=NDEigensystem[{-Laplacian[f[x, y], {x,y}], DirichletCondition[f[x,y], True]}, {f}, {x,y} \Element region, {10}]
where “region” is your mathematical description of the shape of the drum head. This command uses a Finite Element approach and returns the 10 lowest eigen frequencies. Note that you have to take the square root of the frequencies you get from this command to get the audio frequencies.
Here’s a sample of listening to various frequencies on a slowly changing shape:
Simple shapes are easy: a circle? Disk[], a rectangle? ImplicitRegion[-1<=x<=1 && -2<=y<=2, {x,y}]. But what about crazy shapes? And what about shapes that Mathematica can programmatically shift around while it hunts for cool shapes that produce cool spectra?
What we’ve decided to do is to use control points around the edge that Mathematica can make slight adjustments to. When it does, it redraws a smooth, closed curve that includes all the points and it then uses a cool command that turns that border into a region:
region = BoundaryMeshRegion[controlpoints, Line[{1,2,3,4,6,1}]]
The problem is that you have to make sure that the control points are in the right order around the border (say, clockwise, for example). Luckily it turns out that the traveling salesperson problem comes to the rescue here. If you want to find the shortest path visiting all the points in a plane (and returning to the first one), that path will not cross itself and hence will be a proper region border. So:
fst = FindShortestTour[points];
comes to the rescue. So Mathematica does this:
Ok, so let’s say you have six control points. Each one is an x and y value so you have a 12-dimensional optimization problem. What could we use? We’ve decided to use Mathematica’s implementation of an evolutionary algorithm (or genetic algorithm). Really it’s the same thing I was using when trying to see if Mathematica could learn to race around corners. Evolutionary approaches work well where there’s a humongous parameter space and you don’t really know any other way to explore it other than brute force.
The big problem (yes, I’m getting to the title of this post, hold your horses) is that a set of frequencies from a drum head (the result of step 4 above) needs to be converted to a single number that can be used to rank various drum heads in the evolutionary algorithm.
Ok, so we realized that we needed to be able to look at a spectrum from a drum head and rate it on the scale of “is it melodic?” We thought of some interesting approaches. Mostly they centered around measuring how close the frequency spectrum is to an evenly spaced one (which is what a stringed instrument gives you). We ran into lots of potential problems, though, not least was that orchestra chimes have a “missing fundamental” and still sound good.
We also realized that maybe we could handle mostly evenly spaced frequencies if we could determine where to thump the drum head to kill the offending non-evenly-spaced ones.
Ok, so now we had to go back to Mathematica to determine where on a particular drum head you could thump it to control the relative amplitudes of the various frequencies (think about how a stringed instrument sounds very different depending on where you hit it.
Here’s an example of how the frequencies from the shape of Minnesota change their relative amplitudes if you thump in the center of every county in Minnesota (note that the find shortest tour command was used to do that):
Luckily the NDEigensystem gives us the resonant shapes for every resonant frequency so finding the relative amplitude for a given thump location (and shape) really just amounts to doing this integral:
where is just the ith resonant shape and thump(x,y) is the function that describes the thump shape (and location).
It’s taken us a while to find a good way to do this integral fast, but we’re getting there (right now we’re at one second per frequency per shape).
So now we can look for a good candidate of frequencies and then hope there’s a thump location that’ll shut off the bad ones (fingers crossed!).
So then we hit on the way we could pull all of this together (we hope). We’ve decided to let the crowd (you!) help us rate a collection of frequencies and relative amplitudes on a scale of 0 – 5 where 0 is like white noise and 5 is a pure tone. We figured that since we’re making drums for people we ought to let people determine the single number that our evolutionary algorithm needs.
One of the researchers in the math department this summer is working on an artificial neural network to recognize handwriting and my students realized that approach could work here. All we need is to train the network on what are good, bad, and medium sounding collections of frequencies and relative amplitudes.
Luckily Mathematica has recently built in some really powerful functions that implement the major algorithms in neural network theory. The one we’re planning on using is “Predict” which just needs a whole bunch of these:
{{216, 456, 786, 890, 1012}, {0.5, 0.3, 0.6, 0.7, 1}}->2
where the first list of numbers is the random frequencies and the second is the relative amplitudes. It then trains on whatever you give it and then it can be used on future untrained ones.
So, we need your help! Please go to our new site and score a few random sounds on our 5 point scale (decimals are welcome). It just takes 1 second per sound and we’d love to just get a ton to train the neural network. Then our workflow will look like this:
We started developing the training set using Mathematica to generate sounds. This is pretty easy (just use the Play command) but it was tedius and we weren’t generating enough. This notion of crowdsourcing came from my wonderful students so I decided to give it a try over this holiday weekend.
I knew making a database-driven website wouldn’t be a problem (I rail against Blackboard so much because I finally just wrote my own LMS). But I didn’t know how to generate the sounds. So, I decided to dig into the HTML5 audio standards. It turns out that just a few lines of javascript code will generate a sound with a controllable frequency and amplitude:
oscillator$key = context.createOscillator(); gainNode$key = context.createGain(); oscillator$key.frequency.value = $value; currentTime = context.currentTime; oscillator$key.connect(gainNode$key); // Connect sound source 2 to gain node 2 gainNode$key.connect(context.destination); // Connect gain node 2 to output gainNode$key.gain.value = $amps[$key]; oscillator$key.start(currentTime); oscillator$key.stop(currentTime + 1);
where $key is set up as the loop variable (goes from 1 to 5). Feel free to take a look at the html source of our page to see how it all goes together.
So thanks for any help you can give. We really hope we get enough data so that the training is robust.
Thoughts? Here are some starters for you:
I’ve used a Lagrangian approach a ton in my work with students and my posts here. It’s a great way to model the dynamics of a system because you just have to parametrize the kinetic and potential energy of the system and you’re off. No vectors, no free body diagrams, just fun
Here’s the idea in a nutshell:
Hold a ball in your hand. In 2 seconds it needs to be back in your hand. What should you do with the ball during those two seconds to minimize the time integral of the kinetic energy minus the potential energy during the journey?
It’s a fun exercise to do with students. You’re asking them to minimize this integral over two seconds:
When I do this their first guess is to leave the ball in your hand. They like to define the gravitational potential energy there to be zero, and then know the kinetic energy is zero if it doesn’t move so they’ve found an easy way to get a total of zero for the integral. So I challenge them to find a path who’s answer would be negative! It’s a pretty fun exercise, especially if you actually calculate the integrals for their crazy ideas.
The point is that the winner is to throw the ball up so that it’s trajectory, responding simply to gravity, takes 2 seconds (ie throw it up 1.225 meters). The kinetic energy is positive during the whole journey (except for an instant at the top, of course) but the potential energy is positive during the whole journey too.
Calculus of variations teaches us that if you want to minimize an integral like this:
(where is shorthand for the x-velocity) you really just need to integrate this differential equation over the same time integral:
What’s cool is that if the function is KE-PE the equation above becomes Newton’s second law! That’s why this works. You use scalar energy expressions and you get the force equation for every component of motion! Now there are some other cool things like not needing to worry about constraint forces but I won’t worry about that in this post.
Ok, so what happens when you consider relativistic speeds (ie close to the speed of light)? Well, the first thing I did (which, spoiler, didn’t work) was to wrack my brain for an expression for the kinetic energy and plug away. When teaching relativity you get to a point when you’re making the argument with your students that KE isn’t just anymore but is really where gamma is given by:
If you take the limit of that expression for small v’s you get the usual expected result, and that’s certainly what we do right away with our students to make them feel better.
Ok, so I plugged it in and got a relativistic version of Newton’s 2nd law:
Note how the second term on the left side looks a little like “ma” while the right hand side is just the force from a conservative potential energy (U). The extra term on the left hand side is the weird stuff.
Without really thinking about whether that was the right equation, I modeled a constant force system and got this for the velocity
(I set the speed of light to 1). You can see that the speed is forced to obey the cosmic speed limit.
But here’s the problem. The equation above is wrong. That is not the correct relativistic Newton’s 2nd law equation.
So what happened? I plugged in the correct relativistic kinetic energy and the Lagrangian trick (minimizing KE-PE) gave a trajectory that doesn’t match what actually happens! So something’s wrong. Here’s a few possibilities (one is right, see if you can guess before reading the next paragraph):
It turns out it’s the last one. It took me a while of digging around, but this wikipedia article set me straight. The gist of what’s talked about there is this:
Yeah, weird, I know. It’s like “hey, I know what the answer in the back of the book is so I’m going to futz with my early equations until they give me the right answer. So what is the right functional to use? This:
Yep, it’s negative. Yep, it’s not an expression you’ve ever seen before if you’ve studied special relativity. But, guess what, it works! When you plug it in and do the calculus of variations trick you get the right dynamics. Surprise, surprise, given that it was built to do just that.
Here’s the same graph as above but not comparing that prediction with the right dynamics (in red):
It also asymptotes to the cosmic speed limit, just at a different rate.
That’s the question I was really wondering about. Luckily google came to the rescue with this great wikibook article that it found for me. It points out that the kinetic energy portion of the functional you use to make the relativistic dynamics work is really just proportional to the invariant space-time interval:
This is an expression for the “distance” between two distinct events in space-time that is the same for all inertial observers. It’s really cool given all the weird time dilation and length contraction that can go on in the various inertial frames.
So basically the trajectories that actual things follow is designed to make the space-time “jumps” add up to the smallest number. That’s super cool
Your thoughts? Here are a few starters for you:
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