Lucio Prado - Rodrigues, Euler, and Hamilton: The problem of Rotations. - JMM2018 Quaternion

Transcript

someone keeps time for me I'll give you mine would you like to be my talk you have time could you do something for me thank you just for me and I'll do funk you know I'm so different okay as everybody knows the rules we go like why there have a 20-minute slot or you have a 45 and a slot one of our speakers will have a 45 minutes long is mine okay Hey let me just get this one here that's my other sweet I'm so sorry but this gives me Oh okay so here we have okay so this is going to be Euler Rodriguez Euler and Hamilton the problem of rotations so how else we discovered for tournaments in 1818 43 is many of us don't know and he had he went through a long long study and struggle to come up with with quaternions and mainly this photo was algebraic but surely after he made his great breakthrough he found that he could describe rotations by quaternions the geometrical discovery was eventually published in 1850 under the title on attorney ins and the rotation of a solid body and the proceedings of the royal irish academy but it happens it wasn't the only one nor the first one to think about rotations about a things center Leonard Euler Olin Rodrigues amongst others for 2,000 in 1840 oh and Rodriguez published his paper on rotations now my French isn't very good so I'm going to ask Lucio to read this to me my colleague so he will beat the French hooker to read the French okay no to geometry geometry Kent a resistor that is placement statement system so they done the space and there are variations s coordinates Permenter they said displacement consider is independent they cause us one curve and a label to you in Louisville that I know that was awful Journal of X so to make everybody's life easier guess what I did this much I can to translate it into English so it's on the laws of geometry that control the displacement of a solid system in space and the variation of port is producing this displacement considered independently of any causes that produce it this paper has not been translated into English to my knowledge if anybody knows of an English translation I would be really grateful so we had to struggle with it in French okay so Rodriguez's paper was important for several reasons he departed from Euler from from Euler by considering only the geometry of motions kinematics without any regard to the physics that produce there so he wasn't worried about things like the forces etc secondly he describes general motion as a composition quotations by trend above composition of rotations and translations by doing this he showed these combinations form a non commutative group and this was something revolutionary at that time he perceived Hamilton's formula for bike return the bike returned ionic description of rotations by three years before Hamilton discovered quaternions and he introduced a set of parameters which rotations can be calculated by purely rational operators and this is something that even Rodriguez himself said in his paper that he did not believe that anyone else had done this and this has become indispensable to present a computer programmers and and does anybody know why okay well the reason is is computers don't like to do things like square roots computers like to do addition subtraction multiplication and division and that sentiment so if you add in anything more complicated than that then computers start to slow down and that makes computer programmers uncomfortable so Rodriguez and he's even said it in his paper which we translated from the French that I believe this is the first time it has been done in this way okay in the end in the course of his paper and raw vegan portion of his paper Rodriguez apply to proof with the theorem of with the theorem of boiler on motions of three dimensions with a fixed point so Rodri so Rodriguez is proof is essentially boilers but in Oilers food it surrounds herself with other materials such as what we know today called boilers Agnes so if you look at the proof by Euler there is an English translation of this from the Latin you'll see not only is there the fixed-point theorem but there's a lot of other stuff one wrong okay we shall use modern vector notation to squad Rodriguez's composition here therefore pure attorneys about an arbitrary origin of coordinate oh now the reason we're doing this of course we're more familiar with the language of vectors that we are familiar with the language that Rodriguez used on even Rodriguez use tools like projective geometry and things like that that people in his day were very familiar with but we are not as familiar with so to make things a little easier to understand it would be translated into modern languages so let alpha L had me the rotation about the axis L hat so we know these axes they elevated and had me the rotation to angle about access and Allah gamma and had me the rotation to the angle about axis n so Rodriguez was able to prove that the composition law we will say is gamma n gamma n equal to alpha L hat Eden and meant so and this is equivalence and so after much calculation we found that it's equivalent to this where he uses the half angle cosine gamma over 2 is equal cosine alpha over 2 cosine beta over 2 minus n the dog-cart of course he didn't know Duncan but essentially he did the same thing or equivalent to what we would fall on top on ik and and had sine alpha over 2 beta over 2 and that similarly for the vector where he does something very similar to the cross product even he did not know it was called cross-party and came up with something something similar to sine gamma and and had sine alpha over two beta over 2 L hat etcetera etcetera should i okay so this is the other so these are the same formulas that are arise through multiplication of two quaternions so he decided to test this okay so what we did was we took let the paternity and be with scalar and vector part cosine alpha went to and this is a silencer if you've looked at houses 1850 paper you'll see that haven't seem to also use these half angles so cosine alpha over two L sine alpha over two okay similarly for Q n cosine we kill it will say is beta over 2 M hat sine beta opens and assigned eight over two and then similarly for n cosine Q of n cosine gamma over 2 sine at the N sine gamma over two so and then of course we just have it here and we decide okay that we have the cop of the formula and we just the product we do the product and we just simply calculate it out so here working it out just plugging you know plugging it in the definitions and then using commutative laws just commuting going through the whole commuting process and then we find guess what they got the same formula okay using Petronius whether we use Rodriguez's way of looking at it and whether we use the paternity onic way of looking at it we end up with the same relationships so which is the same formula saying that Rodriguez but without attorney ins so we did it both ways we looked at it both ways so again using modern so what happens who did now we're going to front go into what Hamilton did and this is again based on analysis and that would unity to use modern notation okay we do we derived a wide paternity otic formula for rotation because Hamilton use what we what was called by quaternions and so we given a unit vector n and then in three space and an angle theta and we want a formula that could be applied to any vector V to obtain the result of rotating V to end to angle theta so we cool we're going to call the result V prime the vector V Prime so we'll say for any back to top of you we shall write W bracket for the paternity for the quarterly so we use our to our usual quaternion definition I WX plus aw r plus k w sent twenty minutes and then we just multiply out okay so we have W and W Prime and then we just multiply it through using distributive law and we get this relationship this was not that simple believe you enough okay and this is one of the things that learned about when using quaternions this took us a long time to do I mean it looks nice when it's alright now but after we had to struggle through it a long time to get it to work out because all the confusion about the sign when you know the signs and everything like that okay I got a VIN number wait a second I apologize for that so this took us a while to work out to get everything to work out nicely and then in particular okay we have n hat V is equal to negative and had V Plus and had the force P and similarly okay for V and hat minus and had AV n hat plus the V across n hat so we're going to keep in mind for the future that if we but these two together the dot product the first part cancels out and we end up with two and half trophy so now we're going to let W but here we go back to our W and we're going to let W equal to n hat force V and W prime equal equal and and and have any loose and happened one okay so we have okay so we put them together there's we mistaken okay so we put them together and we do the whole process nice working with quaternions directly the by quaternions was not a trivial process that's why one can understand why Gibbs ended up and savvy side ended up modifying for attorney ins because they are not easy to work with when it comes to calculations and as I said my colleague and I we learned this the hard way okay so replacing V and 3 by n V and given given two so we go back and there we have and again this was not that much fun okay so we got n negative envy okay this formula and then we have the minus two and minus the X so all we're doing is taking our results and plugging it into the rotation formulas so now even given theta and had we now form the unit quaternion and this is something that in Hamilton's paper Hamilton actually discusses this and talks about these things in this 1858 so we're going to use this relationship and we're going to use as conjugates so we're going to take Q and Q star so we made the conjugate seven and its conjugate and then we proposed the rotation formula so we propose V is equal to Q the V prime is equal to Q V Q Prime so we have now guess what plugging everything in and warping it up starting to look a little bit familiar and so we go through this whole big calculation again working with scooter nians can be very very which rectly with quaternions can be very very nasty as you could see and we finally using four and six this becomes QV cute cute cute star and we have there we thought there bin working it out okay so finally after working on using out co-star us our trigonometric relationships etc substituting all your basic trigonometry things we recover the formula for V Prime and this formula V prime is equal to V prime cosine theta plus F hat plus B the front the front sine theta plus n hat and had the N hat and then top be had thought V prime times 1 minus cosine theta this is known if you bred papers you look at papers this is known as the well-known Rodrigues location milah that Regas found by jean correct geometric arguments so we come to the conclusion after going through all of these things that you can go back and forth between whether you're looking at it from Rodriguez's point of view whether you're looking at it from houses point of view they come out to be the same thing so Rodriguez is concerned Hamilton and Rodriguez so Rodriguez's concerns with geometric and adamant and analytical mechanics Rodriguez did everything in the real field he did not even consider complex numbers when working out his comfort his calculations Rodriguez and Hamilton came to the same conclusions about rotation but they took very different paths as I showed you and as my colleague we showed you that even though we took different they took different paths at the end the same formulas ended up coming out and they say exactly the same things however they both had have both arrived in a common place using half angle formulas so that was the real revolutionary thing that the two of them did that had not been seen before is using half angles and this is something that Haley made a note to so in 1846 Cayley acknowledges Euler's and Rodriguez's priority and describing orthogonal transformations these just regular transformations to in the letter to the editors of philosophical magazine journal 13 he wrote odds to on certain results relating to quaternions can leave credits Rodriguez with his use of an Angus his transformations Kaley refers to Rodriguez's 1840 paper mentioning that Rodriguez's results were given to Hamilton's attorney ins so it turns out that even though Rodriguez was not a well-known math mathematician and in his day he did something really really extraordinary given the times he was living in and the thing is is that this was something that was really very very interesting and a lot of people are thinking about at that time and as as I mentioned earlier Rodriguez our house in Rodriguez were not the only people to think about rotations when things that are related to these rotational product so here is a timeline and I did a timeline here 1840 to 1845 that includes some other people who contributed to the subject besides gaps everybody know houses and everything so of course to publish it but we know he did everything so we left I left em alone okay so in 1840 Rodriguez was 46 and he wrote his kinematic description of solid movements which is what we just talked about and then Rosman which I spoke about yesterday in 1840 he was 31 he did the theory of tides and he presented this but never it never got properly published Hamilton he was 38 he did the algebra Plutonians that we all know and love and then in 1844 he announced about the by quaternion in quotations to the Irish Royal Society but didn't wasn't didn't publish it until 1850 in 1844 Roslyn was 34 he published his linear theory his theory of when your extensions which is the most famous book where you know he talks about the exterior product 18:44 t4i cysteine okay fine it talks about not to you Nativity of linear transformations and non-punitive ax t as you notice is important to these notation so the fact that we're talking about non commutativity is important and then in 1844 rule who is 29 he wrote as perfect paper on symbolic algebra meaning that you can use symbols with something more than just numbers you can use them to represent objects and then in 1845 oh she who was 56 wrote about the non commutativity of permutations again be important in the rotation theory and lastly Cayley and 24 wrote about two by two by two hyper determinants which were also mean okay so thank you very very much look at I hope you enjoy it and here's my colleague if you have anything to add or co2 what I do thank you you know a favorite thing we talked about and then one is to really are the formulas right to make it a connection is a modern way so this beta times the best source in French is another harder to worry you are capable to resolve this information so we managed to to tease it apart and try to bring it back together I think you for the next thing I think is to make it a relation the notation you just will use the boat across all the dot product that keeps the later so looking at the historical context you see very well meet concise really spontaneous but romantically is easy to understand it because in all its Brodie fashion they have a no picture in order to speak not why we had a so is every six written so they say the geometry I described in the previous section to understand well there he wrote the comma so you need to take the writing this because you don't have any specific figure in yeah we had a matter that was just showing the audience you know here's the paper and there are absolutely no pictures and I know what I saw Lucy O's copy he was growing more constant you can see me I'm doing or trying to understand and not only reading the French but we were also trying to figure you know out how to represent it pictorially because we're talking about rotations and angles which gives itself naturally to pictures and I forgot my pipe cleaner my models but I mean I was with pipe cleaners and I add one more point to here is that he is not joining concerned with the geometry but he also benek annex yes Adam is Lisa main scene there I point out we say there's a bigger difference he's not on the mathematics by the way you approach everything the kid would not have I'd always more oriented toward the algebra take as a reference that complex know exactly okay any any questions for either of us about this yes I was really interested to see their list at the end there you mentioned the paper and Eisenstein from 1840 right yes so in 1844 Eisenstein published unbelievable I know that'll be something like 20 or 25 I can't imagine anybody who's ever published as many papers in one year as Isaac's not published in 1844 so I'm curious do you know what in what context and what paper and context is this together at the very last minute you know this is kind of interesting in the paper from the papers that I've read but I could look at you want email I can look this stuff up and and tell you where this which paper this actually came from because even even Gauss was saying you know all the three greatest mathematicians and he said well Archimedes Newton and ice and steam so if Gauss gave Isis Lee such high praise and and it was really sad because he also liked like flipper died rather young from tuberculosis and it seems like turkey losses now we've cured it but was one of the really really bad things that killed a lot of great potentially great mathematicians but I can look yes this one call maybe it's time to yeah well that you know Jax then that's my cousin Euler for example but yes very good the translation you have to know what you know I didn't know some branch you have to know some German yeah I'm just surprised that there is knowing I was there I we look there we couldn't find one an English translation so we just it would be a google translator and knowing graduate school French I think that's that's what I use the same thing with when I was translating rasam on I use google translator and graduate school German so any other any other questions or comments okay thank you very much okay