Newton's Laws - Professor Raymond Flood

well welcome everybody welcome thank you very much for coming three twos a day to hear about Isaac Newton and some of his laws and essentially today's talks really going to be quite geometric because I'm going to look at the most important work really that Newton produced which was his Principia and which he magnificent book in which he posted which he proved his universal law of gravitation and his laws of motion but first of all I want to give an overview of Isaac Newton's particularly rich complicated and successful life and not only in the Principia that he proved his law of universal gravitation but he also showed our two bodies would move the orbit of two bodies moving on their mutual gravitational attraction he was able to describe the orbits completely and that of course gives rise to the natural question of trying to explain the motion of more than two bodies under gravitational attraction many people tried that over the next couple of hundred years and I want to look at the endeavors of one particular mathematician only Poincare a who took a geometrical approach to try to find the motion of three bodies a simplified case of three bodies moving on to gravitational attraction that's what I mean by it took quite a lot of geometry in today's talk and finally I'm going to look at the question as to whether or not the solar system is stable so if you thought you were coming along for a triviality or not we're going to deal with a very important question and you can amuse and astound your friends with the answer as to whether or not the solar system is stable so you have to stay to the end in order to get the answer to that well and as time Newton was exceptional very exceptional in the extraordinary rapidity with which success came his way by comparison Galileo ended his days persecuted by the Inquisition Rene Descartes died in exile and Stockholm surrounded by hostile scientists an unfriendly doctor while Gottfried Leibniz died largely forgotten and with we are told his secretary as is only Mordor in contrast in April 17 27 the French writer and philosopher Voltaire viewed with astonishment the preparations for the funeral of Sir Isaac Newton believe president of the Royal Society laying stairs in Westminster Abbey for the week preceding its funeral at the ceremony his coffin was borne by two jerks three Earl's and the Lord Chancellor he was buried Voltaire observed like a king who's done well by his subjects no scientists before or since has been so revered few such have been buried with such dignity and high honor the boy from rural Lincolnshire had come far the quotation at the bottom of the slide is from Alexander Pope and written shortly after Newton's death it shows the reverence that Newton's contemporaries felt towards him and in particular for his work in gravitation nature and nature's laws lay hidden night God said let mutant be and all was light in the words of the French physicist and mathematician Laplace since there's only one universe it could be given to only one person to discover it's fundamental law the universal law of gravitation and that's what Newton did in his Principia Mathematica one of the greatest books of all time Newton's interests were not confined to mathematics and I can illustrate some of those by looking at his term in Westminster Abbey it was designed by William Kent and erected in 1731 and it's rich symbolism alludes to some of the fields of Newton's interests on the left is a picture of the memorial and on the right on engraving the beer Spurs a Latin inscription and supports a sarcophagus with large scroll feet and a relief panel and let me read they are a translation of the the inscription it's in Latin and could be translated as follows here is buried Isaac Newton Knight who bear strength of mind almost divine and mathematical principles peculiarly his own explored the course and figures of the planets the paths of comets the tides of the sea with the similarities and rays of light and what no other scholar has previously imagined the properties of the colors thus produced diligent sagacious and faithful in his expositions of nature antiquity and the Holy Scriptures he vindicated by his philosophy the majesty of God mighty and good and expressed the simplicity of the gospel in his manners mortals rejoice that there has existed such and so great an ornament to the human race and we can also see something about his range of influence range of activities if we look at the sarcophagus and it depicts boys using instruments related to Newton's mathematical and optical work including the telescope and the prism and this activity is master of the mint the boys playing on the left are playing with this reflecting telescope on the right we have a furnace with newly minted coins that's alluding to his time as warden and master of the mint on the middle here there is one boy who's weighing the universe which is what we are able to do to find the relative masses of the different planets there's the Sun and hanging down here are all the other planets in the solar system and then on the right just to the right of it there's another boy playing with the prison and mutants discovering that white light is a mixture of of colors above the sarcophagus coming back this is why I use the sketch because it's a little bit clearer there as a reclining figure of mutant and classical costume his right elbow resting on several books representing his great works it's works in mathematics the Principia and celestial dynamics optics and also on divinity and chronology on the background there's a pyramid on which there's a celestial globe with the signs of the zodiac the constellations on the path of the comet of 1680 comets were quite important and the story I'm going to tell later on on the top of the globe sits a figure of astronomy leaning upon a book so let me briefly now outline muteness life it took place against a backdrop of only three locations Lincolnshire Cambridge and London the man who's very famous for comparing himself with a boy playing with pebbles on the beach not noticing the ocean of truth art before him Newton is unlikely ever to have seen the sea he was born Christmas Day 1642 you're nuts old dear doll start by the Julian calendar and force in England for the rest of Europe using the modern Gregorian calendar the date was the 4th of January 1643 his father was an illiterate but fairly prosperous human and he died three months before Newton was born Newton's mother remarried when he was three and for the next eight years Newton lived with his grandmother apart from his mother and all Newton's recent biographers have seen this separation from his mother between the ages of three and ten as crucial in helping to form the suspicious tortured and neurotic personality of the adult Isaac Newton the mother and son were together for only a few years before he came to the free grammar school in Grantham 7 miles away and we knew little about the type of education he would have received but he would have learned merely latin grammar mainly latin grammar that's what grammar schools were for and more than a modicum of basic arithmetic under little years many stories of Sir Isaac Newton theorized 'ok newton school days were put a bite not least by Sir Isaac Newton himself telling off his mechanical inventiveness this experimental ingenuity Anna's on easy relations with his fellow pupils and he tells the story of making a small windmill that actually ground-floor powered by a mice and litter life he was to make the equipment for his experiments for example he made the equipment that enabled him to grind his old lenses for his optics experiments as Newton himself remarked if I'd stared for other people to make my tools and other things for me I had never made anything of it 17 it was time for him to return home and manage the estate and apparently this was a total failure his mind was for the problems full of things he wanted to think about and he knew interest and earthly matters in hand and the stories told probably apocryphal that he was leading a horse up a hill and the horse slipped its bridle and he went on leading the bridle up the hill well how do I wait for them this and other reasons he was sent to Trinity College Cambridge which he joined in 1661 and which became his home for most of the next 35 years to begin with he was a subscibers rather strangely because as his mother was really quite wealthy by this time meaning that he earned his keep by serving the fellows and wealthier students cleaning their boots waiting at table and emptying their chamber pots Newton was essentially self-taught at university by his own self-directed reading and early in 1655 he became a Bachelor of Arts but that was just because he beat up into university for four years was a formality the more crucial milestone was passed the year before when he was elected a scholar of Trinity and somewhat surprisingly because there was no evidence then of any distinction but anyway his position was not guaranteed for another four years and the studious a sub sizer ceased then in the summer of 1665 the dreaded plague reached Cambridge and the university closed I'm this was the last widespread outbreak of bubonic plague in England Newton returned to linka sure where he spent most of the next two years until the spring of 1660 seven and we have mutants account written over fifty years later of his investigations during those two years and I'm going to share that with you don't worry about the details just get a feeling for the range and extent of the successes some of the points that he makes I'll come back to you later so this is a kind of the two plague years when he was on his own back in Lincolnshire in the beginning of the year 1665 I find the method of approximating series and the rule for reducing any dignity of any binomial into such a series I'll show you an example shortly the same year in May I find the method of tangents of Gregory enthusiasts and in November had the direct method of fluxions and the next year in January had the theory of colors and in May following I had entrance into the inverse method of fluxions what we would now call differentiation and integration on the same year I began to think of gravity extending to the orb of the moon and deduced the forces which keep the planets in there or be as the squares of their distances from the centres about which they revolve the universal law of gravitation and thereby compared the force requisite to keep the moon inner orb with the force of gravity at the surface of the earth and find them answer pretty nearly and all this was in the two plague years of 65 to 66 for in those days I was in the prime of my age for invention and minded mathematics and philosophy more than at any time since well of course it's not impossible to exactly corroboree at these details but the spirit of the account seems right and an extraordinary short time of two years you laid the foundations of celestial dynamics optics and mathematics let me show you one of his manuscript pages from this period from 1665 he quite a neat hand and what he is doing here is he's culling the area under this ship this curve which is a hyperbola and the areas under this curve the hyperbola are in fact calculating logarithms and he's calculating a logarithm an area under the curve to 55 decimal places so if you're going to do it correctly and let me show you how he does it because this is a very powerful technique I'm going to zoom in on line three down here up there I mean and if you look at line three on the left hand side you have an expression which gives you the IRIAF which gives you which is the hyperbola the equation hyperbola it's a squared he writes out as a times a over a plus X and what he does and you can do it using if you know about geometric progressions as he writes this art as an infinite series right so it's like a polynomial if you know by then but it keeps going on forever there's the etcetera down here so he's taken an expression and he's written a tight as an infinite series a series of other terms but the thing is these other terms only involve powers of X and he knows how to find the area under a curve which is given by a power of X scene it was hard integrate parts of X he sort of says it here multiplied by X and divided therefore multiplied by X and divided by the number of its dimension which you may remember so the integrator part of X X to the five years becomes X to the power of 6 over 6 so he is now able to do the integration of these things term by term is what we do so you have a complicated expression it Brit rights that are just an infinite series and simpler things he's able to do something to each of those simpler things and then he adds them up which is why he goes to the 55 decimal places and the reference to the binomial theorem earlier on during the plague years give him another technique of being able to write certain expressions or in terms of infinite series to do to apply this technique to them and I've said it was the same year he had two binomial theorem and this also during this period that the story of Newton and the Apple originated that we've got two sources for that one of them was his niece who looked after him in later years and the other the 18th century royal strana mur astronomer royal James Bradley who had no one Newton personally I'm very of course the whole point of the Apple story is that seeing an Apple fall you realized that the force that draws the Apple to the earth is the same universal force that keeps the moon in orbit around the Earth the earth and the planets in orbit around the Sun and the lowest moon writings about this we're not to appear until the Principia in 1687 the initial ideas came from these plague years so let's continue on with the brief biography University of Cambridge reopened in 1667 and Newton returned to Trinity College soon after that autumn he was elected to a fellowship of the college and two years later he became at the age of only twenty-six Lucasian Professor of mathematics the position he held for the next 32 years now the first optical work of Newton to become widely known was not as theoretical inquiries into the nature of colors but his reflecting telescope and it caused an absolute sensation when it arrived in London lit 1672 and ensured Newton's election to the Royal Society Newton as I mentioned before grid manual skills and constructed the telescope from self grinding the mirrors and making the tools necessary for that as well new skills came in very useful for another activity that he was spending a lot of time on his work in alchemy and in chemistry unless the decade went on he was getting more and more interested in theology as well and the 1680s is the decade in which the Principia appeared the decade in which Edmond wholly invaded the Cambridge lair of the sleeping giant and squeezed the work out of its reluctant author a world that was beginning to forget about Isaac Newton accept and select scientific circles awoke to find the world changed by him and almost from the moment of its publication even those who refused to accept a central concept of action as a distance recognized the Principia as an epoch-making book and gave us author an international reputation and you can understand why people didn't want to accept action at a distance people were trying to get rid of the magic and the mystery out of science and mathematics and here was some of the explaining motion by means of a force for which there was no carrier unlike de cartes theory of planetary motion were the planets were swept around by vortices which seems much more reasonable if you think of the way the leaves outside I mean blown about in the wind but one of the things you didn't keep here was to demolish de cartes vortex theory so at the end of the decade Newton continued his transformation to your public figure by providing the backbone to the University in its rejection of King James the second demand that a Benedictine monk should be admitted to the degree of Master of Arts without the appropriate exercises and oaths and this resulted in his election in 1689 as a Member of Parliament for Cambridge and he seems to have enjoyed his time in London and that was one strand in why he decided to leave Cambridge the other relates to the mental breakdown he suffered in 1693 and there been a lot of explanations given for this most common when I've read has the destitute of mercury poisoning and I think analysis of his hair some of you may know better than me did show that there was mercury in his body but I think it was more due to overwork and disillusionment in the early 1690s what he was trying to do was to weave together his interests in a whole variety of areas mathematics dynamics optics alchemy and theology and tried to bring them into a coherent framework but by 1693 it withdrawn as theological publications it scuttled plans to publish works of mathematics and optics and hesitated over plans for a second edition of the Principia he also reached a climax of disillusionment in his alchemical researches in the summer of 1693 1693 probably was the summit of his intellectual activity and apart from some work he did after that on lunar theory he didn't inaugurate any new investigation of importance and he devoted the remaining years of his life to reworking the results of earlier endeavors and became warden of the mint in 1696 on the last thirty years of his life or where the secret of retire scholar became an interventional public figure attaining and ruthlessly weeding par he first put them into writes a Newton who could do nothing half-heartedly took charge of the record each which was needed to stabilize the monetary crisis in the economy he was an extraordinary efficient administrator and a shrewd political operator and he acquired power and influence until 1700 I became master of the mint in his early years in London Newton paid little attention to the Royal Society which was going through an unhappy and directionless fears but then in 1703 in March of 1703 Robert Hooke died and Robert Hooke was a major irritant to Newton because he had criticized Newton about his theory of colors and had also claimed laid claim to the inverse-square law that appears in the universal law of gravitation but public recognition was coming thick and fast for example in 1705 he was knighted by Queen Anne and when he eventually died in his 85th year was a sine signal for a display of pomp and pageantry and a wealth of poems statues medallions and other communications so outlets brief biographical sketch let me go on to some of his laws Newton's laws essentially those contained as mathematical principles of natural philosophy and as I've said it was Edmund holy of holies Comet for him who visited a Cambridge and coaxed and cajoled Newton to publish the pink hippy brink appear in fact Holly even paired for the work to be published well when visiting Holly asked uten what he thought the curve would be that would be described by the planets supporting the force of attraction towards the Sun to be reciprocal to the square of their distance from it Sir Isaac replied immediately that there would be an ellipsis the doctor struck with joy and amazement asked him how he knew it why said he our calculators were found dr. Holly asked him for his calculation without any further delay Sir Isaac looked among his papers but could not find it but he promised him to renew it and and in fulfilling this promise Newton eventually wrote the Principia and the title is Latin for mathematical principles of natural philosophy and the book is a byte a mathematical investigation of the forces and the motion of bodies both on earth and in the heavens Newton was the first person to unify terrestrial and celestial mechanics his approach in the Principia does not use calculus he uses geometry and proportion with forces velocities accelerations distances and times all represented by lines and areas and I want to convince you of that when we look at one of the propositions in the pink area the core proposition but first of all he started with his three laws of motion and the first one is relatively straightforward if you do nothing nothing changes but there is content to the first law because it says that what a force does is a force produces a change in motion so that if something is moving uniformly and no forces acting on it it continues to move uniformly if the body is still stationary no force acts upon it then it stays where it is the second law gives you a relationship between force and change of motion forces produce change of motion and it's a proportionality if you double the force then you double the change in motion that is produced doubling the force doubles the acceleration and then the third one which he probably got through a study of collisions saying that to any action there is an equal and opposite reaction and Newton's Cradle was a device that constructed to illustrate the third law following a hint in the Principia and from the wave here and I think it's still surprising when I see it you you lift one of the balls on the side and let it drop down and instead of the other four moving right what happens is that the one on the other side jump site and that's because the first one comes die hits the next one and when it hits it it stops the next one moves along hits the one beside it and stops the next one moves along hits the one decided it stops and then the last one has got nothing to stop it so it swings I so using the action reaction and here's a different take on it here which which I might try with my five grandchildren at Christmas of their parents allow me but I doubt it alright so now we know about forces and changes in motion what we want to do is to get force and the force we're going to get is the universal law of gravitation this law states that the gravitational attraction between two masses varies directly as the product of the masses other sizes and inversely as the square of the distance separating them so everybody in this lecture theater is attracting each other gravitationally and between me and you if we were to move twice as far away the gravitational attraction we dropped back water if we were to move three times further and it would drop by ninth so goo Stein us the square of the distance inversely as the square of the distance I'm thus gravitation attraction that causes objects such as apples to fall to the earth and also explains the moon and let me show you this if I made to explain the motion of the moon which is a book published in fact 1728 afternoon died but he was working on it's a simpler version of his book three of the of the Principia it's called a treatise of the system of the world and it contains this diagram and the right-hand side of the whole book is available I've given a source where you can get there and it's quite accessible alright so let me just blow up that diagram a little bit more now what we want to do is to show that the motion of the moon can be explained by the similar way to the motion of an apple on earth and what we are going to imagine is that we have a mountain up here V and we're at the top of a mountain and we're going to throw an object from that top of the mountain horizontally and we throw it with a syrup and we're going to throw it with increasing speeds so first of all we throw it and of course it falls to earth and it ends up at E all right then we throw it a little bit faster and it still falls to earth under the ends up at eight then we throw it a little faster and it falls to earth but this time it's gonna fail under the end self at F then we throw it a little faster again and then it's falling to earth and it ends up I think that's G yes should they shouldn't F G then we throw it just a little bit faster allan falls to earth again but the earth is curving away from it at the same rate that it is falling so a cup right run back to where it started again and assuming there are no retarding forces or resisting forces it will go round again but it's just a limit of the same behavior of things falling to earth so the motion of the moon caused by the same gravitational force we might on the moon go right here can be explained by the same gravitational forces apples falling to earth and Newton and of course did the calculation with the inverse square law and showed that the numbers are correct that the rate of fall of the moon is given by what it should be given it's that much further away than something falling on the surface of the earth but you can see qualitatively why it is the case that the gravitation explains the motion of the moon so let's do a little bit of geometry then and what we're going to do is to show you our Newton used Kepler's second law know Kepler's second law this was derived experimentally from astronomical observations and what Kepler find was that if you've got a planet going round the Sun let's say something that is orbiting and you draw a line from that planet an imaginary line from that planet to the Sun and you look at the areas that that line sweep sight then in equal times it's going to sweep out equal areas so it's a quantitative way really of saying that the planet moves faster we're nearer the Sun than it does went further away from the Sun so I'm saying here that these three areas this one on this one on this one are all equal equal areas in equal times and you can see why that tells you that the planet is going faster here the if in one minute it has to sweeper at the same area because it's closer to the Sun it means that this curved part has to be longer than when it's over here in other words it's going faster in not minute what Newton did in the pink area was to prove this result for a force like this between a planet and and something like the Sun and let me just show it to you and then I'll try to explain what it means so this is Brooke one proposition one theorem one this just comes after the three laws already shared with you and let me show you what it means let me go to the diagram down here the orbit is and I'll say more about the orbit then the plant is going from A to B B to C C to D D to E e to F and it's moving on there an attractive force towards this s here so the force is an attraction towards s no I what Newton's theorem one proposition one of book one says is that the following areas are going to be the same equal areas in equal time so tich divide your time up into equal parts let's say a minute or a second or whatever and then in the first minute the particle goes from A to B and therefore we have this area s a be in the second minute the planet goes from B to C and what we're saying is that the area si be the first area first red triangle is equal to this one SBC and then in the next minute it goes from C to D and we're saying that the same area swept out so that the two previous areas are also equal to s CD and you're getting the pattern by noise in the next minute it goes from D to e and we're asserting that this triangle here that's got the same area as the previous three and then the last thing on this diagram it goes from E to F and therefore that area s F it's the same as the previous once okay let me explain the strange thing about the orbit because this is this is where he's very very clever what he asserts is that in a minute the planet oh we were taking our equal time goes from A to B then if it were left alone in the next minute it would go from capital B up to small C but what he imagines is that at act B there's a tug a gravitational tug an attractive tug towards s and the planet is diverted from its path the straight-line path that will continue on by his first law and instead of going from B up to small C it goes from B up to capital C then everything left alone and we continue on from capital C up to small D but it gets another tug at C which diverts it from going from C or D small D to C up to capital D and the same thing happens at D the same thing happens that he seems thing happens to Dave so he's discretized time and he has a gravitational tugs acting at B C D E and F which is why the orbit is composed of those straight line segments unless I say what we want to do is to show that all of those triangles I colored in a moment ago are equal and the same if I can show that the first two are equal the same technique will work for the rest of them so the two that I want to show equal are here on the Left si B and here s B C and we're going to do it by the usual technique of showing that each of them is equal to another triangle the area of another triangle and if two things are equal to the same thing then they must be equal to each other and I'm going to show that each of these red ones is equal to the green one right so we've got the two red triangles down here si B and then we've got s B C and we're going to show that each of them is equal to the green triangle s B small C s be small see under if they must be equal to each other all right so to do this you should really turn your head on the side but you would look very foolish doing that so I'm just going to rotate so this is here taking the left hand picture and rotating it so that this line here is horizontal so now how could we show that the red and the green triangles have got the same area well we've got a red and a green triangle their bases are the same the red piece here is the piece of the red triangle the green one line here is the base of the green triangle they're equal because it's the distance the planet travels in a minute notice that the two triangles look at the same height their vertex is at the same point now if you cast your minds back you remember that the area of a triangle is half the BS by the height so these triangles have got the same base they both got the same height so their areas are going to be the same so that red triangle is equal to the green one now let's do the other one and again I've rotated it and this time the red one and the green triangle have got the CMBS so that's half of it done already and then if you look at the heights well the height of that one is up to capital C in the height of that one and so it's a little C and Newton can show using his laws that this line C to small C is parallel to the line s B so here we have two triangles that both got the same base and they both of the same height because their vertices lie on a line which is parallel to the piers so again you've got the rear triangle it's the same as the green triangle so let's just go back through it we've noise showing that the red is equal to green R is equal to green and back here is telling us that those two areas are now going to be to see him and what Newton then does is he uses a limiting argument and he imagines time the time intervals to become smaller and smaller and smaller increasing the number of triangles as he says ad infinitum he ends up then getting a curved orbit and the general result that the line joining the planner to the point at which the attraction occurs is going to sweep out equal areas in equal times and but you can see it it's very much a geometric argument a natural fact it doesn't use very complicated geometry just showing that what the area of a triangle is it's the framework as I went to bite to approach it that's so brilliant and in fact in the next proposition he uses the same diagram this diagram here to prove the converse result that if the line joining something which is orbiting to a fixed point sweeps out equal areas under equal times then it's moving under an attraction to that point so we see the converse but what he now has is a way of representing time by a geometrical thing by an area time is now represented by an area and in proposition 11 problem 6 he answers the question given the shape of the orbit an ellipse something is orbiting an ellipse on the fixed center towards which the force responsible for the shape of the orbit point how does the magnitude of that force vary with the distance of that force from the center and the answer is the force must obey an inverse square law if something is traveling in an ellipse under an attractive force towards a fixed point then it has to be undergoing an attractive force which is an inverse square law if you had a chance to look at what is happening here what he is doing is he's trying to find out what the forces which corresponds to finding out what the acceleration of the orbiting body is but an acceleration as a change of motion a change of motion he represents by the length of a line a change of motion in a certain time it represents the time by the area of one of those triangle so all of this argument and here is trying to approximate are trying to say what the ratio ratio is of a length of lines compared to the size of an area know calculus whatever just using geometric and situations in order to did and as hi he represents time that has the basis of it and he also showed in other parts of the Principia that if you've got two bodies moving around front of gravitational attraction they'll and they'll orbit in ellipses with up with the mutual center of gravity so want to leave Newton for a while and move on to hundred years because Newton solved the problem of two bodies moving under mutual gravitational attraction and what I want to do now is to look at more than two bodies at three bodies all right so two hundred years later 1889 king oscar ii of sweden who's an enthusiastic patron of mathematics he offered a price of two and a half thousand swedish krons for a memoir in any of four given topics one of which was predicting the future motion of a system of bodies moving under mutual gravitational attraction this is what he's saying here on the left hand side if you were to read it out given a system of arbitrary many mass points that attract each other according to newton's laws under the assumption that no two points ever collide tried to find a representation of the coordinates of each point as a series in a variable that some known function of time which converges which gives you an answer so find out what's happening in the future if you know that positions on the speed of the bodies at this present moment all right and the attempt on this problem by only Poincare a is very important has been very important influential so let me tell you now about pong correct 1854 to 1912 and he's really viewed as one of the great geniuses of all time probably being the last person to cover the whole range of mathematics found is the theory of several complex variables and algebraic topology and one of his problems down to brick topology was only solved the centuries one of the clay problems upon correc injector and pond Cray respond that the king Oscar's challenge by tackling a special case of the problem when there are only three bodies but not only are there only three bodies he takes one of them to be an infinitesimal little dust particle which will have which he takes to have no gravitational effect upon the other two bodies but is itself gravitationally influenced by the other two body so you can see the cleverness trying to reduce the problem enough so you may be able to get a handle on it but at the same time preserving some of the complexity of it and what he hoped is that he would eventually be able to generalize his approach to you know three bodies which are mutually gravitationally acting on each other and then to more than not there and what him to sure is the approach that he took all right the stories told in June Barrow Greensburg on pong curry and the three body problem right now what Newton's laws and his law of universal gravitation of gravitation what they do they can be coached in terms of differential equations and that's equations involving rates of change and the solutions of those equations is going to give you the motion of the bodies and they're very hard to solve and what concrete did was to develop methods for describing properties of the solutions of the differential equations without solving them and this qualitative approach to differential equations is exceptionally important and pong craze core idea was to explore the geometry or of the topology of the solutions and to do this we need to introduce the notion of a of phears space so suppose we have three particles they're moving under gravitational attraction well then each particle has got a position and a speed in order to specify the position you need to give me the three numbers that tell me where it is perhaps using Cartesian coordinates and you need to give me three numbers telling me its speed and the X and the y and the z direction so in order to specify the position and the speed of one particle you have to give me three the sorry the three numbers and to specify the speed and the three numbers to specify the position okay and we could three particles so how many numbers do I have to have I've got six for each of them and three times six is going to be 18 not right well 3 times 6 is 18 not certain correct but where it's at where it relates to so the position of the particle will correspond to a point an 18 dimensional space right I'm eating dimensional space it's just all the lists you can write of 18 numbers in list but quite hard to visualize I'll give you that there so it's a multi-dimensional space and called fear space and here we have what pong Kure did now the black is what I'm thinking of as sphere sphere so this is the phase space of some system and every position of the system corresponds to your point in that space now let's look and see what kind of behavior you could have for the solutions and what we're able to think of is that as the system evolves under its and the laws that are governing it we're going to get a trajectory we're going to get a path in phase space every configuration of the system corresponds to a path at your point in phase space as the system evolves we're going to get a path in fear space now if it so happens that when you look at that path and follow it around you return to a point where you've been before you're going to have a periodic solution because once it's returned to somewhere that it's been before it will follow the same path round and round and what plonker a did was to introduce a lower dimensional surface if you can reduce the dimensions in this case it's very good and that low dimensional surface is called a pong section and what he does as he starts the system off a different points on the plonker a section so here I'm starting it off at the point G this is the green one and it goes round and it eventually comes back to the point G again so if that is the behavior then you've got a periodic solution everything will come back to where it was and in particular the thing would be stable because it's returned to where it was but on the other hand it may be that you started off at a point X and when it returns it's not to the point X it's to another point P of X and then when it returns again it's not to the point P of X it's to another point p2 of X all right and what wonk ray wanted to do was to try to detect those loops that are for the green ones there and to see when or not the thing returned to the point returns to its original its original position okay well plunk refined that for some situations it did return to the point where it started so that was good they were periodic solutions and the system was going to be stable you know it's stable because it's returning to some position that's been added in the past so it's just going to mimic its behavior all over again but then he find there were other solutions where it didn't return quite to where it was before but it returned to a point on a small loop about where it was one before and every time he came round it took a small step around that loop and that's okay that's fairly reasonable behavior and and panca I could deal with that he could answer King Oscars question about stability because he was able to say it's what its behavior was so he was awarded the prize but well the paper was being prepared and this isn't June barrows greensburg tale to the story and the insight she's found out about it one of the referees was unable to follow the arguments and contrary to what plonker a thought even a small change in the initial conditions plonker was used in the approximation argument he was saying of trajectory start off from points that are quite close then the succeeding behavior will remain quite close but that turns are not to be the case and the results were not what he expected what Whangarei had discovered was the basis of modern-day chaos theory that even with deterministic laws the resulting motion can be irregular and effectively unpredictable and in particular it has the sensitivity to initial conditions it has this and butterfly effect as it's sometimes called so let me summarize what I've said here key thing was to reexpress the way you think of the problem so instead of thinking of three bodies moving in three dimensions you think of the system being represented by a single point but in a higher dimensional space fear space then you try to find Punk resections within this fear space and see whether or not if something starts off on the Poincare section it returns to where it was that would be a periodic solution or slightly worse but still manageable if it doesn't return to the same point but returns to some point on a small loop and then take steps around that loop as it comes back and forth that's what's called a quasi periodic solution but unfortunately things can get worse than that they can get a lot worse than out there and there are solutions there are behavior which are chaotic so the two if you start the system off in two positions that are very close to each other the trajectories well essentially subsequently diverge so you're wanting to know is the solar system stable or not and this is work that was done in 2000 2009 in nature by Jack Lascar and Mikhail Gastineau from the Paris observatory the papers called the existence of collisional trajectories of Mercury Mars and Venus with the earth and what they did was to use powerful computing techniques to simulate the evolution of the solar system over a five Giga year period that's 5 billion years five thousand million years and they took this time span because it's comparable to the life expectancy of the Sun before the Sun becomes a red giant they used two thousand five hundred and one scenarios and each there was a slight change in the starting conditions so they might move mercury by a meter this is because upon carries discovery that small differences can have a substantial effect and what they find in these simulations was that in 1% of the solutions it led to a large increase in Mercury's eccentricity and an increased large enough to allow collisions with Venus as a Sun and one of those high eccentricity solutions a subsequent decrease in and Mercury's eccentricity induced a transfer of angular momentum and from the giant planets that destabilizes all the planets that are closer in to the Sun and the earth would end up crashing into of Mercury Mars or Venus end up colliding with the earth to happen and about three point three four Giga years which is three thousand three hundred and forty million years from now so as Ian Stewart says when he discusses it in this very nice book the great mathematical problems and the chapter on orbital chaos and the three body problems the answer to the question is the solar system stable probably not on the incest but we won't be around to find out and that doesn't just mean us in this room but what he says is that's going to be humankind's good to be mankind so I think I find that simultaneously reassuring and deeply disappointing so I've lived question time for questions this time which was my plan so thank you very much for coming along you'll be able to astound and amaze your friends at your next dinner party talking about the solar system do come along for the Christmas treat for all there's Exponential's remember what I said last time everybody goes away with a free exponent for function so thank you very much you

Channel: Gresham College

Views: 29,004

Rating: 4.8278146 out of 5

Keywords: math, maths, maths lecture, math talk, mathematics, newton, isaac newton, principia, gravity, voltaire, robert hooke, raymond flood, gresham, gresham geometry, geometry, gresham lecture, free lecture, free

Id: zC_DP-KKhw8

Length: 51min 0sec (3060 seconds)

Published: Wed Oct 29 2014