Leapfrogging vortices for the Gross-Pitaevskii equation

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Leapfrogging vortices for the Gross-Pitaevskii equation
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We will present a rigorous derivation of the leapfrogging mechanism for the 3D axisymmetric Gross-Pitaveskii equation (joint with R.L. Jerrard), and point out the difficulties that arise in the related context of the incompressible Euler equation.
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the time and time and the right to do so thank you and thanks to the organizer is pleased to be here them so this this year and that the real mentioned in the 2nd part of the talk is interfering with Upjohn regarding the improving phenomenon for the cost Pitofsky creation I've been working with the the government and the meal trying to the extent that this analysis are that this analysis to the United Laura question and that we have failed so far I read some of the earliest start explaining some of start with some basic things on the on the artillery question and try to do give the intuition so 1st what what's what's important is what are the difficulties that are still there there and then I will show there is a four-course pedestrian expand the pool for the new receiver so where the gaps are for the for question and and maybe that will give some guidance to some of the worst that someone has to make some progress and so the the point so I read stuff is there is there to be quite questions in so that the 2 the underlying question which IRA timidity in Nevada City formulations so lawmaker is a function here and stay on off and he's a velocity for a few of the fluids which is from the vault city through the use of of printed the year Green function of the passion and as you know this question when the north to Ezeiza transport equations of the initial evolved to Citi Field is just transported by the flow which is given by the velocity so think of particles to evolve them this diversity fear that and then you just transport the the toxicity city by this by the solution of this problem this with an infinite number and the 1st because it is a it is a time question or functions of the multi-city of conserved to grab total mass but there are other interesting a conserved quantities in particular the center of mass journalists can hear the anger of momentum which is sometimes written in terms of the center of mass centered on the center of mass ready and and the energy in Europe and there are many much more conserved quantities in the 1 of the consequences of these at the conserved quantity is among the many consequences of this occurrence of quantities is the fact that when you have a positive office city In addition time that it remains somehow concentrated 4 times with the has compatible the at times equal 0 than most of its masts will stay in in abound the region for time because that depends only on conserved quantities and and very easy Whittle To see admitted of course it requires that all men gathered voters is a constant sign otherwise and it doesn't work so I spend the mosque as a part which is in the border around the center of Mosul may under the compliment now that the part which is on the compliment you just since it is next season distance from at least or from the center of you can minimize 1 like this and then and there and seeing the 1 of the conserved quantities of fear the anger of momentum and so therefore you you have this lower bond on the massive multi-city contained in the morals of Major's often tell that the center of mass which does not move from the fund which is a concept called the for the Idaho equation in terms of the terms of the total amount of capital are sufficiently big can make that this as small as he wishes so it is a weak form of contemplation of democracy the of this only works for the this whenever an implication for middle this only works if all my guys assigned and because otherwise you can't do that OK and their solutions of course which are known not to stay a compact and in space when on solutions are if you take a patch with the person about with minus those 2 will behave like a troubling parent and so they would not stay localized stay here grows to 0 in space but you could say whether they are still localized 1 close ,comma each other but you could cook up a different things like that you would have to to pay that escape 1 from each other through the different signs and so that there is no way that that it would be localized so so this is something which is there for which the sign of the Protestant it is it is these caution and they're not the kind of solution I will look at is precisely such kinds of initial data where I would put the scale Epsilon and so I would consider patch which are this is just little to to put upon during the problem I could also use the distance between patch and make it big but it is sometimes this way so they would behave with a diameter of sigh subsidence so that they have positive mask the Omega would be like a side too -minus toast to the In those French so if you do that of course the annually rewind this estimate for 1 batch then you know you have that up to what effect on its side of the square with the masses still contained in the bowl of of religious war of is what this of course only works if you take 1 patch because because the smallness of the smallness of battle that the
slowness of and of course and it expects square is more on the form and its consequent coastal points but if you take 2 of them than effect square more here is not common the small there and so this is a very simple enlisting does not work if you take care if you take multiple batches if we can just use the environment in that case to 2 tell about and would be to say that such structure and remaining time possibly with emotion that was that of study that but it's not sufficiently using violence to do what must prevented the they're being back in the eighties is to say well them b what we are going to say so that the bill some are always there is a linear role that give them their passports the only got to the V and that this is the only question is upon spot equation if initially your city can be divided into 3 I want Buchan a 2nd batch I transported each of dispatched then I'm away to spilled the multi-city at later times for and this is just because it's upon spot a question and so and the Bills have always singular are on the ideology of cancel when when dispatch is going to move by the IRA equation I I can consider that the speed of the speed which is generated by himself by viewers last and look at it as the single object it but then there is also the speed at which is it which is invited by the other parched and since initially they are apt to find distance they all 1 distance than as long as there remain like that I can treat this other terms as a as this most and are as Egyptians so what they did was just now even if you have multiple patch just consider 1 of those patch it will it will end and that move it was a thrill where there is a lot velocity which is given by the Bills our itself and then the views of our 0 implied by the other patches you just see it as a false Intel has hasn't each it's forcing too many digits as long as the 2 patches do not come grows 1 to the other he was formally this is an abstract as an abstract flow and then you compute the time directive of the Alliance of what was in my hands full pure white question and you just get at the time and that attacked content directive of the center of mass noise just moved by by the Falstaff and and the important thing is the the only the younger momentum here that satisfies this equation and if you notice that this quantity is 0 just because this depend on X socially integrated this on X by definition of BTU gives forget about these 2 out of the out of the into Gaza this Illinois you compare this with this and you make the difference with effort x and f at the center of mass so since at his Lipschitz the distance you can bounded by X minus speedy and then you get the X minus speech at the square Tanzim Miguel which is which is empty so that the anger momentum and such kind of froze the site is is is it would constitute foreign formed Monday inequality so if it wasn't silent square initially means it remains of size of all the time square the false times of of all the way but of course there was there and that I essentially is the fact that there is no interaction between the 2 world and and so so unique kind to to Woodstock and to be able to say that if they are initially at finite distance then destroyed all time that William indicates in and this is a difficult that is something which is not easy to polarize the electorate a difficult but in order to walk of of Moscow and brilliantly ending is exactly what the pulled so if you start this patches that time that emphasize abstinence what a therefore they are form their precise share didn't need to be disks and that at at a distance which is on the 1 hand then for times of all the while they still remain separated and that the other longer outsize and Simon after a positive time for them to become approved at the Met where this is going to an open Program for the developmentAsian problem to establish something of pages 1 whole flock in the voters to go and and when you but once you once they are able to prove that the 2 of the Baltic states remain at older 1 distance 1 from the other than diving in their motion is just looking at the at the at the PDM and and exchanging the old McCovery that time and the fact that there are concentrated just say that approximate this just a consistency fuel old once you know that the vote the patches of concentrated you emotional and the the so-called final Texas stems from the the from the folded just a little this is the difficult part of the pool and this is the common students is a constant this is putting on the air in this system is homage to an end to it which is available within the Khryakov unbeaten in here In actually of course it has to do with the energy of the year the order of energy of such configurations and which I would which I there and the reason why I wrote it he was that they the Evertonian of the .period systems here only depends on the relative positions of the patch and 1 with each other so is the center of mass of the patch i 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somewhere you can't read the different batches a separate case will work that can be reflected in the way in the fact that the momentum of the energy someone independent of position of the part that we something would be different in in the in the symmetric Tribune in hospital in them too so this this is just the the interval of 1 of the extent of that at the time of week L all that 1 as big as it is has been powerfully studied that depending on the undersigned so went in there if the size of the alumni the geologist saying there than in the of the collision Parliament's nor few different signs and you know you have those solutions to the collapse of some of this is a report that the only thing that goes on In this is finite time so yeah there is that the skin OK and yes well enough Corso is that this this year I'm by my going toward unity air works for fall and the Thai army township of patches in particular it does not use a tour in the summer optimizing property of the patch on gay community the number tight budget and is so somehow it's not before if you want to To make about areas of sale Amendment to Saudi don't entirely it does nothing to do with that in the sense that it it does not use a at Of course the but it it does not use the fact that for example 1 would be an optimum of the energy at a constant momentum and finally just uses the fact that if you sufficiently farther apart than that they do not end just long-range none induction I want pursuant units will be able to say more if the patches somehow at some certainly their behavior and in part cannot there are some optimum offers some conserved quantities and some constraints and service it those objects of all along and there is the optimum notifying members and if you there if you consider the sister of batch then its ability as being there has been studied and in particular 1 and pulled the 90th proved that if you consider a patch which is initially sufficiently close to Basescu last batch is another 1 that meets the state's closing and 1 4 time but also because this is just follows former inequalities that on concern over the on her story on yes on this concept quantity just the the Inc and he was suddenly method that I would like to make a living in mind Parliament was a recent action of sometimes on this looks like a very good so that the secular Bojan only days here is the is the is the country patch this looks like a nun degeneracy uh condition defined by the user respect conduct a it actually isn't and the reason is date the lack of differential ability of the structure of the manifold on which you if you look at the use of the money Ford of extensions is not something which is a nice evict of structure and the perhaps the easiest way to see it is that is considered the most expansion this this is upon data now another quality that would be needed to dig a hole here and to put it here of course so you in this you can doing going 1 way you can do it with the press but you cannot do it with a minor so if this is a problem at the direction of evaluation the opposite of this valuation is not permitted to was allowed there's not the it's not that there's no tangent space and there and you can see this inequality as another generosity inequality and the 2nd dimension of something this is the main at the main difference and 1 other way too it 2 to see that this would be absurd to think unity in this waste and 1 almighty and to normally would you you would just get a square here and there and the ball for him and that he was out of this is not the door in London did not see as indigenous it is indigenous so this is 1 of the reasons why I always start with respect to was the best that I represent later is is badly behaved is the fact that the India the choir albeit on most expensive is the set of of candidates from enough stuffed fodder for batch does not have at least for me and 1 on which is the 1 that had been cylindrical tavern with and money for structure so this was a stability during the 1 an initial spots that walk rose to work crews to the circle expansion cities and guy in the English often ending in paperwork to proved that it is actually a robot and after beating estimates but never prove which relies on under different concept quantities and ease of this was based on the energy and the use conservation of momentum in the Indian government to and again here I with this factor to which some of them in 1 of the main trouble and also the other 1 known is this 17 you I we would like to to avoid being so now I got to where that from the progeny something about Baltic strings so it's not into the it's in pretty during the look at the analysis in entreaty axis results were out here so I will be the only question in intraday gain newly covered the middle city from the with the U.S. oral and I woke up only consider multi cities that are independent of so I'm looking saying they can coordinates so busy warranty terms will receive this is our 0 this because each of the system of playing which of things that we can coordinate and forgot about it them so only guys that is independent of the time aligned with BT in that case the velocities it is annexed Austria in this in this part playing age and they're in the year the I like question translates into integration with the EU former the Dow which again is at times but aggression that at times but aggression not far off only get that on off of you divide should have written it differently if you divide this aggression by Are you would have DDT of get it over our is the reason the gradient of this quantity and therefore this is what this is .period spotted that divides as responsible altered by investor fears which is not of week get diamonds 385 look at it as dozen different in H the case
this mix things to be different but still you and the fact that this quantities is transported so if you go to values of our the Omega is increasing what why wrote the at the other question is as you might think a Bachelet the batch like this a lot of physically and was going to happen to excel here was a bachelor here corresponds to his will 2 2 win hundreds of trying to punch that I wrote this is just a mistake I would be at the Idaho questioning in in the week forms for HIV before falling on the tiny Omega which is exists wake and and you can play a little bit was the right inside using the years so this is the expression of the fact that the Omega is the call of the music of of this of this form is and this is the fact that the divergent lead as a three-dimensional effect Orazio which translates loosely In the end the US space age and then you get an equation week fulminations from overnight which well on the right hand side you want to get the OK you don't have the Omega anymore and and so what can we guess about the evidence this is there the way I prefer not to understand the emotional takes things formation I think the came to understand because there and we know it is a prime spot former governor Oslo so this is a passion it's going to stay patch this this locally concentrated and there we want to go to to guess its motion so to guess its motion it suffices to take a function fire which is anathema as a fine fine function going so if I take the fight to be a finer than the Hindustan Times that of secondary matters a fight in the present all and I kept likened them off sufficiently far away where there is no and uh where viewers decrees all right when they say and I have to to understand these 2 forcing terms in order to guess what will be the motion of the of the smaller others evolved expansion is a lover of the mistakes rings and the fact that what the video scenario in 3 D looks locally like the views of our owing to the weather for accessing across it looks like the abuse of power into the hands of lioness more patch at and the diversity of that is close to it will look like that like it like it 1 of the US times at times into Delaware often say 1 of wall where always the distance to the sense of touch and not on the spot and so if you look at if you look at it you see that this would be like 1 of our square finds a "quotation mark square after death so this would indicate to something Nunzio it to integrate precisely to of the of the cut of distancing of the departure is 1 of the most integrated whereas this 1 here is that 1 of our house finest close the Dow 1 of our society two-time that 1 integrates 2 0 at because incident the demand so this will give me that the time that of the position Omega is essentially the IOC and this is 1 of find function this is the main talent we're going to see direction With speed which is proportionate to 1 over distance To the into the center and of the wages of the analysts here and there multiplied by the regarded maker of the width of the text so this is a if you take into account the the master of the year the master of the patch which is for 1 year and 1 to years you get demeanor although in the in the speed is this slammed our house not so the intensity divided by this courageous of the analysts multiplied by the number of the thickness of the patch which we assumed the expects identities that the attentions OK indeed this is something that and lots already knew but he didn't do the computation was the question office to form nationally just use the uh just computed the very precise asymptotic folder velocity across to the tune of any of you that the main profit this 1 to concert in some in some sense because it would only make the little what it on its head but your idea and a precise computation and the even to the next although which was which were actually in a tree the existence of exact the most explains can be so exact in the sense that exact timing with solutions so solutions that do not deforming in time except for those from the Vatican translation can be constructed by maximizing the idea of energy which I wrote here in the office city formulation and of course find that the momentum is fixed division to big there's nothing and that and the fact that the or our our transport of what it was initially missiles also something which have to impose otherwise the maximization problem becomes the previous year's infinity and OK so this was a long known was an idea of to use such a such optimizing constrained minimization to get the exact timing waves and it there's a lot of work which Indian makes a very precise the fact that this problem has a solution provided these larger than actually did the momentum of the involve Texas In the end it has the shape of fixing and so on and on and on different so this is for an exact accounting with solution under a different direction Benedicto Kennedy and not pulled that at or most maximizes of this program behave like you like explained in we knew weeks sense in the sense that I am there are there is a point close to where we were you expect and and the patch which has most of its mass there at it's so therefore it is a good week we come out the other information in this form is going in the dispatchers going with the with the right speed the asymptotic it so that is been done for one-off explains and the extension to the cases of 2 or more explains which would be the exactly prevent on the point of text system that that Muster will improve the energy giant as not being done for the system In an open question so this is something which I wanted to just the middle of next week to show that this is an extremely this seems to be an extremely stable solution of Odai question is whether the carrier who proposed it to B and it's just with call then and which are which I applied find myself and as you can see it looks very stable rates can go up to 10 or 20
meters in the lab without changing the original from there is nothing very precise in the distance I just ideas was actually my doctors builds who built the Horned Lizard thousands of any goes immediately it goes really fast without we don't without changing shape and that in the 2nd
half of this was the destiny of the city for freer flows was in this 1858 58 took over and lots in the 2nd part is at the bottom half which is not preceded by computations in it comes under UN it be foresees wants who will be the behavior of the interaction between the Baltic swings and so he claims that if you take 2 of them this 1 in his direction and that can come close what's going to happen is that the exterior 1 is going to to grow it's the following 1 will shrink since it will shrink Europe have a smaller operators and so it would be quicker because the speed is 1 of of windows and catch that 1 person front and then they will exchange their on that 1 would be behind them so they will play this leapfrogging 1 will pass inside the other 3 video that was made by Indian and Singaporean citizens care so this is this is a Latin view of what's happening and you can see it is the 1 who wanted prodding motion they thought this was a very very dedicated staff to reproduce there in the lab at an excellent students from 1 ground across to this year that came to me because they had to do do with details
and and then we tied together to to use these same and sued to measure of some and so in the end they were right but they were not the SPC rather uneven and meaning many facilities and that is you can't see or not see 1 1 of the leapfrogging motions with sit that and the parties which was built with the handling and listen very human dissidents as beautiful the previous so this is the phenomenon that we would like you would like to explain how to United
Overseas and as I said that there will be a new offense for the for the underlying question and I will show you how it goes solely was potentially aggression and 2 2 1 2 2 mentioned the link between the 2 so at the cost at St question is it is not enough the focusing shredding I question it has a parameter outside of it which is fixing the question that's 1 of the main difference reside or work in the previous 1 into the audio you can fix it in the initial daytime because responsible bolted the between the interim inside it is also a length scale at silence so you will see that the Baltics tubes for was best will have some affixed excellent uh escape here where small Albert was a difficult thing proving that a concentrated enough to secure remains concentrated some that would be the easy part in there was Pitofsky aggression but there there will be some parts that will take you to be is you so you easily ease with values in the complex and there and so it is the Amazonian fulfill this against Orlando energy and the energy which proposes the following when you ever you hear you computer it's a Jekyll and in the following way so maybe the easiest way to understand this formalized to think of you as being a so sinking since its complex I think it as a modernist exponential phase then familiar computes the core guarantee jail view which is you cross the line you is just the gradient visited the gradient of the face of all square the weighting of the face barriers that you see here and there was the density equation and energy there is a penalization when it scientists more disabilities edition of the fact that mothers of you is not close to 1 so you should think that for 4 functions that fears that have abounded the energy of confluent energy also be close to 1 in most of the domain so the current is essentially the gradient of the phase and there at the end they developed a city that defined be best used account of this of this gallant and so you see in particular that if if you have at your face and his evil ways exactly equal to the old and the Jacoby and calibrated it is identical equal to the sum of the main difference here with respect to iron out is that in order when you have the to city You a cover of the velocity of the velocities is completely uh defined by devolved to cities will be was the last of it is not the case and must tasty thing must be it uh you see if you are pure phase the it identical easy all but they're there but their diversity can be and not be kindly gentleman so there is some additional and degree of your freedom you can think of it also we mention that later on the fact that somehow was the testy as early as a month compressible component wrestler was compressed as a compressible component whereas I was was compressed this is something you mentioned so as I have written hear the week formulation of the of the Gilespie at that was thinking another the week formation because it does not characterizes that I've written over weak form a week identity for the evolution of the Jacoby and jail view when you was and is an axis symmetric solution of the hospitality equation this is the equation where this is the capital of the exact form the doesn't have such an interest but that the point is that I have written it here in a form that resembles very much what I would be fulfilled the underlying questions and actually so much that I care write these to equation here this is philosophy diversity and this is for all was the exact same capital as just that it acts on different objects for the Boston-based equation it acts on the 4 gradient of you which contains most days what I could do little jail view the current but also the none of the none compressible past whereas a unified I question is just act on on the and so in in a sense to to close the question from was Penske unit also an equation falls a for the conference in the past and this is the 1 what which is because jail you is not to die vengeance is not divergent free-fall Archer viewing innocently :colon coordinates but it is or most diverges free in which you can see here it said I've put the absent purpose this quantity is part of the energy we you take its it's square is the potential part of the energy so this aggression tells you that on average in time far-ranging time the diverges of Jay-Z's at and this is the link with the encompassing the final opportunity this additional
decryption this is the last whereas the gain was the fact that the landscape was fixed but also in other energy which you can see here than ever of the of the existence of lost explains the new here I just woke up what I had for the underlying question you can get exact small fixed-wing so out by maximizing the energy and the momentum constraints for the rusty Penske you can get exact struggling rocked explains by minimizing the energy and the the exact equivalent constraints so so this is analogy but also a very big difference because 1 is a maximization problem and the other 1 is a minimization problem so others of the energy does use somehow that there is a sudden fall for Colorado and maximize India in the set of fees that encompasses burning fuel compressibility then you are minimized to 9 energy but transitory so far sharper differences 1 is amended a maximization and the other 1 is a minimization OK so uh OK how will we will restate the interaction of Baltic scene in it and it will be in which Beijing said muscular prevented it down Alice's father of advertising left expansion we will use the energy and so we use greatly the fact that some objects of optimizes of energy and fixed moment when you that they're in a crucial away so we cannot work with this very that setting inside to explain precisely what our mission that which we look like and they there and that's so to do that I consider an number over an hour oriented in enough to which I see as a matter think of it as a as a line of current and capital J and I consider all the Nextel distribution which is to by this is something also which is important for it s scheme 1 of the main differences on ovens discretization philosophy that Sky and which you not need a floor fight off so I consider up to 5 times the integration along the coast of X of so this is a victim of this Commission's gesticulations of along the coast C you know when you have a line of counterfeiting clear-thinking dose of excellent magnetic and you have an induction capital B which satisfies these beverages speak with you in the counties is exactly Captain James virtually government views are not anywhere near the induction and you can go to the vector potential which satisfies these sets of aggression and then you get that connect plus a creation of the effect of potential is equal to just now now if you are instantly got a call they meant something just 1 point here and there and I was playing a little game and then I will tell you what is the idiotic and the equivalent of a jock must fall pointed out the reason Deakin longer along the south going in there and so the capital gains uses solution of of the Victoria Cross equation was wheezes Yocum died and signed in which you can divide in flavor can look at it and you can tell you what the capital is even was complete and taken to London this is this is something that is that is that can be done but that is not really the most important of the thing is that since there is a tool by here we can the defined cost betrays the vector of the view that will be related to capital and in the sense that is that it's vaulted city will be precisely at this singular sucker OK and what you do basically is using this formula for what about the obvious something just because violence in the capital called it is that what you say is that the gradient of the phase of you it's just a and all the of the of the potential of a this would behave like a like number close to the singularity is faced with a like a typical diagnostics said into did you get this equation the sets of the Commission for use with colleagues to buy them time the directors of J 0 this change of of metric so this is the equivalent of topics exports now I have to say what is the equivalent of the sister of patches of rough patches were the year the year the optimizer thus far energy in the optimizer Scientology and against Orlando framework unknown and our good offices as well so they have it they have this cannot form there they behave like exponentially IT down this is into the air the and they have a profile which is like this interpol find this is the moderates of you will look like that on the singularity it would not be a X for for the mothers of expert you regularize by something which is a list of candidates and this is energetic and optimizing says is the best way to regularize
8 single month explains in the context of the boasted that he equation and you define your field just just in this way so this would be the equivalent of a of a disk of fixed disk in to the life you want to take many of them since you are essentially in a modular Martin's you just multiply them so this would be the single out 1 multiplied the complex fusion and the the single ones you you can't you also missed and so no you have defined this way you which as a member of 102 which has a number of Baltic swings at different points and you can compute the expansion for its energy and written here and also checked that it's not the city is close to the sum of the masses and here is the difference with respect to identical in either of the other 1 moment was and that your article on the death was the youth wanted it to measure these the spreading of office the English protested dull now that the U.S. overtures function is there is something which is well behaved and in and which which also has some advantages because it attracts more the fact that smaller parts wanted go to infinity for the W minus 1 1 along with rest more part goes to infinity you see the distance at which it goes whereas for the and 1 on it just it's not there but if does not make their 1 on does not make the difference between being a distance 1 ending at distance 1 thousand From Census Census something this kind of estimates that would be nice to have in order lost to Wonderland and 1 a distance is and then 2nd important thing here is that the main town in the in the energy now depends of course on the position of the points and not only this is the equivalent of the interaction and log expire minus X J which we added to the known that is a main term hit here which depends on the on the position of 2 of the 6 for the 1st extracts so since it across the desk is Hamilton inflow for the year for all the years was the dusky questions as the and the 1 would I expect that the position of the year evolved extremes of baby emit an inflow for the limit energy which would computing and which is a chance to which I just work here and this would actually be the content of the theory might just mentioned that this role here which is which I cordially probing the a old III as there are in addition to the 2 young Estonian as a conserved quantity which is the momentum which is just the sum of the squares of the IDI of the of the Baltic strains and that actually not all the Audi but for all the food was beaten last year we have seen that there was a constellation of this which was the equivalent of the uh of the anger of momentum for orders and therefore all her patches in the you have done the same in the same asymptotic this which an owner or owners don't pay attention to something like this sorry OK so another statement of the theorem I claim that their rights and the right staring to see the interactions between the officers and is this 1 so when the Baltics scores at a distance Of all the 1 restaurant overlooks so I wrote them as a Senate amendment .period in not and in the distance he is half of 1 of the growth of log some if they are closer than the effects of of being afraid he will not be enough to be seen if they have to close they would behave like that to be like straight vortex filaments and you 1 would only cover the the equivalent of them afterward and Peruvian dinners out and got to follow a path than they were just behave like Baltic swings we say this too few interactions between them and they would not be an interesting the meat of the and that the series of so so this is an estimation on the initial data and I assume that I have abounded but I let those points it lost by the limits what what I think to be the name of game and I am in the statement of the theorem I I assume that the BDI absence of S remains bonded this is something that you always have for some time and we receive a or even 4 times of of 1 minute into the bigger In summary I assume that initially my not seduce concentrated there close to those points and this is what I call the concentrations and I assume that the energy of my initial data it not storage bigger than that what I expect to be the optimal energy for such configuration of .period look that and this is that this is the optimal 1 it will be at at although it off 1 of the consequences of the theory just expect assume that I have not too much energy and this is what I call the excess and this year says that is most the concentrations can end the excess are not to begin a initially then then there may not too big for for a time of older ones In this case I can make as small as I wish my by my initial data and we have this this loss which we cannot tell which comes from the technicalities in the pool that there has to be something here and and the point is that we have an exponential here but with no single party its and there's no it's not 1 of them excited about something that he will unless there was a 1 at square In the question so far although 1 we can say that the behavior of the for fans of all the 1 we consider the behavior the cost desk equation is essentially the same as the behavior of the funding for the another leapfrogging
over the eye depicted here it's it's his anagram when there are only 2 Baltic springs and that is so that they are full free parameters because each each A 1 and A 2 will have 2 degrees of freedom but because of the 1st you can uh where you have the momentum that is 1 and there there is also a translation invited survive those 2 points and I moved them by a fixed affixed on busy direction that the motion will be equally Lyons was this this and this Council mission so I can just look at the difference between the attitudes of the 2 underlying and use it as a as a function to this communist you and the other 1 said was the conservation of the momentum so that an appeal was the sum of the the square radius of the 2 so I can define this environment 8 . 5 this month did some user was the sort it out if it stays close to 1 0 minus 1 means that 1 of the 2 what they say is much bigger than the other 1 and then when it dies you know it just means that they have the same at the same time time eyes and so if you brought the Amazonian absent in terms of the time and Zeidan you get this phase diagrams and so you see that here in the center of the region where the solutions of periodic which means that so the time goes from positive to negative the positive negative which is exactly what you you observed in the video in the sense that sometimes 1 underlying available and the other 1 is beginning and ending think they catch each other but there are different regions here so this would correspond to the place where there and the Baltics things have to much different Idei initially and so they were in time to see it 1 is going close after to the other 1 but there it has made the difference is to begin the it's not catch by the other 1 and it would flow inside for example taken very small Wilander that big 1 informed it would cost fool but it is not sufficient is not officially close to .period truly with In longtime would escape with so some of them online Regency where have 1 really come closer that's what I will accelerate it will not last inside that they will finish like like this so controls them that what goes on in this direction you can also at least numerical you look at the case that the leapfrogging systems for 3 or more the most extreme gossamer very nice examples like like from the point of Texas like from the point of Texas systems the bookcase idling at half time and it's some slides on the clitoris and my time is over legislation to say that is 1 would like to do the same for all of the difficult he is the concentration answer the main question and I think that was a good way to try to 2 the package his replaced the other 1 arm India in the year of the gala so 1 Peruvian Engineer estimates by the norm which is a more adapted to catch the past of dust lost a city that would that would want to go to the infinity is it possible to do it too is Durand of Lipschitz we've tried that this is actually wrong but that 1 may think of other ways to measure of the expansion of electricity and instrumentation problem in order that this is a difficult question from which we have fax questions will all right what were you see on the face of this time so I really don't like it and some of the food offerings of the use of force against the militants the only 1 yesterday that the amended and this this requirement final so if if this is of size Cape Times Square hotel room Gibson this will be essentially exponential-of case of these are actually very flat thinks so the motion is election it's it's better to get environment but it doesn't look like something that does this but a lot of wind case begins just something like that you don't see anything followed and it exchanges you wait for a long time and took stands in their way when you get close enough to to do to be a token McHugh and he is losing money so many Reitzel in this In this furor over with In this year and we can actually go to times that after long beloved site and in what will be the rest of his life it's really depends on the concept which will be informed if you really want to have to go that but this requires a time of although 1 in this in the skating so principle we can if we take it saddened that very big very eyes more sorry we can see met many of the projects many of those that it would weaken follower is not just 1 part of the National we a little more serious than the government's chief of police the other in a moment of my life because of pollution in the city the cooperation of to it was then that the yes there may have been in videophone so soon after the tour the 2 friendly .period said there both going in the same direction and the entire now if you take a different signs in the office said today they will collide with depending on where they have live and this is this is also the violin and there were life seems to be much more complicated than that case a what medical you would think there is something which you don't like because you he start in the inner city and situation isn't the solution is in the direction can instantly but life is never seen in all this is put of this video is very popular in the fluid mechanics community because you see you will recover somewhat explains which each of them as hard for 4 of the past which instantly this and also this is a great experiment is clear this information is another lot explains reaches the scanning this pilgrimage monetary good time to take a position to president to city thank them as