Introduction to Topological Fluid Dynamics - Lecture 1 (of 7)

to introduce our speaker professor Minh to recover he's a professor of mathematical physics of the University of we had to become a teacher he his research interests include classical field theory especially in the area of classical theoretical mechanics dynamic systems including but economics and natural practical Magnus von destruct structural complexity he had often his his fifty degree from the University of Cambridge Trinity College his supervisor was professor peace moment but Peters maybe not one of the leaders of the area the area of it has being afforded many distinctions including the night price JT night price of Cambridge uni the immigration fellowship of jsps of Japan visiting scholarship of a my rt3 and Laguna senior research fellowship Institute for scientific and interchange easy it is show that karate is easy eating as well as some other visiting fellowship that this being professor Patrick analysis distinctions from the year of 2016 he has become just the professor of our university so it is a Greek language from him to give us what they chuckles about dynamics here may not have a suggestion for their opponents please just right here we will just if you had not maybe two days after after giving up actually budget do you think I'm talking about that all together very shortly [Music] well thank you very much thank you for the invitation of course the instance of theoretical physics in particular represented here and she knew as part also of the regime Dublin International College I I thank you very much because it's a great pleasure to be here and I know I've been always welcomed with heart and that makes me very happy so thank you for being here so these lectures will follow the kind of new revolution that we are living these days due to the great achievements and the successes of recent approach of topological theory to physical sciences you all know that this has had a great impact in string theory through the work of Ed Witten and others and I am quite happy to tell you that this approach is actually much older is an approach rooted in the work of Kelvin 1868 when he wanted to present to produce a theory of everything until it arrum a theory of matter exactly as we would call it now a theory of fundamental physics he conceived the idea that vortices were like infinitesimally small particles we could could paraphrase that saying that is like a string in a fluid medium is like fundamental energy field at invisible scale and the idea is extraordinary the idea 1860s he had the idea that a vortex ring could link or not in space and because of the permanence of ideal fluid no dissipation then topology had to be conserved first thing second thing if topology is conserved suppose you have a knot and you have a different type of knot of different complexity then the two knots had to have a discrete amount of different energy so through this idea he envisaged the possibility to interpret discritization of energy through topology this is a incredible idea 1860s because he not only understood that through that thinking was possible to determine a spectrum of fundamental energy states but it could also Envisat the possibility that these states were independent from the metric you see so many ideas already there that we now attribute maybe to more than scientists okay so we we all know that this theory didn't survive in the eighteen he had a number of outstanding followers including JJ Thompson JJ Thompson was a brilliant mathematician in Trinity and he worked on mathematical aspects of this vortex atom theory twenty years later 1818 and the more I was working and the more things came uneasy and finally he dropped the theory to become a physicist and in 1906 he got the Nobel Prize in Physics for the discovery of the electron his first works were on the mathematical foundation of the vortex at on theory so this is very briefly the story about the topological fluid mechanics that has gone shadow'd has gone down after that period because of the great successes of physics and the interests in fluid mechanics are now takes etcetera etcetera etc but we inherit from that time these approach the approach that through feel the theory as we say now we can investigate fundamental aspects if not of the fundamental nature but at least of of nature so I'm going to present you just the introduction to this field that is now very rich these E in the last 20 years we saw a very exciting new discoveries coming from very very talented young mathematician and physicists and so these field that survived through periods of ups and downs is now resurrecting and has having great impetus now I will draw your attention to a number of facts that are also relevant for mathematical problems and I apologize if I will focus on mathematical aspects that are interesting for physical problems but more mathematical in character so forgive me for this being a short course I need however some background material so I am obliged to start from scratch in order that everybody is happy with what we are going to do and then paste will increase after the second day so from tomorrow onward we go a bit faster on more recent results so forgive me if I start from really the beginning in the beginning is a material that you find in standard books but has been revised so I'm not going to present really the standard way of some of these topics I will try to present you emphasis on aspects that are relevant later on so I'm preparing note lecture notes for you so I still have to prepare my last lecture and I want you to have all the set ready and since the set starts with an index the index is not ready yet and so please wait for few more days and you will have all the lecture notes now the lecture notes contain full proofs of what I'm going to mention so that'd be much more material than what I can present to you I will skip some proofs I will do all on some proofs but in the notes you will have a much deeper presentation so also historically I told you what I wanted to say there is a little bit more in the notes I will skip some basic definitions but I want I want to duel on starting from a theorem that is a very important is not so important for the topics of topological fluid mechanics but is a tool that has been very important in developing many ideas in this field so I have to go through that first but before going to technical aspect let me just draw you a few picture and the pictures are stimulating our imagination so suppose we have I will be extremely unprecedent just to give you an idea of what we want to tackle one problem is for instance something like this I have to look at my pictures because I don't draw things very nicely sometimes and we have something like this and this get linked with this part and it goes back like this okay suppose you have something like this and imagine that this is made of some material maybe plastiline okay you can deform it and you want to play with this material by deformation continuous deformation and you want to reduce this material to this material okay can you can you do this without any cuts no you think no right so the answer is that you can you can buy continuous deformation of this material imagine is plastiline you can deform this to this so this is a challenge right it's a challenge because it defies our imagination we believe it's not possible but it is possible you can deform continuously without cuts this structure to that structure and well one way to do it is to you know you can deform this to make it a very thick very thick okay and then well we can use this to enlarge it a lot you know we can we can make it very very large very very large through here and here yeah and then we can go this side all around the other and reduce this to this shape I have a sequence here a much better drawn that what I did and I'm sorry you cannot see it but you will be in the notes so I prefer not to waste time to leave you the curiosity to have a look at the drawing and step by step how to deform this to that well there is another situation continuous deformation there is another situation where we may start with fluid we may put some light fluid on top so suppose this is light fluid that stays on a heavier fluid if we reverse the situation this is stable right because gravity is a pool of gravity working downwards and imagine now that this part of the fluid is made by a heavier fluid and for some simple reason for simplicity we assume that that's a certain point this collapses okay we know this oil and water or vinegar and oil etc etc we know that the two liquids oil and vinegar are hardly miscible if we want to mix them we have to steer them up a lot to put a lot of energy okay now suppose that this fluid is heavier but not so much so that after some time as time passes the fluid collapse here and at a certain point floats on in the bulk of the other fluid of the lighter fluid then there is a trace and then these trains as time goes to infinity disappears and suppose the fluid is heavy enough to be balanced by gravity at this stage then again if you look at the situation at the beginning and at the end of where the fluid here that goes down like this as time passes then you have a change of topology of this ambient space without cuts how is possible is a continuous deformation there is another famous example and this this is quite recent and has been done experi mentally and these you know you imagine to drop something a drop and this drop goes into the water and then there is a reaction and that you get a reaction from the water so this is water and there is a cusp for me and by changing is air air and maybe some particular liquid is not entirely water you may put some glycerin you may change some particular physical properties of this but you can get a cusp a cusp and this cusp in the infinitely thin here with an extremely thin radius of curvature these problems share a comet one aspect is what in mathematics is called a singularity singularities singularities they share also the problem that in this case there is no change of topology here there is no change of topology under certain assumption and here there is no change of topology except that there is a class but there is a age of a point in space where the radius of curvature can be extremely small extremely small how small or very small tend to minor something meters so small that we go to scales of quantum physics so that classical analysis gives you the wrong answer because you get to a limit where you get wrong numbers for for quantum aspects you go below certain scales that are absurd now all these problems share common features continues change of shape without cuts and emergence of singularities in finite time maybe so these aspects are central to modern research in mathematics and indeed the problem of gelareh t formation in finite time under equation of fluid dynamics is a well-known clay problem of the clay institute is one of the eight and solve the problems of seven and solve problems of the millennium it's a 1 million dollar prize for this ok so this is just to give you an idea that these are challenges and represent all these phenomena challenges for mathematics so let's introduce some tools first of all I will focus on to keep things simple on ideal fluid dynamics ideal context ideal is of course an idealization of the real world but we may capture phenomena that to a certain degree of approximation are relevant for finer aspects so I will assume a continuum hypothesis everything is continuous till a very small scale how small any scale we like any scale so you know the problem of viscosity if you go below a certain scale then can we define viscosity ok so this is the first assumption continuity and associated with that let me just sketch very briefly what I wanted to to stress is that we will focus on certain flow maps so I will consider the motion of a fluid to say from a point x0 to a point at and x0 denotes the initial position X will be at a certain time and this is a flow map that transformed a structure a vortex a magnetic field some brings a point in the fluid to the new position now this flow map I constrain myself to work in a set of flow maps that are like this I take these flow maps to be continuous smooth with the new large enough so that we have a inverse map by minus 1 that gives me access 0 through the investment and the T if we have T and this is for any T finite in a finite interval to new to a number yeah yes yes now this means this means that we can always go back to x0 right we had this problem of the custom remember each point when when the surface produced this cusp each point of water goes up up up up and at a certain point we should be able from this 5 minus 1 to distinguish here every single point that collapses to one point which contradicts what I said you can reverse this experiment with the usual experiment you know tap water tap water you you shut up date the the flow of water and at a certain point from the water forms a drop there is water here at the certain point of the drop start to detaches from the fluid what happens to this point this breaks down we have no longer inverse mapping well-defined we have problem with the inverse map so you see as we define our fluid to be continuous with the illness that exists at all time we immediately cut out many many important problems main important problems of real life but also mathematically mathematically here we may have problems for a singularity for the inverse of the flow map that is ill-defined or even worse in some natural circumstances we have instead of a cusp we have an angle and the angle is to some extent is the same problem where is the tangent at the angular point is ill-defined we have a problem there too okay so I will confine myself it can be perfectly well defined yeah I know I know it depends on the context it depends on the context exactly what I'm saying is that what you will see we take this as our working set of flow maps and by doing this we cut out many many many problems that we don't know how to handle how to handle let me let me let me go back to the point suppose we have two field lines the intersect exactly what is the physics there what is the mathematics there can we talk about the tangent at a point there this is a normal problem tackled by many context of physics is called reconnection the state of a field line that goes through the other are the instant of the intersection there is difficulty to tackle this problem because we don't have the mathematics so my emphasis here is that we are using old-style mathematics but we need also to think at the same time of something new in mathematics to have new techniques to handle these problems problems of singularities that are associated with changes of topology ok before I mentioned in the first diagram I told you there is no change of topologies continuous deformation at all times haha what is the singular it depends on the context it depends on the context yes depends on what you're talking about I mean there are many ways to to talk about singularity for example when this breaks down when the inverse breaks down when there is no when there is no no inverse to the flow map then you have a singularity because you cannot identify suppose you have a system like this to field lines that are certain suppose are you oriented these field lines and suppose that now they touch this is in space can you go back to here or can you go back to this situation which one this is maybe previous time this is after time or the other way around but there is no way to know it from this because you have no inverse yeah is one of the key points of singularity yeah losing the inverse and so he said if you morphism is is a continuous deformation that is differentiable at all times so differ morphism but differ morphism may not have the inverse map all right so this is just a little preamble okay let's start with the first tool I told you we are going to use some tools he has no time I everybody is at home with the concept of course of velocity of acceleration acceleration we know how to define acceleration in space and time in classical physics I will use this notation D for the total derivative of the quantity suppose that I consider ax ax as my space variable and I can say that this da DT then DX DT then DX DT will be if X will be a function of say a arc length and time suppose I go to velocity okay I take acceleration is the same thing suppose I take the velocity U and u is function of X and T then I have acceleration a defined as d u DT and this D u DT is given by two contributions as you know we have a contribution in terms of time and the contribution in terms of space all right are we are we happy about this we have u dot grad u this is called the temporal derivative and this is convective transport derivatives okay this is standard yeah you use this notation taps yeah okay okay all right I will use I will use this one I'll try to use that one all right this is yeah this is the total derivative D DT total derivative with respect to time okay so this is important right because there is a nonlinear part and the linear part this is the transport of U and the in space and this is the transport so to speak of you in time one other basic notion is the notion of field line so I will stick to u u is the velocity field velocity field this as I said is the acceleration the other the other fundamental concept is the concept of a field line so let me use velocity I will call streamlined I will call streamline a line in space in r3 where at each point you is defined to be identified with the unit tangent okay so if we prescribe the unit tangent at each point at each point of this line say this line see we have the unit tangent T t hat because his unit and t hat is made of three components t1 t2 t3 then you will have three components little u v and w then the definition of a streamline a velocity line is simply given by u over if you like DX over u equal dy over V and this is equal to DZ over W if if the space X X is given by X Y Z this is the definition of a streamline okay now notice that we may divide everything we may divide everything by D s D s where s is a coordinate is a coordinate here and we get the DX D s as a t1 D Y D s as a teacher and these add es has a t3 so we identify as I said the velocity U with the unit tangent T okay so this is a vector field line I use it the word streamline I use it for the velocity I can use any vector field magnetic field so how do i define a magnetic field the same thing okay I have a B the magnetic field that is identified with the unit tangent on a curve C in space I'm cutting short a little bit of a little bit of introductory notes on this so field lines will be very important for us we will talk about the collection of field lines and now I'd like to introduce you a kinematic results due to Reynolds so this is a kinematic transport theorem this is useful because we want to apply these derivatives the total derivative to quantities for example mass that are defined through an integral ok mass is just the integral of the density over the volume and we want to use this derivative on integrals on integral quantities so we want to know how this quantity works ok so we need these theorem and the theorem I want to to give a little little proof of this because it's so fundamental so let me is based on is the development of lightness rule so it's not so extraordinarily unrelated with the typical integration so the theorem says the following let G a certain quantity function of space and time we are in r3 of course the some fluid property unspecified for the moment and we take it per unit volume then the standard derivative D DT of this quantity G of X and T in d3x sorry I'm you follow me if I keep using this otherwise I tend to I tend to confuse myself continuously well for instance is not in so necessary at the moment I just want to show you that the what we are going to do is that these derivative enters here and we have a partial derivative d g dt over the volume d 3x plus the surface t of this volume g times of course this is always the same G times say W W is like the velocity at the moment I'm not thinking exactly as the fluid velocity U that's why I use a different letter it's just a theorem of kinematics of mathematics it's not of physics I will apply it to physics in a moment some velocity V unit vector new normal to the surface d2x okay so this I will give you a little proof of this because we need that this and after this proof we can move quickly on applications so the proof of this is just rather elementary and you have to consider the volume and this volume is function of time it changes and as the time changes T then there is a new surface that is formed at time T plus DT so the proof of this is rather simple you have the integral of G V on the 3x and this is at time T plus DT okay because we want to construct the differential DT working on this system so we have the integral on the volume that has now changed at T plus DT of this G that is now X and the T plus DT in d3x and this is the integral of V T plus DT of what well we expand Taylor expansion as normal in books every everybody does this we Taylor expand this is the first very first time that I have to stop slow down and tell you remember we Taylor expand we assume that we can tailor expand always remember that problem of singularities etc etc you see how how creep in in the fundamentals of what we are doing we just started a few minutes and we already notice that we expand we will have a tool we use this tool everything is fine books and books and books but if here we have a problem because there is a singularity somewhere immediately we have a problem of the foundations in the tools we are going to use for the rest of the course okay so we Taylor expand that and this is a G of X of T plus DG d t xt plus well we can oh DT sorry T plus an order of magnitude of this DT squared okay something like this and this is D 3x now we need we need to evaluate this integral and so the V at time T plus DT is so this I will call it just the volume just to focus on this part and this is just the integral on DV of V of T plus the contribution the variation of V T in DV and this is just the integral on the V of V of T plus now this variation the variation of the volume is due to the variation of the surface and so we have to take into account the surface contribution so we have s of T and this is just some velocity V not necessarily the fluid velocity because the volume may deform due to some other reason so V which act which acts on the unit vector that I assume is pointing outwardly so let's say new okay and this is d2 yes okay so we put all this together now so we have integral of G or the volume V X and T indeed 3x this becomes a sorry this is at T plus DT this is the intro of V T of again the Taylor expansion we used before let me write it as G Plus V G DT in DT plus the order DT squared and these was on the 3x not yet it will yeah yeah okay and then we have what we have this is on V and then we have to move on on s so we have plus integral s of t li g + VG DT DT plus V order DT squared and this is D 3x and now we can conclude integral over V of T G + plus the higher term contribution VT d g dt in the 3x + surface s T G x double-dot new I will emphasize I will put it in this form over there and this is the s just to keep things differently and this is DT plus an order DT squared you know what I'm going to do I'm just taking the limit so I'm taking the limit I have T I have V at T DT and then I'm doing - the V at T as a delta T goes to zero so we have the following the DT of integral of G exactly the statement v 3x over V of T this is the limit as delta T goes to 0 of 1 over delta T of that difference and that difference is that the integral on V of Gd 3x - sir this calculated at T plus delta T - said the integral on V of G the 3x okay and this is calculated at T so we proved so we proved that this statement ok we prove these things this one yes this comes from here this part here comes from here this part here is just is just I'm grouping up G sorry I'm grouping up the gdt here and in G in D s if I'm correct let me see if it I might have escaped sorry I lost on the notes what I've done okay here and this in DT plus the rest yes yeah yes yes this is I omitted here all the expansion yeah I just show that that part is due to the movement of the surface because the volume changes and then I apply this argument to the G expansion or meeting I all the terms because this is a D to the S is a d2x in DT so as I order and I order oh sure sorry yes ah no no no no this is just the product of G times the scalar product of V new w is some is arbitrary at the moment it will be velocity field at the moment is just a field that makes makes the volume changes so imagine that the volume is subject to motion that is not necessarily coinciding with the fluid motion you may have a fluid that behaves that suppose is charged then particles go in one direction and the fluid goes in the other the fluid as a velocity U and the particles may charge may react to some electric field or magnetic field so for the moment for the moment is just a vector field then for simplicity we identify this vector field with you I can do it instantly in a moment and then you give a sense of this change of volume due to the motion of the fluid a fluid that is not electrically conducting I'm not considering magnetic fields I'm not considering electric field I'm just considering neutrally charged if you like neutral fluid okay let me let me remind you well if you take this theorem and you consider I want to leave it there and they want to just give you the idea that you may now consider V as a material volume okay in material volume meaning in fluid mechanics a volume made by the same particles all the time imagine that you color you color with ink this particle or this particle are like coffee powder then they remain the same in the volume hmm the volume is a material volume so that this particle that has certain property for example are colored by blue that remain always the same with the same blue this concept is very important and useful because we will talk about field lines that remain with the property of being streamline or being vortex line of being magnetic lines forever made as material lines but the same properties all the time so if if we V of T denotes material volume material volume then of course we can extend this and V coincide with the fluid velocity U as I give it before then this immediately becomes the transport theorem for fluid classical fluid mechanics where V is a material volume so we still have the same now the DDT becomes the DDT in this case becomes DDT of G of X T in d3x volume V T and will almost the same with this exception of understanding that now U is the velocity field and and the V is is a material volume okay s T and this is G times for the moment is as color is still s color but I will show you we are going fast now G of U dot nu D 2 X D 2 X is d s over there so from here we add to this so the kinematic transport theorem for fluid mechanics becomes this one and this one is a very fundamental tool because you see how it works you take the Lagrangian derivative of the quantity under integration and you perform you allow to perform partial derivatives inside remember VT is changing but is made by the same particles and these particles are moving according to the velocity U ok so this tool is fundamental in doing many derivations now I will be quick because this is the hardest point when we are here now we can use this tool and you will recognize immediately this tool in many derivations in many derivations I will use this tool if g is a scalar quantity as it is now and the represent density we know that mass is conserved and I will apply DDT of this quantity of mass to show that there is no change of mass in fluid and we get the conservation of mass immediately that is one from the mental equation we will we will get to continuty equation but before doing this I want to show you I want to mention not to show you but to mention that if I'm not going to do it is trivial is very simple imagine that you have a fixed unit vector in space fixed okay you choose a vector in this direction of unit length and you multiply this vector through this equation all through so you get inside because it doesn't depend on time is fixed and you get g times this unit vector this g times the unit vector becomes am now moving becomes this he's a G with a unit vector and the unit vector goes inside everything so you immediately have a theorem after the theorem we had for a scalar quantity we had also we have a theorem for a vector quantity actually even more if this G we put per for fun two hours two hours is just not the standard mathematics but I'm just playing with symbols just to remind you that not only we have the theorem first color we have a theorem for vectors we have the theorem for tensors for tensile quantities is the same thing why because we just go through multiplication of something that is fixed in space and time so is a way to prove that this theorem holds for any quantity in any dimension good this is a starting point we have a tool to use for a quantity that is either a scalar a vector or a tensor now surely you noted here we have a surface integration surface integration come on one of the first thing you have to remember in standard background education of math for physics is to use what the Gauss discovered we have an integration of a surface so we have integrate we can convert this into the divergence right it's called the divergence theorem because the new on the surface gives you a direction along which you act so you have a flux a flux through the surface a flux through the surface and you use divergence theorem so you can convert this surface integral into the divergence equivalent of it all right so we do that so already you saw quickly once we got this result how quickly we can now get this total derivative working on integral quantities scalar vector tensors and now we do one further step we transform this by using divergence theorem and then we are almost done because we have a tool that is working throughout fluid mechanics repeatedly remember that we had to go through this Taylor expansion at the point okay so these two features one is that there is a little point that is so to speak not so worrying but is worrying if you have a different frame of mind and the other point is that you have a tool that is very powerful and you get all the theorems in basically all the fundamental theorems in fluid mechanics let me show you how to convert that into the divergence form yes yes yes yes but this one so this is a is is a material volume and so you can convert the standard derivative the ordinary derivative with this one because you considered the Lagrangian you follow Lagrangian Li I did I skipped of course difference between Alerian and Lagrangian coordinates pardon me green function we get to the green function yeah because you see the G no no it's not quite there we get to the green function we will I am going in that direction yes tomorrow not today tomorrow will be green function time and singularity time a little bit all right so [Music] ah okay so this is important is a fundamental aspect okay I have to invent a way to explain this one way to explain this or to remember this is the following suppose I don't know suppose you have a river and you have a boat yeah and there is a there is a bridge over this okay and there is a bridge and this bridge there is a person here looking at the boat and the boat is another person there and this person we call it Euler Euler and this person we call it Lagaan I hope it works I don't know she's I'm improvising sometimes it doesn't work oiler is you know you have WeChat right so is we chatting he's sending information to a friend and maybe he said laganja is a beautiful lady I don't know is a beautiful man who knows what an Euler wants to say I'm seeing like Roger going away so is sending information about the ground coordinates coordinates so the coordinates this position is the oil Arian position and as time suppose the boat goes in this direction as x goes this is signaling the position of lagron okay but lagoons has a different view L'Orange you know moves with the fluid imagine that the ground is not really on a both these on a molecule of the fluid he's floating so the information the Lagrangian with E we chopped to somebody else is ascending is the reference the distance from here so Euler gives information in terms of as time changes and coordinate changes but lagoons gives the information in terms of the initial position the initial position it's zero I was here and now I'm moving away from here or from some reference point from where oiler is so all the information is at zero and T at zero is the initial point and from that point initial time T zero the initial position and from that initial position he sends the information as so you have to know you have to know how the motion is with respect to some initial position on a molecule that is very bad for computational reasons for computational reasons it's very bad that's why Lagrangian mechanics developed up to a point and then was dropped especially in this last century because you want some reference fixed and the motion taking place in your numerical box but if you want to follow if you want to follow the powder of coffee you are on the coffee powder and you see all the world around you moving that is very very hard that is very hard to report back to the information it's not very hard in principle you are there you moving we are on this planet and we see all the universe around us that somehow is moving so to speak but if you want to report back with respect to some reference frame fixed reference frame is hard because you have to know this and you have to construct the map you have to know the map at each time it's much easier this one because you measure the position at each time but here you have to have inverse math all the times so this is the Lagrangian viewpoint this is the illyrian viewpoint and then is reflected when you integrate the equation you may have a velocity you want to integrate velocity in order to have position when you integrate velocity then you have to take care of these two approaches one is just integrating with respect to time you get immediately the position and the other one is not enough because you have to know that the function that depends on X 0 X depends on X 0 so this is more complicated so this is the view we often in many aspects I will present you we use a Lagrangian viewpoint I told you it is not fashionable it's not fashion mode because computationally is very hard why we use the Lagrangian viewpoint is simple because I want to stay close to the structure say vortex I want to be on the vortex I don't care if the vortex hits the river hits a building meteorologists care about that meteorologists want to know if the vortex hits a building so they have to have a reference frame fixed but I want to stay on the vortex I want to see how the vortex evolves change its shape ok so I will be moving Lagrangian Li and all I'm going to say will be on these structures suppose we take magnetic not I want to see how the not change shape is staying on the north and not outside the node not with a fixed reference so this was a very popular at a certain point with the number of people some for example Khushi was keen on the ground coordinates and Lagrange was not keen on Lagrangian coordinates both are due to Euler so it depends when you use say the Cauchy approach you find quantities that you may see invariant properties because you move you know I move on this planet hurt and I recognize this planet Earth as my home I know the structure of this planet Earth I know everything of this planet Earth I can see if it is changing or not if it is breaking up or not but if I'm away and I'm studying a planet I have to report back this information all the time so invariant properties are easier to study not knowledge only because I can see staying on the structure if it changed some topology for instance is not Euler is definitely you have to have this flow map the inverse of this flow map all the times that makes problems in in reconstructing the position the position because what you're thinking now is oil urgently you think as an oiler you want to know this boat where is going but I don't care where I'm doing I'm caring if the boat breaks up or not I want to stay on the boat and I can reconstruct the motion close to the boat with respect to the river I was with respect to this boat to the shore to the bridge to some reference but I sometimes I don't care you see I don't I do I may not care I may not care if you are away in the sea you may want to take care of your boat if the boat is working fine know at the moment is still kinematics you're right yeah we get to dynamics in a moment ah I I don't I don't I'm not so sure about that because it depends on the problems certainly if you concern I rephrase it my way if if I'm considering invariant properties of a object say the boat then I am Lagrangian because measuring changes here measuring changes here can tell me if these changes are occurring or not but if somebody wants to know here what changes are occurring here you see what I have to do I'm moving all the changes I have I have to tell this is changing to this person in terms of the distance from the bridge I have to add some more information because I'm caring about the bridge now so I need this flow map inverse I want to say the boat you know is taking water this is a change and with respect to the distance but if I care about the water if the boat is taking water or not I don't care about the distance at all I just care about if it takes water or not if it doesn't take water I'm I'm happy and suppose I don't want to go anywhere I just want to float suppose then then it's okay the boat is fine I'm not taking water I don't I don't know I'm not quite sure where I'm going but it's fine but if it's taking water I need help I need to communicate my position and I need to communicate how much water I'm taking in and the people that listen they have to compute two things the water I'm taking and if I'm going away where I'm going to be with the amount of water I'm going to take because I need two informations so of course if you are being somebody it's easier to have oil area viewpoint the oral Aryan viewpoint means you are mapping from from a fixed reference frame whatever happens from here I just need to know if there is water or not so suppose this is the example suppose the water is is empty and is floating well this is the invariant property I want if I'm taking water this is no longer conserved quantity and so this is not invariant with respect to motion of course is not invariant also in order terms but in Euler terms to have the same amount of information I need just more information because I have to report back all the time you see the difference so computationally it's much more advantageous to stay here and to see what happens because computationally computationally if you are Lagrange you have to calculate you have to have a reference system here that calculates your invariant properties and then you have to report back but to report back you have to know the evolution of the fluid there is a similar problem I my first one of the first problems I considered when I was your age is exactly of this type of problem at the time there was a beginning of string theory and my first paper was actually in this context it was the emerging string theory of the time in in the 80s late eighties and and the the street was a vortex and adults and these vortices could be studied either with a reference frame on the vortex all from outside and so I started to consider the Lagrangian viewpoint to stay on this string on this string you see the motion and perturbation of but then you have to report back sometimes Oh both yeah you can you can choose both dynamically is not the same is a is a is a is a composition of functions oh of course oh sure sure sure sure is the same is the same because you can transform one system into the other so in principle is not not not a problem the Lagrangian should have information about the initial position all the times is the first very first mapping I I gave you this one X 0 to X and I have to know this map and I asked for the existence of the inverse at all times that saves me that assumption save oh you have you have you have you have extra turns yes it reminds me yeah you know better than me because it's not my field but is what is called the Thomas precession no Thomas precession is when instead of the electron you are the system of reference is yeah yes this is material now is taking care of that problem yeah so this this problem will accompany us and I will have time to make more more comments because of course a vortex will induce a motion fluid motion and it depends where you are if you are on the vortex or if you are on the fluid of course the the physical phenomenon is the same but the way you perceive dynamics is different the way you perceive it yeah yeah okay so I want just to so this is a little story about Euler so remember Lagrange's on a molecule and is moving and Euler is fixed and he's looking at the world that is changing okay this is just helps you maybe to remember this joy and this is the so called Lagrangian derivative because it combines together the time variation and the space variation we will come back to this because it's important when you say that something is constant its constant with respect to this variation it doesn't mean that has a constant value it means that the sum of the variation in times and space balance out okay all right so of course of course we have the quantity from this we can say a corollary the corollary is that if V of T is a material volume then derivative with respect to time of this quantity G as I said it can be a scalar or a vector the moment is still a scalar if you like is just the integral V T of DG DT plus I apply divergence theorem to that one and I have the divergence of G you in d3x I transform the surface integral into a volume integral and this is just the basic relation yeah one last step is just you know easy on the blackboard to do it please repeat the exercise of multiplying everything by a fixed vector and then you immediately transformed these it the same equation into an equation that works for for a vector what we'll do we use this dual aspects when Jesus scalar we use mass and we apply this and when G is a velocity itself we get the conservation of linear momentum I stop here for the moment we take a break and then we resume okay you

Channel: Renzo Ricca

Views: 3,013

Rating: 5 out of 5

Keywords: Topological Fluid Dynamics, Fluid Flows, Topological Equivalence, Euler Equations, Knotted Fields, Helicity, Vortex Knots, Magnetic Braids, Magnetic Relaxation, Groundstate Energy Spectrum, Structural Complexity

Id: wfbdqjlX0gE

Length: 81min 10sec (4870 seconds)

Published: Mon Jan 07 2019