# How physics solves a math problem (and a 3D graphics problem)

## Метаданные

- **Канал:** Mathemaniac
- **YouTube:** https://www.youtube.com/watch?v=Cvs8iRqG6lg

## Содержание

### [0:00](https://www.youtube.com/watch?v=Cvs8iRqG6lg) Segment 1 (00:00 - 05:00)

if a surface is bounded by two circles one on top of the other what is the configuration that gives the least surface area this question belongs to a class of problems known as plate problem which asks for any given boundary what is the surface with that boundary that has the least surface area the solution is called a minimal surface because it has minimal surface area this might just be a fun maths problem to play around but animal surfaces do occur in nature if you dip metal wires Twisted in some shape into a soap solution and then take it out a soap film will be formed the film itself forms a minimal surface so it seems like nature can solve this problem for us but how does nature or more precisely physics solve this problem once we know that we can build some algorithms to solve plateau's problem by simulating the physics however there is a bit of a disclaimer before we start the minimal surfaces we will discuss in the remainder of this video are only local Minima in the sense that if you slightly distort the surface you end up with more area but it's not necessarily a global minimum with that let's see how physics forces the soap film to take the shape of a minimal surface let's say this is the current configuration of the soap film and we want to figure out how a particular Point evolves to do that we consider the forces acting on an infinitesimally small piece of the surface close to the point generally surface tension will try to pull this piece of surface outwards this comes from attractive forces pulling the piece towards other parts of the surface it turns out that the force acting on a very small piece of the boundary say with length DL has a magnitude to gamma multiplied by the length where gamma is the surface tension a constant that depends on the material in this case soap while two is because we are actually dealing with two layers or two surfaces here as an illustration of this a soap film like this has one side facing the outside air but there is also another side facing the cavity in the middle so for our purposes there are two surfaces or two interfaces so that's the magnitude done what about the direction well the force should be tangent to the surface so it should lie on the tangent plane this means the force is perpendicular to n the normal Vector at this point it will make sense shortly why we want to make this observation on the other hand the force should be perpendicular to the boundary let's name this small Vector along the boundary t with length DL so T is a unit Vector that just encodes the direction so f is perpendicular to both t and n and because t and n are also perpendicular to each other the direction is given by the cross product of t and n combined with the magnitude we found previously we see that the Force Vector f is 2 gamma * T CR n DL but this is just for a very small part of the boundary if we want to consider the effects along the full boundary then we need to integrate essentially summing up all these contributions from Tiny bits of the boundary the next step is to compute this integral which we will do in a minute after Computing the integral we will divide this total four by the area of this small piece of surface so that we get pressure the reason we consider pressure rather than force is that if you shrink the surface towards the Red Dot the net force becomes closer and closer to zero due to symmetry while net force from surface tension goes to zero pressure that is the force per area turns out to tend to a finite generically nonzero Vector this allows is to Define pressure acting at a point on the surface so from now on we will consider this as an infinitesimally small piece of the surface to compute the limit so we will focus on the net force first and divide by area later we can separate the vector n into a sum of two vectors n not and Delta n where n not is the normal Vector in the middle of that piece of the surface and the

### [5:00](https://www.youtube.com/watch?v=Cvs8iRqG6lg&t=300s) Segment 2 (05:00 - 10:00)

Delta n would be the difference between the normal Vector at the boundary and that in the middle the reason we want to separate is that this part sees n not as a constant Vector in a sense that it doesn't depend on where you are at the boundary so you can pull this out of the integrant and we are left with an integral of TDL cross n not but what is this integral well it is just adding up these small vectors along the boundary because they form a closed loop adding them up just yields a zero Vector so this part of the integral vanishes but the total force on this piece of surface isn't just T cross n KN sure this part is zero as we just discussed but we still have the other integral to compute now remember we are considering this an infinitesimally small piece of the surface so we will do some approximation if you are worried about the inaccuracy of the result remember that we want to find the limit of this integral per area on the surface so even if we have an approximation as long as the eror per area goes to zero we are okay with only using an approximation of the integral so how do we approximate this integral the trick is to realize that both T and Delta n can be close ly approximated to lie on the tangent plane at the Red Dot for the vector T it is a bit more believable after all this purple piece is so small that even if we project onto the tangent Lane T hasn't changed much and in fact the same in this illustration What about Delta n the difference between n at the boundary and the N knot in the middle the normal Vector is confined to have its tip on the unit sphere because it has length one the normal vectors n would in general deviate from n knot a little bit and the change Delta n can be approximated to lie on the tangent plane of the sphere however this is perpendicular to n knot which is in turn perpendicular to the tangent plane of the surface at that point so we can EAS easily translate that Vector back onto the tangent plane of the surface with all that both the vectors T and Delta n can be well approximated to lie on the tangent plane so let's focus on the tangent plane the small piece of the surface can now be projected entirely flat on this tangent plane and the vector T * DL will be a small Vector going around boundary and Delta n is just some Vector on the plane the magnitude of the cross product would be the product of the lengths of these vectors time sin Theta being the angle between the vectors equivalently if we Define a vector B perpendicular to this boundary then because the angle between Delta n and B is now Pi / 2 minus Theta we can rewrite the magnitude as the product of the length and cosine of this included angle in other words this magnitude can be Rewritten as the dot product the dotproduct of B and Delta n * DL so that's the magnitude of the vector what about the direction of the cross product because both T and Delta n are on a plane the direction must be perpendicular to this plane in this case if you follow the right hand rule it points into the screen if we go back to the whole picture and use the right hand rule again more precisely the direction is opposite to n not most importantly the direction is constant so during the approximation we can pull it out of the integral and we are left with the integral B do Delta n the dot product itself can be viewed as the flux from Delta N Out of the small piece of the boundary the integral wants us to add up all the flux contributions along the boundary so the integral is the total flux of Delta n across the boundary ultimately we want to compute the limit of this integral per area when the area tends to zero which is almost exactly the definition of Divergence or you can use the Divergence Theorem to see this Divergence is double what's called mean curvature of the surface at

### [10:00](https://www.youtube.com/watch?v=Cvs8iRqG6lg&t=600s) Segment 3 (10:00 - 15:00)

the point and is denoted as 2 h mean curvature was mentioned in the previous video for those who have watched the previous video see if you can figure out why this Divergence is double the mean curvature perhaps also using this video on the intuition of Trace anyway the whole story is to First compute the net force on a small piece of the surface which can then be approximated with an integral that can be interpreted as the total flux of Delta n across the boundary when we finally divide by area to obtain the pressure at the point we know that the flux per area tends to 2 AG 2 times the mean curvature so the pressure at a point is perpendicular to the surface with Direction n not and the magnitude is proportional to the mean curvature the soap film evolves under this pressure this calculation only considers the surface tension in general you might have to consider something like atmospheric pressure however in the case of soap films both surfaces face the air so the air pressure from both sides cancel and this is really all the pressure there is but why does this mean that the surfaces formed will have minimal surface area here is some intuition the mean curvature is proportional to the Divergence of Delta n if the normal vectors at the neighboring points Point generally away from n like in this illustration then Delta N Point generally away from the origin so the Divergence is positive the negative sign means the pressure acts in the opposite direction of n KN at the Red Dot but in general should be n the normal noral Vector at that point when pressure is applied so the pressure acting on this piece of the surface Point generally downward more precisely while the neighboring normal vectors point away from n notot because of this negative sign pressure from different places Point into each other and push this region inwards so the area decreases on the other hand let's say the surface looks like this instead in which case the neighboring normal vectors Point into each other and towards and not so the Divergence of Delta n which generally points back towards the origin is negative so the mean curvature being half of that is also negative but the negative sign suggest that the pressure points in the same direction as n so the pressure vectors also Point into each other again decreasing area whichever case it is the pressure vectors Point into each other to decrease the area this means that if the soap film reaches the minimal surface configuration the pressure is zero otherwise the pressure keeps decreasing the area this suggests that the mean curvature on minimal surfaces need to vanish anyway this is how nature of physics solves plateau's problem how do we actually simulate this process to solve plateau's problem ourselves naively you might think that pressure is force per area and from Newton's Laws FAL ma pressure should be proportional to the acceleration of the Red Dot if this is true then simulating this would involve setting the acceleration the second derivative of the position Vector X to be negative beta hn for some positive constant beta such an evolution is named hyperbolic mean curvature flow and seems to be how soap films actually behave however there are physical systems where it is the velocity that is proportional to mean curvature which leads to the first derivative being negative beta hn instead and the resulting evolution is just called the mean curvature flow this turns out to be what's happening in an annealing pure metal like aluminium which I honestly don't know too much about regardless the general procedure to solving plateau's problem would be to start with some initial surface where we denote the position vectors to be X and then you have two choices either you evolve the velocity to be proportional to mean curvature in which case you choose mean curvature flow or the acceleration is proportional

### [15:00](https://www.youtube.com/watch?v=Cvs8iRqG6lg&t=900s) Segment 4 (15:00 - 17:00)

to mean curvature that is you choose the hyperbolic mean curvature flow at the end we hopefully approach the steady state solution which is a minimal surface having mean curvature zero however choosing that initial surface requires some Ingenuity or at least some luck because if you don't choose your initial surface carefully like in this case where the neck is a bit too thin there can be singularities being formed in this case the soap film pinches off to form two surfaces the conditions for existence of such singularities are a hot topic of research on the mathematical side of things even on the physical side of things there is still plenty of research into exactly how this instability arises perhaps a bit more surprisingly this procedure is also useful in computer Graphics if we choose the path of mean curvature flow but not go all the way through the surface will just become smoother there is a great stack exchange answer on how to implement it on Mathematica Linked In the description the reason for this smoothing behavior is a bit complicated but intuitively if the surface has a huge bump in it then the neighboring normal vectors deviate away from n a lot so the mean curvature has a very large magnitude the pressure which is proportional to H then points down also at a great magnitude thereby smoothing this bump quickly so even though we start off with a fairly simple sounding problem the plateau problem where we want to find the configuration that has the least surface area there are still lots to explore we even somehow also solve problem in 3D Graphics given the richness of this subject it might be surprising to know that actually there is some kind of general formula for minimal Services something that we will explore in the next video as always thanks to these patrons and don't forget to like subscribe and comment see you next time

---
*Источник: https://ekstraktznaniy.ru/video/40211*