hello everyone and welcome back now in the previous video we were introduced to the five-step method that you should use when you approach problems in mathematical modeling and in particular we saw how to apply this approach to a very simple problem of falling goggles now what I would like to do is zoom out just a little bit and talk a little bit more about the components of mathematical modeling and in particular over the course of this lecture series we are going to talk about two main classes of models now the first class is called deterministic models now deterministic models they mean that there is no Randomness there are no random variables involved and what it means to be a deterministic model is that the future is entirely determined by how things start right now okay so think about a pendulum arm that's swinging well what I can do is I can pull up the pendulum arm and I can leave that thing to swing and if you came and did that in an hour or a year from now and you pulled it up to exactly the right spot that I pulled it up to and you let it go you will see exactly the same swinging motion that's because the system is completely deterministic there is no element of randomness of all involved in that so what this means is that typically when we are solving deterministic models we are going to use a lot of calculus we're going to use some linear algebra we're going to use differential equations so in particular ordinary differential equations these are the typical ways that we model deterministic systems now if deterministic systems don't involve any random variables then clearly there is a class of models that do and those are called stochastic models now stochastic models means that there is an element of Randomness involved right so this might be something like playing a board game if you have to roll the dice right so just because it's my turn and I started on the go Square in Monopoly I am not going to do exactly the same thing that you did when you started on the go square and that's because the randomness comes from whatever comes up in the dice if we have a hundred people lined up it's possible that there are 36 different outcomes based on the values that come up with those dice now of course if deterministic models use a lot of calculus and linear algebra and ordinary differential equations stochastic models they use a lot of probability and statistics right so that Randomness causes us to have to be a little bit imprecise in how we talk about things we say this is what should happen this is what is expected to happen but of course the randomness means that some very outlandish things are possible right so for example if you are on the Monopoly uh if you are on go uh starting a game of Monopoly then rolling two dice the highest probability thing that you can do is roll a seven now I believe this would put you on a community chest Square so I say you know the highest probability event that's going to come up would be landing on community chest but sixes and eights are also high probability rolls so you do have a high probability of rolling on those so you might be you might be tempted to say something like you are expected to roll uh between 5 and 10. but of course you have the low probability of event of rolling Snake Eyes you could roll two ones that is entirely possible within a stochastic framework okay so as we progress through the class we're going to start with optimization and uh then move into dynamical systems that's primarily going to be focused around deterministic now the nice thing about deterministic is they are relatively simple to solve they have an answer right the answer is three the answer is pi right there is a specific answer that people are looking for now towards the end of the class we're going to start we're going to talk about stochastic models things like Markov chains which again have very specific answers but they have to be asked in a different way you have to say you know what is the probability of finding someone at this Square after 10 turns right so you have to be careful about how you phrase these things uh these questions in the five-step method when it comes to stochastic and deterministic models now those are just the kinds of models you have the question is uh you know what can you build into a model well building a model you might want to use something like conservation laws so if you have taken a physics class before you are definitely familiar with conservation laws right physics is just brimming with conservation laws it's really how we understand a lot of the laws of reality so conservation of momentum for example um these are things that we sort of build into models because we know they have to be there and they don't just have to come from physics for example imagine we have 10 animals in a pen and we want to sort of try and understand their movements and their behavior in the pen well if we're looking at this on a short enough time scale there's not going to be births or deaths so the number of animals is going to be conserved there will always be 10 animals so that is a type of conservation that can be built into these models now of course along with uh conservation laws another very very powerful thing that we can do is we can put in constituent relations now what is static constituent relations I mean that's a very very fancy way of saying known equations right so things like Newton's Second Law force is equal to mass times acceleration That's a classic right physics problems use this to derive even very very complicated models things like the navier Stokes equations just coming from forces you go and mass times acceleration another classic things like hooke's law right so this is the this is related to Springs and the force required to stretch a spring out but you can have something coming from maybe chemistry right so this is the ideal gas law and there have been many many attempts that people have made to try and come up with these types of laws when trying to understand um uh sociological relations right so types of responses language death models right there are all kinds of these constituent relations that are hanging around and this goes to the B of the ABCs of math modeling borrow borrow borrow right we want to be able to draw upon what others have already done for us okay so once you actually make a model what is it that you want to do well of course you want to answer the question right this is the class so you want to sit back you know you want the full marks you want to say I want to get perfect on this problem I want the right answer okay that's great right that is a wonderful approach to succeeding as a student but how do you succeed as a mathematical modeler well you still have a question and you might have a specific answer in mind right so you know where are all of the animals in my pen after two hours after six hours after six days right these are these are quantifiable questions but what you should really be doing once you build a mathematical model conducting numerous studies to see if things sort of make sense they should be qualitative and quantitative studies that say okay even if it gives me the answer that I sort of expected think about my air resistance problem from the previous uh lecture you know what happens if I tweak things a little bit right what happens if my ski lift is 20 meters in the air instead of 10 meters what if it's only 5 meters right the thing still makes sense right and there's that kind of robustness that kind of tweaking still makes sense in my model now the other thing that you should aim for is to have your parameter values so these are your sort of constants that you are taking in your model might be your C and your Alpha from the previous example it might even just be the acceleration due to gravity you want to be able to find these things either in the literature or through experiments you want to know what those values are right because think about what I did in the previous lecture I said I don't know what the air resistance is actually for my experiment so I'm going to do things really generally and so therefore I can't really answer the problem anymore I can give you an answer in terms of C but I don't know what c is well that doesn't answer the question for me so of course I need to be able to pull back and ask myself you know what is C did somebody measure it probably I or is there a way to measure it then get out there and get it okay now the last thing and the most important thing units okay unit analysis is one of the uh most cumbersome and most frustrating thing that you can do in a physics class and we in mathematics absolutely hate units we try our very best to get rid of units through a process that is referred to as non-dimensionalization so before I get to the non-dimensionalization process I want to talk about how units are a good way to know if you put together the model correctly right so let me give you a little example so let's look at this example I give you an ordinary differential equation I said I've got this DX by DT so the derivative of x with respect to time is equal to a some constant times the position X okay well here's what we know okay so since I am clearly looking at uh knowns and unknowns here X is going to be measured in meters for example so that's good now the derivative is a rate and so it is going to be meters per unit time in my case let's imagine it's seconds so time is measured in seconds so this is uh meters per second because time is measured in seconds what are the units of a what are the units of this constant a right so again if I go back to what I just said I want to figure out what these constants are because I want to be able to give definite answers but clearly you know if I give uh a in Newtons for example or kilometers is that going to be consistent right does that even make sense do I have to do some conversions to the right units well if we look at these things as a sort of fractions this is maybe M over s right meters per second which is equal to some unit times meters sort of rearrange all of these things and this means that this is measured in one over seconds so this is a per unit time relation right this is again a rate here and what this tells you is that either if you are given a and it is not measured in one over s then you have either built the model wrong or you have measured a wrong this gives you a self-consistency check when you are going through that model building phase right this tells you whether or not things actually make sense right we cannot compare apples to oranges we have to have this unit consistency this is the best way especially when it comes to physics models to determine if you put things together in the right way okay now as I mentioned once we get all of the units to make sense the next task is to get rid of them we try and get rid of as many constants as we possibly can through a process that's known as non-dimensionalization now I'm going to give you an example of this again I'm going to use differential equations primarily because that's you know my sort of favorite subject but it's also a good way to sort of see how these things work okay so let's imagine I've got a constant a times the derivative of a function plus a constant B times the original function and then C times some function f of T So F of t function in this uh X might be your position DX DT might be your velocity a b and c might be some measured constants that you put together to build this mathematical model okay so let's not worry too much about the mathematical model itself let's think of a way or let's let's discuss a way to get rid of a b and c okay I don't want them in there I don't want to have to talk about them because as we saw with the previous example with the following goggles the C really sort of muddies things up so if we can get it out of there we can get ourselves into a realm where we can just do the math that's what we want here's what I'm going to do I'm going to say let okay I'm gonna say let x equal to a constant Chi and then times say XC and I'm going to let time equal to Tau times Tau C okay so what I'm going to do here is I'm going to introduce new variables these are the TC and the XC variables and I'm going to do it in such a way that I am going to rescale time okay so maybe time is originally measured in seconds what I'm going to do is I'm going to find a new unit of time that gets rid of a bunch of these constants okay so these are sort of you think of these things as sort of natural units you're sort of scaling everything so that they everything becomes one okay so instead of talking about one second in the future maybe I talk about Pi seconds in the future because that's when I was able to measure a lot of my things okay well what does this actually do well okay my derivative here if I plug in x and t i get Chi XC divided by Tau OTC now Tau and T uh these things are going to be uh chi sorry Chi and Tau are going to be variables of time XC and TC are going to be what I scale time according to so they're going to be constants so that means that I get XC and TC coming out and I get D Chi D Tau so what I would like is to have a new variable Chi that is a function of Tau which is the natural time scale and I'm going to do it by finding the right XC and TCS okay so into the differential equation into the ode when I put this into the ode well I get a XC over t c and then times D chi d Tau that is a DX DT that's all that I've done substitution so far Plus and then I get b x c and then times Chi all right that's just replacing x with Chi XC all I'm doing is putting the constants together out front of these variables and then similarly I get PL uh sorry equals c f of and now it's going to be uh Tau TC Okay so this is it is I understand this can be a confused Computing process when you first uh learn about it but the goal here is to get rid of all of the constants right now I've got a whole mess of them but remember I'm allowed to choose what XC and TC actually are so first thing I'm going to do is I'm going to get rid of this I'm going to multiply it all the way through so I get D chi d Tau and then Plus b t c over a x c I and now this is equal to I get C T C over a XC and what I'm going to do is that I am going to call this thing now capital F of Tau okay so what I did here I said capital F of tau is is actually the old function f of Tau times TC okay so nothing too too fancy here all it is just a redefinition remember this is now a function of Tau I just want it to look like a function of Tau okay so what is it what was the goal the goal was to get rid of all of the constants so I want this equal to one and I want this equal to 1. that would be awesome that's what I want to happen so the question is how do we do this well if we are going to do this we need this isn't here that's my mistake pardon me this is divided off from right here so the first equation we get is B times T C over a is equal to 1. remember A and B are given to you so this tells me that I need to choose T C To Be A over B A and B are given to me and so that means that the natural time scale is measured in A over B that's what that tells me and it also tells me that if I choose to rescale time by A over B then I completely get rid of all of the constants on the left hand side of my differential equation I now have a derivative of a function plus the original function no multiples no twos no threes no A's no B's nothing okay then the other one is C T C over a x c is equal to one in this case now I've already got TC so this actually gives me uh C C over b x c is equal to one which tells me that XC is equal to C over B and what was the point by choosing those two values remember I got to choose how I rescale the variables I got to choose TC I got to choose XC I chose them in such a way so that my ordinary differential equation becomes the derivative of the new variable CHI with respect to sorry Tau pardon me plus the original function Chi and this is equal to a function Tau it's not so much nicer to look at there's no A's no b's no C's nothing no TC's no xc's just a derivative of a function plus the function itself is equal to another function now twos no Pi's no exponentials nothing nice and simple that is what we call non-dimensionalization in mathematics I got rid of all of the constants by moving to these natural time and length scales okay so this is all related to what's called Buckingham Pi's theorem and it's a way of telling you how you can non-dimensionalize a system in such a way that you can get rid of as many constants as possible by moving to these sort of natural scales just like what we did here okay I know that this is a hard process right this takes a lot of practice but hopefully as we progress through the class you know these are the kind of things that you'll be able to do you'll be able to distinguish what is a deterministic versus what is the stochastic model and that means that you'll be able to distinguish how you're going to actually solve that model what methods you're going to use and you know maybe as you're building these models for yourself you're going to start thinking about you know the sort of ingredients the components that you might want to use so of course you want to start with conservation laws they're some of the most important things in the entire world and they are the most important thing to start with building a model and then of course constituent relations we have all of these equations hanging around of course we want to be able to use them Now units give you the sort of self-check they tell you if things actually made sense or not just like what we saw right here you know we had to have a measured in per seconds and if it's not measured in per seconds then something is wrong it's either measured incorrectly or it's measured with the wrong units or maybe you made a mistake and we should probably assume that we made the mistake a lot of the time in building the model it's a nice self-consistency as soon as you have all of the model built get rid of as much as you can make it as simple and pretty to look at and easy to analyze as you possibly can okay when we come back in the next lecture we'll start talking about optimization the first of three units that we will progress through in this lecture Series so I'll see you in the next lecture