1 00:00:00,000 --> 00:00:01,950 The following content is provided 2 00:00:01,950 --> 00:00:06,060 by MIT OpenCourseWare under a Creative Commons license. 3 00:00:06,060 --> 00:00:08,230 Additional information about our license 4 00:00:08,230 --> 00:00:10,380 and MIT OpenCourseWare in general 5 00:00:10,380 --> 00:00:11,930 is available at ocw.mit.edu. 6 00:00:17,090 --> 00:00:19,790 PROFESSOR: Let's get started. 7 00:00:19,790 --> 00:00:25,510 Could I start first with an announcement of a talk 8 00:00:25,510 --> 00:00:27,670 this afternoon. 9 00:00:27,670 --> 00:00:32,350 I know your schedules are full, but -- 10 00:00:32,350 --> 00:00:37,020 the abstract for the talk I think will go up on the top 11 00:00:37,020 --> 00:00:38,730 of our web page. 12 00:00:42,300 --> 00:00:45,200 It's a whole range of other applications 13 00:00:45,200 --> 00:00:49,550 that I would hope to get to, by an expert. 14 00:00:49,550 --> 00:00:52,140 Professor Rannacher, his applications 15 00:00:52,140 --> 00:00:54,750 include optimal control, which is certainly 16 00:00:54,750 --> 00:00:58,610 a big area of optimization. 17 00:00:58,610 --> 00:01:04,160 Actually, MathWorks, we think of them as doing linear algebra, 18 00:01:04,160 --> 00:01:08,910 but their number one customer is the control theory world, 19 00:01:08,910 --> 00:01:14,070 and it totally connects with everything we're doing. 20 00:01:14,070 --> 00:01:20,220 He's also interested in adaptive meshing for finite element 21 00:01:20,220 --> 00:01:25,700 or other methods -- how to refine the mesh where it pays 22 00:01:25,700 --> 00:01:32,720 off in arrow problems, all sorts of problems. 23 00:01:32,720 --> 00:01:38,350 The math behind it is this same saddle point structure, 24 00:01:38,350 --> 00:01:42,270 same KK -- when he says KKT equations, 25 00:01:42,270 --> 00:01:44,390 that's our two equations. 26 00:01:44,390 --> 00:01:47,670 I've been calling them, sometimes, Kuhn-Tucker, 27 00:01:47,670 --> 00:01:49,830 but after Kuhn and Tucker, it was 28 00:01:49,830 --> 00:01:55,370 noticed that a graduate student named Karush, also a K, 29 00:01:55,370 --> 00:01:59,740 had written a Masters thesis in which these important equations 30 00:01:59,740 --> 00:02:00,740 appeared. 31 00:02:00,740 --> 00:02:03,590 So now, they're often called KKT equations. 32 00:02:03,590 --> 00:02:05,920 I think Rannacher will do that. 33 00:02:05,920 --> 00:02:08,220 Anyway, I don't know if you're free, 34 00:02:08,220 --> 00:02:17,300 but it'll be a full talk in this program of computational design 35 00:02:17,300 --> 00:02:18,360 and optimization. 36 00:02:18,360 --> 00:02:22,100 I don't know if you know MIT's new masters degree 37 00:02:22,100 --> 00:02:24,590 program in CDO. 38 00:02:24,590 --> 00:02:29,800 So it's mostly engineering, a little optimization 39 00:02:29,800 --> 00:02:32,160 that's down in Operations Research, 40 00:02:32,160 --> 00:02:35,140 and a couple of guys in math. 41 00:02:35,140 --> 00:02:38,490 So that's a talk this afternoon, which 42 00:02:38,490 --> 00:02:43,560 will be right on target for this area 43 00:02:43,560 --> 00:02:48,690 and bring up applications that are highly important. 44 00:02:48,690 --> 00:02:54,652 Well, so I thought today, my job is going 45 00:02:54,652 --> 00:02:55,860 to be pretty straightforward. 46 00:02:59,420 --> 00:03:02,300 I want to do now the continuous problem. 47 00:03:02,300 --> 00:03:05,020 So I have functions as unknowns. 48 00:03:05,020 --> 00:03:09,360 I have integrals as inner products. 49 00:03:09,360 --> 00:03:12,410 But I still have a minimization problem. 50 00:03:12,410 --> 00:03:15,100 So I have a minimization problem and I'll call, 51 00:03:15,100 --> 00:03:17,300 it's often a potential energy, so 52 00:03:17,300 --> 00:03:21,460 let me use P. Our unknown function is u, 53 00:03:21,460 --> 00:03:26,620 so that's a function u of x; u of x, y; u of x, y, z. 54 00:03:26,620 --> 00:03:30,320 And instead of inner products we have integrals, 55 00:03:30,320 --> 00:03:31,810 so there's a c of x. 56 00:03:35,010 --> 00:03:36,960 This is going to be a pure quadratic, 57 00:03:36,960 --> 00:03:39,770 so it's going to lead me to a linear equation. 58 00:03:43,540 --> 00:03:47,250 In between comes something important -- 59 00:03:47,250 --> 00:03:53,250 what people now call the weak form of the equation. 60 00:03:53,250 --> 00:03:59,580 So, this is would be the simplest 61 00:03:59,580 --> 00:04:07,042 example I could put forward of the calculus of variations. 62 00:04:07,042 --> 00:04:08,500 So that's what we're talking about. 63 00:04:08,500 --> 00:04:10,240 Calculus of variations. 64 00:04:14,320 --> 00:04:17,360 What's the derivative of P with respect to u? 65 00:04:17,360 --> 00:04:19,120 Somehow, that's what we have to find, 66 00:04:19,120 --> 00:04:22,810 and we're going to set it to zero to minimize. 67 00:04:22,810 --> 00:04:24,800 Then why do we know it's a minimum? 68 00:04:24,800 --> 00:04:26,860 Well, that's always the second -- 69 00:04:26,860 --> 00:04:32,060 the quadratic terms here are going to be positive. 70 00:04:32,060 --> 00:04:34,980 So we have a positive definite problem. 71 00:04:38,590 --> 00:04:43,200 Positive definite means things go this way, convex, 72 00:04:43,200 --> 00:04:47,560 and you locate the bottom, the minimum 73 00:04:47,560 --> 00:04:50,130 is where the derivative is zero. 74 00:04:50,130 --> 00:04:51,410 But what's the derivative? 75 00:04:51,410 --> 00:04:53,680 That's the question. 76 00:04:53,680 --> 00:04:56,800 It's not even called derivative in this subject, 77 00:04:56,800 --> 00:04:59,110 it's called the first variation. 78 00:04:59,110 --> 00:05:01,630 Instead of saying first derivative, 79 00:05:01,630 --> 00:05:03,740 I'll say first variation. 80 00:05:03,740 --> 00:05:09,700 And instead of writing dP/du, I'll write -- 81 00:05:09,700 --> 00:05:10,980 where can I write it? 82 00:05:10,980 --> 00:05:14,455 So this is key word, so this is going 83 00:05:14,455 --> 00:05:18,660 to be the first variation, and I'm 84 00:05:18,660 --> 00:05:24,080 going to write it with a different d, a Greek delta, 85 00:05:24,080 --> 00:05:25,490 dP/du. 86 00:05:25,490 --> 00:05:27,510 It's just sort of a reminder that we're 87 00:05:27,510 --> 00:05:32,422 dealing with functions and integrals of functions 88 00:05:32,422 --> 00:05:33,300 and so on. 89 00:05:33,300 --> 00:05:36,210 So it's just change the notation in the name 90 00:05:36,210 --> 00:05:42,150 a little as a trigger to the memory. 91 00:05:42,150 --> 00:05:47,840 But this board really has a lot -- 92 00:05:47,840 --> 00:05:50,230 not all the details are here, of course. 93 00:05:50,230 --> 00:05:56,030 This is a summary of what the calculus of variations does. 94 00:05:56,030 --> 00:06:00,660 So it takes a minimization problem. 95 00:06:00,660 --> 00:06:03,840 So I'm looking for a function u of x. 96 00:06:03,840 --> 00:06:08,720 And always since we're in continuous problems, 97 00:06:08,720 --> 00:06:10,490 we have boundary conditions. 98 00:06:10,490 --> 00:06:17,390 So as always, those could be -- let me imagine that those are 99 00:06:17,390 --> 00:06:19,800 the boundary conditions. 100 00:06:19,800 --> 00:06:23,950 So I would call those essential conditions. 101 00:06:23,950 --> 00:06:27,600 Every function that's allowed into the minimum 102 00:06:27,600 --> 00:06:30,050 has to satisfy the boundary condition. 103 00:06:30,050 --> 00:06:35,039 So it's a minimum over all u with the boundary conditions, 104 00:06:35,039 --> 00:06:36,330 with those boundary conditions. 105 00:06:40,719 --> 00:06:42,510 There are two kinds of boundary conditions, 106 00:06:42,510 --> 00:06:45,940 and maybe I'll postpone thinking about boundary conditions 107 00:06:45,940 --> 00:06:48,740 till I get the equations, the differential 108 00:06:48,740 --> 00:06:51,970 equation inside the interval. 109 00:06:51,970 --> 00:06:54,360 So all these intervals go from zero to one. 110 00:06:54,360 --> 00:06:56,010 I won't put that. 111 00:06:56,010 --> 00:06:56,710 We're in 1D. 112 00:06:59,300 --> 00:07:03,890 So, how do you find the function, u of x? 113 00:07:03,890 --> 00:07:06,990 that stands for du/dx. 114 00:07:06,990 --> 00:07:10,340 So the given data for the problem 115 00:07:10,340 --> 00:07:12,985 is some load, some source term, f 116 00:07:12,985 --> 00:07:16,820 of x, which is going to show up on the right-hand side, 117 00:07:16,820 --> 00:07:19,740 and some coefficient c of x, which is going 118 00:07:19,740 --> 00:07:22,180 to show up in the equation. 119 00:07:22,180 --> 00:07:25,570 They depend on x, in general. 120 00:07:25,570 --> 00:07:30,330 Many, many, many physical problems look like this. 121 00:07:30,330 --> 00:07:33,700 Sort of steady-state problems, I would say. 122 00:07:33,700 --> 00:07:36,330 I'm not talking about Navier-Stokes fluid flow 123 00:07:36,330 --> 00:07:45,340 convection, I'm talking about static problems, first of all. 124 00:07:45,340 --> 00:07:53,030 So here's the general idea of the calculus of variations. 125 00:07:53,030 --> 00:07:56,050 It's the same as the general idea of calculus. 126 00:07:56,050 --> 00:08:03,860 How do you identify a minimum in calculus? 127 00:08:03,860 --> 00:08:09,440 If the minimum is at u, you perturb it a little, 128 00:08:09,440 --> 00:08:15,720 by some delta u, that I'm going to call v to have just one 129 00:08:15,720 --> 00:08:17,970 letter instead of two here. 130 00:08:17,970 --> 00:08:22,260 You say OK, if I look at that neighboring point, u plus delta 131 00:08:22,260 --> 00:08:29,490 u, my quantity is bigger. 132 00:08:29,490 --> 00:08:31,460 The minimum is at u. 133 00:08:35,220 --> 00:08:39,370 So we're remembering calculus -- I guess I'm saying I've written 134 00:08:39,370 --> 00:08:39,980 that here. 135 00:08:39,980 --> 00:08:42,470 Compare u with u plus v, which you could 136 00:08:42,470 --> 00:08:44,920 think of as u plus delta u. 137 00:08:44,920 --> 00:08:50,790 It's like -- v you might think of as a small movement away 138 00:08:50,790 --> 00:08:53,390 from the best function. 139 00:08:53,390 --> 00:08:55,740 In calculus it's a small movement 140 00:08:55,740 --> 00:08:57,650 away from the best point. 141 00:08:57,650 --> 00:09:01,300 So let me draw the calculus. 142 00:09:01,300 --> 00:09:04,950 If you think of this blackboard as being function space instead 143 00:09:04,950 --> 00:09:08,730 of just a blackboard, then I'm doing calculus of variations. 144 00:09:08,730 --> 00:09:10,540 But let me just do calculus here. 145 00:09:10,540 --> 00:09:13,330 So there's the minimum, at u. 146 00:09:13,330 --> 00:09:16,930 And here is u plus v near it. 147 00:09:16,930 --> 00:09:20,740 Could be on this side or it could be on this side. 148 00:09:20,740 --> 00:09:24,020 Those are both u plus v. Well, you maybe 149 00:09:24,020 --> 00:09:26,100 want me to call one of them u minus v. 150 00:09:26,100 --> 00:09:31,870 But the point is v could have either sign. 151 00:09:31,870 --> 00:09:36,230 I'm looking at minimum sort of inside, where I can 152 00:09:36,230 --> 00:09:39,410 go to the right or the left. 153 00:09:39,410 --> 00:09:41,240 So what's the deal? 154 00:09:41,240 --> 00:09:46,090 Well, that point is then that at that point or at that point 155 00:09:46,090 --> 00:09:50,430 or at any of these other points, P of u plus v 156 00:09:50,430 --> 00:09:54,400 is bigger then what it is at the minimum. 157 00:09:54,400 --> 00:09:58,110 That tells us that u is the winner. 158 00:09:58,110 --> 00:10:04,040 Now how do we get an equation out of that? 159 00:10:04,040 --> 00:10:05,860 Calculus comes in now. 160 00:10:05,860 --> 00:10:10,800 We expand this thing -- this is some small movement away from 161 00:10:10,800 --> 00:10:11,890 u. 162 00:10:11,890 --> 00:10:15,220 So we expand it, we look at the leading term -- well, 163 00:10:15,220 --> 00:10:18,420 of course, the leading term is P of u. 164 00:10:18,420 --> 00:10:22,730 Then what is the next term? 165 00:10:22,730 --> 00:10:27,190 What's the first-order, first variation 166 00:10:27,190 --> 00:10:32,910 in P when I vary u to u plus v? 167 00:10:32,910 --> 00:10:37,550 Well, it's the whole point of calculus, actually. 168 00:10:37,550 --> 00:10:42,290 The central point of calculus is that this is some function 169 00:10:42,290 --> 00:10:49,940 that we call P prime of u times v, plus order of v squared. 170 00:10:52,960 --> 00:10:57,420 When v is small, v squared is very small. 171 00:10:57,420 --> 00:10:59,640 So what's our equation? 172 00:10:59,640 --> 00:11:03,490 Well, if this has to be bigger than P of u -- 173 00:11:03,490 --> 00:11:07,050 I could just cancel P of u -- so this thing has to be bigger 174 00:11:07,050 --> 00:11:09,600 equal zero now. 175 00:11:09,600 --> 00:11:12,090 I've squeezed it in a corner, but since it's calculus 176 00:11:12,090 --> 00:11:14,580 we kind of remember it. 177 00:11:14,580 --> 00:11:19,205 Also, it's easy to learn calculus and forget 178 00:11:19,205 --> 00:11:20,460 the main point. 179 00:11:20,460 --> 00:11:23,250 So this has to be greater equal zero. 180 00:11:23,250 --> 00:11:26,530 Now this is going to be -- if v is small, 181 00:11:26,530 --> 00:11:29,630 this is going to be very small, so it won't help. 182 00:11:29,630 --> 00:11:35,910 So this thing had better be zero, right? 183 00:11:35,910 --> 00:11:38,750 That had better be zero. 184 00:11:38,750 --> 00:11:42,700 Because if it isn't zero, I could take v of the right sign 185 00:11:42,700 --> 00:11:47,040 to make it positive, and take v small, so 186 00:11:47,040 --> 00:11:48,610 that this would dominate this. 187 00:11:51,440 --> 00:11:53,730 Maybe I wanted to take v of the right sign 188 00:11:53,730 --> 00:11:57,190 to make that negative anyway. 189 00:11:57,190 --> 00:12:00,930 I need P prime of u to be zero. 190 00:12:00,930 --> 00:12:04,300 So that's what I end up with, of course, as everybody knew. 191 00:12:04,300 --> 00:12:08,810 That P prime of u had to be zero. 192 00:12:08,810 --> 00:12:13,130 So in that tiny picture, I've remembered what we know. 193 00:12:13,130 --> 00:12:19,920 Now let me come back to what we have 194 00:12:19,920 --> 00:12:21,310 to do when we have functions. 195 00:12:23,900 --> 00:12:26,690 So what happened? 196 00:12:26,690 --> 00:12:31,060 I compare P of u -- u is now a function, u of x -- 197 00:12:31,060 --> 00:12:39,360 with P of u plus v. So I plug in u plus v, 198 00:12:39,360 --> 00:12:42,840 I compare with what I get with only u, 199 00:12:42,840 --> 00:12:44,020 and what's the difference? 200 00:12:44,020 --> 00:12:47,670 I look at the difference and the difference 201 00:12:47,670 --> 00:12:56,724 will have a linear term from u prime plus v prime squared 202 00:12:56,724 --> 00:12:59,265 and then I'm going to take away the u prime squared because I 203 00:12:59,265 --> 00:13:01,070 gotta compare the two. 204 00:13:01,070 --> 00:13:02,070 So what's left? 205 00:13:02,070 --> 00:13:04,600 There will be a 2 u prime, v prime. 206 00:13:04,600 --> 00:13:07,500 I'm maybe just being lazy here. 207 00:13:07,500 --> 00:13:09,740 I'm asking you to do it mentally and then 208 00:13:09,740 --> 00:13:11,720 I'll do it a little better. 209 00:13:11,720 --> 00:13:15,170 So the difference in this comparison is a 2 u 210 00:13:15,170 --> 00:13:18,670 prime v prime times c, and the 2's cancel. 211 00:13:18,670 --> 00:13:22,170 There's the difference right there. 212 00:13:22,170 --> 00:13:24,540 What's the difference over in this term? 213 00:13:24,540 --> 00:13:26,150 Well, I have that term and then I 214 00:13:26,150 --> 00:13:29,370 have the same term with u plus v, and then when I subtract, 215 00:13:29,370 --> 00:13:34,020 I just have the term with v. 216 00:13:34,020 --> 00:13:46,610 So this is the dP/du that has to be zero for every v. 217 00:13:46,610 --> 00:13:49,440 Now I'm really saying the important thing. 218 00:13:49,440 --> 00:13:56,470 This weak form is like saying this 219 00:13:56,470 --> 00:14:02,990 has to be zero for every v. Then, of course, 220 00:14:02,990 --> 00:14:06,800 in this scalar case, it was like a very small step 221 00:14:06,800 --> 00:14:09,580 to decide, well, if this is zero for every v, 222 00:14:09,580 --> 00:14:11,720 then that's zero, which is the strong form. 223 00:14:15,320 --> 00:14:16,540 Are you with me? 224 00:14:16,540 --> 00:14:20,190 So we have a minimum problem, minimize P, 225 00:14:20,190 --> 00:14:25,600 we have a weak form, the first variation, 226 00:14:25,600 --> 00:14:27,760 the first derivative, the first-order term 227 00:14:27,760 --> 00:14:33,730 has to be zero for every v. Then if it's zero for every v, that 228 00:14:33,730 --> 00:14:39,630 forces the derivative to be zero and that's the strong form. 229 00:14:39,630 --> 00:14:47,570 Now over here it took more space. 230 00:14:47,570 --> 00:14:50,600 But the ultimate idea is the same. 231 00:14:50,600 --> 00:14:53,550 We looked at P of u plus v compared 232 00:14:53,550 --> 00:14:56,480 with P of u, subtracted, looked at the linear term 233 00:14:56,480 --> 00:14:59,550 and there it is, the first variation. 234 00:14:59,550 --> 00:15:05,990 That has to be zero for every v. I'll just 235 00:15:05,990 --> 00:15:07,590 mentioned boundary conditions now, 236 00:15:07,590 --> 00:15:10,440 as long as we're at this weak form. 237 00:15:10,440 --> 00:15:15,570 Don't think of this weak form as just some mathematical nonsense 238 00:15:15,570 --> 00:15:17,680 to get to the differential equation, 239 00:15:17,680 --> 00:15:20,860 because the weak form is the foundation for the finite 240 00:15:20,860 --> 00:15:25,730 element method -- all sorts of discrete methods, 241 00:15:25,730 --> 00:15:30,510 discretization methods will begin with the weak form, 242 00:15:30,510 --> 00:15:36,980 the weighted integral form, rather than the strong form. 243 00:15:36,980 --> 00:15:39,720 I was just going to say a word about boundary conditions. 244 00:15:39,720 --> 00:15:41,810 What are the boundary conditions on v? 245 00:15:46,460 --> 00:15:55,730 So you could say well, v stands for a virtual displacement. 246 00:15:55,730 --> 00:15:59,510 Virtual meaning kind of we just imagining it, 247 00:15:59,510 --> 00:16:05,700 it's a displacement that we can imagine moving by that amount, 248 00:16:05,700 --> 00:16:11,690 but the whole point is the nature fix the minimum. 249 00:16:11,690 --> 00:16:12,940 What's the boundary condition? 250 00:16:12,940 --> 00:16:17,270 Well, all the candidates have to satisfy these boundary 251 00:16:17,270 --> 00:16:17,900 conditions. 252 00:16:17,900 --> 00:16:22,870 So I have to have u plus v at zero also has to equal a, 253 00:16:22,870 --> 00:16:27,050 and u plus v at 1 also has to equal b, 254 00:16:27,050 --> 00:16:31,370 if I took these simple boundary conditions. 255 00:16:31,370 --> 00:16:36,190 So, by subtraction, I learn the boundary conditions 256 00:16:36,190 --> 00:16:44,880 on v. When I say all v, I mean v of zero has to be what? 257 00:16:44,880 --> 00:16:46,740 Zero. 258 00:16:46,740 --> 00:16:49,800 And v of 1 has to be zero -- when we had those boundary 259 00:16:49,800 --> 00:16:50,809 conditions. 260 00:16:50,809 --> 00:16:53,350 Different problems could bring different boundary conditions, 261 00:16:53,350 --> 00:16:58,560 of course, but this is easier than -- simplest. 262 00:16:58,560 --> 00:17:01,350 So when I say all v, I mean every function 263 00:17:01,350 --> 00:17:05,930 that starts at zero and ends at zero 264 00:17:05,930 --> 00:17:07,870 is a candidate in this weak form. 265 00:17:10,450 --> 00:17:14,900 I have to get the answer zero for all those functions. 266 00:17:14,900 --> 00:17:21,320 Now somehow, I want to get to this point, 267 00:17:21,320 --> 00:17:27,710 the differential equation, and why don't we give it the name 268 00:17:27,710 --> 00:17:34,100 that everybody -- the two guys' names, Euler-Lagrange -- well, 269 00:17:34,100 --> 00:17:35,320 pretty famous names. 270 00:17:37,850 --> 00:17:39,350 This is the Euler-Lagrange equation. 271 00:17:45,580 --> 00:17:49,330 So maybe before where I said Kuhn-Tucker or something, 272 00:17:49,330 --> 00:17:52,060 if I'm talking about differential equations, 273 00:17:52,060 --> 00:17:55,160 I go back to these guys. 274 00:17:55,160 --> 00:18:00,720 Now, how did I get from weak form to strong form? 275 00:18:00,720 --> 00:18:03,130 That's a key. 276 00:18:03,130 --> 00:18:07,540 If you see these two steps from the minimum principle 277 00:18:07,540 --> 00:18:13,820 to the weak form, that's just, again, plug in u plus v, 278 00:18:13,820 --> 00:18:18,430 subtract and take the linear part. 279 00:18:18,430 --> 00:18:22,100 Then it's true for all v's. 280 00:18:22,100 --> 00:18:23,780 Now, how do I get from here to here? 281 00:18:26,710 --> 00:18:32,780 Notice that this form is an integrated form. 282 00:18:32,780 --> 00:18:35,370 This form is at every point. 283 00:18:35,370 --> 00:18:39,620 So it's much stronger and much more demanding. 284 00:18:42,910 --> 00:18:46,060 You could say OK, that gives us the equation -- 285 00:18:46,060 --> 00:18:49,450 that's the equation as we usually see them. 286 00:18:49,450 --> 00:18:53,060 I'll do 2D and that'll be Laplace's equation or somebody 287 00:18:53,060 --> 00:18:59,100 else's equation, but minimal surface equation, 288 00:18:59,100 --> 00:19:00,610 all sorts of equations. 289 00:19:00,610 --> 00:19:01,590 Everything. 290 00:19:01,590 --> 00:19:04,460 All sorts of applications including 291 00:19:04,460 --> 00:19:07,080 this afternoon's lecture. 292 00:19:07,080 --> 00:19:08,820 How to get from here to here? 293 00:19:08,820 --> 00:19:14,440 Well, there's one trick in advanced calculus, actually. 294 00:19:14,440 --> 00:19:17,050 The most important trick in advanced calculus 295 00:19:17,050 --> 00:19:20,550 is integration by parts. 296 00:19:20,550 --> 00:19:23,070 Well, we use those words, integration by parts, 297 00:19:23,070 --> 00:19:24,120 in 1D here. 298 00:19:26,720 --> 00:19:28,330 So I'll do that. 299 00:19:28,330 --> 00:19:30,870 We use maybe somebody else's name -- 300 00:19:30,870 --> 00:19:33,000 Green's formula or Green's theorem, 301 00:19:33,000 --> 00:19:39,060 or Green-Gauss or the divergence theorem or whatever, 302 00:19:39,060 --> 00:19:40,930 in more dimensions. 303 00:19:40,930 --> 00:19:43,740 But 1D, I just integrate by parts. 304 00:19:43,740 --> 00:19:45,760 Do you remember how integration by parts goes? 305 00:19:51,860 --> 00:19:58,120 I want to get v by itself, but I got v prime, 306 00:19:58,120 --> 00:20:02,260 because when I plugged in u plus v here, out came -- 307 00:20:02,260 --> 00:20:04,920 it's the derivative so I've got a derivative. 308 00:20:04,920 --> 00:20:07,250 So how do I get rid of a derivative? 309 00:20:07,250 --> 00:20:08,770 Integrate by parts. 310 00:20:08,770 --> 00:20:12,570 Take the derivative off of v -- can I just do that this quick 311 00:20:12,570 --> 00:20:14,170 way? 312 00:20:14,170 --> 00:20:19,090 Put the derivative onto -- I almost said onto u. 313 00:20:22,070 --> 00:20:25,670 I'm doing integration by parts and saying how important it is 314 00:20:25,670 --> 00:20:29,140 but not writing out every step. 315 00:20:29,140 --> 00:20:33,260 So I take the derivative off of this and I put it onto this, 316 00:20:33,260 --> 00:20:42,530 and it's gotta be -- and a minus sign appears when I do that. 317 00:20:42,530 --> 00:20:46,130 Where the heck am I gonna put that minus sign? 318 00:20:46,130 --> 00:20:48,210 Right in there. 319 00:20:48,210 --> 00:20:51,950 Minus. 320 00:20:51,950 --> 00:20:55,060 Then everybody knows that there's also a boundary term, 321 00:20:55,060 --> 00:20:56,770 right? 322 00:20:56,770 --> 00:20:59,660 So I have to squeeze somewhere in this boundary term. 323 00:20:59,660 --> 00:21:02,380 So now that I've done an integration by parts, 324 00:21:02,380 --> 00:21:09,930 the boundary term will be the v times the c u prime 325 00:21:09,930 --> 00:21:16,165 at the ends of the interval. 326 00:21:16,165 --> 00:21:17,040 I think that's right. 327 00:21:24,440 --> 00:21:28,670 What we hope is that that goes away. 328 00:21:28,670 --> 00:21:32,230 Well, of course, if it doesn't, then we have to -- 329 00:21:32,230 --> 00:21:35,520 that there's a good reason that we don't want it to, but here 330 00:21:35,520 --> 00:21:37,490 it's nice if it goes away. 331 00:21:37,490 --> 00:21:40,630 You see that it does, because we just 332 00:21:40,630 --> 00:21:43,110 decided that the boundary conditions on v 333 00:21:43,110 --> 00:21:48,070 were zero at both ends. 334 00:21:48,070 --> 00:21:52,900 So, v being zero at both ends kills that term. 335 00:21:52,900 --> 00:21:55,410 So now, do you see what I have here? 336 00:21:55,410 --> 00:21:59,250 I could write it a little more cleanly. 337 00:21:59,250 --> 00:22:02,910 The whole point is that the v -- I now have v there and I have v 338 00:22:02,910 --> 00:22:05,180 there so I can factor v out of this. 339 00:22:09,080 --> 00:22:09,900 Just put it there. 340 00:22:14,100 --> 00:22:15,120 That was a good move. 341 00:22:18,020 --> 00:22:20,500 This minus sign is still in here. 342 00:22:20,500 --> 00:22:28,170 So I now have the integral of some function times v is zero. 343 00:22:28,170 --> 00:22:30,240 That's what I'm looking for. 344 00:22:30,240 --> 00:22:36,710 The integral of some function, some stuff, times v, 345 00:22:36,710 --> 00:22:40,230 and v can be anything -- is zero. 346 00:22:40,230 --> 00:22:44,200 What happens now? 347 00:22:44,200 --> 00:22:47,360 This integral has to be zero for every v. 348 00:22:47,360 --> 00:22:50,930 So if this stuff had a little bump up, 349 00:22:50,930 --> 00:22:54,030 I could take a v to have the same bump 350 00:22:54,030 --> 00:22:57,640 and the integral wouldn't be zero. 351 00:22:57,640 --> 00:23:01,740 So this stuff can't bump up, it can't bump down, 352 00:23:01,740 --> 00:23:02,840 it can't do anything. 353 00:23:02,840 --> 00:23:06,560 It has to be zero and that's the strong form. 354 00:23:06,560 --> 00:23:10,030 So the strong form is with this minus sign in there, 355 00:23:10,030 --> 00:23:16,120 minus the derivative -- see, an extra derivative came onto 356 00:23:16,120 --> 00:23:20,230 the c u prime because it came off the v, 357 00:23:20,230 --> 00:23:25,980 and the f was just sitting there in the linear, 358 00:23:25,980 --> 00:23:28,310 in the no derivative. 359 00:23:33,640 --> 00:23:35,630 So, do you see that pattern? 360 00:23:35,630 --> 00:23:39,060 You may have seen it before, but calculus variations 361 00:23:39,060 --> 00:23:41,560 have sort of disappeared as a subject 362 00:23:41,560 --> 00:23:50,590 to teach in advanced calculus. 363 00:23:50,590 --> 00:23:54,110 It used to be here in courses that Professor Hildebrand 364 00:23:54,110 --> 00:23:55,420 taught. 365 00:23:55,420 --> 00:23:59,560 But actually it comes back because we so much need 366 00:23:59,560 --> 00:24:05,600 the weak form in finite elements and other methods. 367 00:24:05,600 --> 00:24:14,250 What I wrote over here is the discrete equivalent. 368 00:24:14,250 --> 00:24:24,580 I can't resist looking at the matrix form, for two reasons. 369 00:24:24,580 --> 00:24:27,300 First, it's simpler and it copies this. 370 00:24:27,300 --> 00:24:29,960 Do you see how this is a copy of that? 371 00:24:29,960 --> 00:24:32,760 That matrix form is supposed to be 372 00:24:32,760 --> 00:24:36,560 exactly analogous to this continuous form. 373 00:24:36,560 --> 00:24:37,880 Why is that? 374 00:24:37,880 --> 00:24:42,540 Because u transpose f, that's an inner product of u with f -- 375 00:24:42,540 --> 00:24:44,040 that's what this is. 376 00:24:44,040 --> 00:24:45,320 That integral. 377 00:24:45,320 --> 00:24:52,610 A*u is u prime in this analogy. 378 00:24:52,610 --> 00:24:57,950 So this is u prime times c times u prime with the 1/2, 379 00:24:57,950 --> 00:25:02,711 and that, again, that transpose is telling us inner product 380 00:25:02,711 --> 00:25:03,210 integral. 381 00:25:06,910 --> 00:25:11,210 So if I forget u prime and think of it as a matrix problem, 382 00:25:11,210 --> 00:25:13,860 that's my minimum problem for matrices. 383 00:25:20,730 --> 00:25:25,560 I want to find an equation for the winning u. 384 00:25:25,560 --> 00:25:29,710 In the end, this is going to be the equation. 385 00:25:29,710 --> 00:25:31,850 That's the equation that minimizes that. 386 00:25:34,750 --> 00:25:39,390 Half of 18.085 was about this problem. 387 00:25:39,390 --> 00:25:41,979 Well, I concentrated in 18.085 on this one, 388 00:25:41,979 --> 00:25:44,520 because minimum principles are just that little bit trickier, 389 00:25:44,520 --> 00:25:47,160 so that's 18.086. 390 00:25:47,160 --> 00:25:52,650 And then, in between, something people seldom 391 00:25:52,650 --> 00:25:55,470 write about but, of course, it's going to work, 392 00:25:55,470 --> 00:26:07,070 is that I change u to u plus v, I multiply it out, I subtract, 393 00:26:07,070 --> 00:26:09,960 I look at the term linear in v, and that's it. 394 00:26:14,280 --> 00:26:18,390 That would be if I make that just a minus sign 395 00:26:18,390 --> 00:26:20,650 and put it all together the way I did here, 396 00:26:20,650 --> 00:26:22,900 that's the same thing. 397 00:26:22,900 --> 00:26:24,270 So this is the weak form. 398 00:26:24,270 --> 00:26:27,824 This is the minimum form, this is the weak form, 399 00:26:27,824 --> 00:26:28,990 and this is the strong form. 400 00:26:36,900 --> 00:26:40,020 You see that weak form? 401 00:26:40,020 --> 00:26:48,560 Somehow in the discrete case, it's pretty clear that -- 402 00:26:48,560 --> 00:26:49,290 let's see. 403 00:26:49,290 --> 00:26:55,440 I could write this as -- you see, it's u transpose, 404 00:26:55,440 --> 00:27:00,770 A transpose C*A*v equal f transpose v. 405 00:27:00,770 --> 00:27:04,280 The conclusion is if this holds for every v, 406 00:27:04,280 --> 00:27:07,820 then this is the same as this. 407 00:27:07,820 --> 00:27:11,560 If two things have the same inner product with every vector 408 00:27:11,560 --> 00:27:14,630 v, they're the same, and that's the strong form. 409 00:27:17,410 --> 00:27:20,730 You'd have to transpose the whole thing, but no problem. 410 00:27:23,530 --> 00:27:32,490 So now I guess I've tried to give the main sequence of logic 411 00:27:32,490 --> 00:27:37,260 in the continuous case, and it's parallel 412 00:27:37,260 --> 00:27:42,020 in the discrete case for this example. 413 00:27:42,020 --> 00:27:45,320 For this specific example because it's the easiest. 414 00:27:45,320 --> 00:27:54,460 Let me do what Euler and Lagrange did by extending 415 00:27:54,460 --> 00:27:57,170 to a larger class of examples. 416 00:27:57,170 --> 00:28:01,680 So now our minimization is still an integral -- 417 00:28:01,680 --> 00:28:04,880 I'll still stay in 1D, I'll still keep these boundary 418 00:28:04,880 --> 00:28:10,620 conditions, but I'm going to allow some more general 419 00:28:10,620 --> 00:28:13,390 expression here. 420 00:28:13,390 --> 00:28:17,040 Instead of that pure quadratic, this could be whatever. 421 00:28:19,630 --> 00:28:21,650 Now I'm going to do calculus of variations. 422 00:28:26,340 --> 00:28:27,950 Still in 1D. 423 00:28:27,950 --> 00:28:33,700 Calculus of variations, minimize the integral 424 00:28:33,700 --> 00:28:45,270 of some function of u and u prime with the boundary 425 00:28:45,270 --> 00:28:47,170 conditions, and I'll keep those nice. 426 00:28:47,170 --> 00:28:50,150 So that integral's still 0 to 1 and I'll 427 00:28:50,150 --> 00:28:54,050 keep these nice boundary conditions just 428 00:28:54,050 --> 00:28:56,570 to make my life easy. 429 00:28:56,570 --> 00:29:01,080 So that will lead to v of zero being zero, and v of 1 430 00:29:01,080 --> 00:29:02,770 being zero. 431 00:29:10,930 --> 00:29:12,690 What do I have to do? 432 00:29:12,690 --> 00:29:16,460 Again, I have to plug in u plus v 433 00:29:16,460 --> 00:29:20,350 and compare this result with the same thing having u plus v. 434 00:29:20,350 --> 00:29:24,840 So essentially I've got to compare F 435 00:29:24,840 --> 00:29:31,900 at u plus v, u plus u prime plus v prime with F of u and u 436 00:29:31,900 --> 00:29:34,350 prime. 437 00:29:34,350 --> 00:29:40,550 I have to find the leading term in the difference. 438 00:29:44,720 --> 00:29:46,450 So I'll just find out leading term, 439 00:29:46,450 --> 00:29:50,170 and then will come the integral. 440 00:29:50,170 --> 00:29:53,040 But the first job is really the leading term, 441 00:29:53,040 --> 00:29:56,790 and it's calculus, of course. 442 00:29:56,790 --> 00:29:59,730 Now can we do that one? 443 00:29:59,730 --> 00:30:00,970 It's pure calculus. 444 00:30:06,090 --> 00:30:11,130 I have a function at two variables, 445 00:30:11,130 --> 00:30:13,610 the function of u and u prime. 446 00:30:13,610 --> 00:30:15,920 Actually, I did here. 447 00:30:15,920 --> 00:30:19,050 I had a u prime there and a u there. 448 00:30:26,650 --> 00:30:30,000 Once I write it down you're going to say sure, of course, 449 00:30:30,000 --> 00:30:32,220 I knew that. 450 00:30:32,220 --> 00:30:34,050 So I have a function of two variables 451 00:30:34,050 --> 00:30:39,070 and I'm looking for a little change. 452 00:30:39,070 --> 00:30:48,700 So a little change in the first argument produces the dF -- 453 00:30:48,700 --> 00:30:55,040 the derivative of F with respect to that first argument times 454 00:30:55,040 --> 00:31:01,160 the delta u, which is what I'm calling v. 455 00:31:01,160 --> 00:31:05,431 That's the part that the dependence on u is responsible 456 00:31:05,431 --> 00:31:05,930 for. 457 00:31:05,930 --> 00:31:09,400 Now there's also a dependence on u prime. 458 00:31:09,400 --> 00:31:13,340 So I have the derivative of F with respect 459 00:31:13,340 --> 00:31:19,400 to u prime times the little movement in u prime, which 460 00:31:19,400 --> 00:31:20,200 is v prime. 461 00:31:26,540 --> 00:31:29,800 I can't leave it with an equal there, 462 00:31:29,800 --> 00:31:34,080 because that's only the linearized part, 463 00:31:34,080 --> 00:31:36,630 but that's all I really care about. 464 00:31:36,630 --> 00:31:43,700 This is order of v squared and v prime squared. 465 00:31:43,700 --> 00:31:48,920 Higher order, which is not going to -- 466 00:31:48,920 --> 00:31:52,750 when I think of v as small, v prime as small, 467 00:31:52,750 --> 00:31:58,200 then the linear part dominates. 468 00:31:58,200 --> 00:32:04,247 So can you see what dP/du is now? 469 00:32:04,247 --> 00:32:05,330 Now I've got to integrate. 470 00:32:09,170 --> 00:32:12,510 I integrate this, that very same thing. 471 00:32:12,510 --> 00:32:18,460 This -- dx. 472 00:32:18,460 --> 00:32:28,320 That has to be zero for all v. You don't mind if I lazily 473 00:32:28,320 --> 00:32:35,110 don't copy that into the -- that's the weak form. 474 00:32:38,750 --> 00:32:43,930 This was the minimum form, now I've got to the weak form. 475 00:32:43,930 --> 00:32:46,310 This is the first variation. 476 00:32:46,310 --> 00:32:56,470 The integral of the change in F, which has two components, when 477 00:32:56,470 --> 00:32:59,510 there's a little change in u. 478 00:33:04,960 --> 00:33:08,170 I should be doing an example, but allow 479 00:33:08,170 --> 00:33:14,280 me to just keep going here until we get to the strong form. 480 00:33:14,280 --> 00:33:16,690 So this is the weak form for every v. 481 00:33:16,690 --> 00:33:19,600 Let me just repeat that the weak form is quite important 482 00:33:19,600 --> 00:33:22,880 because, in the finite element method, 483 00:33:22,880 --> 00:33:27,280 we have the v's are the test functions, 484 00:33:27,280 --> 00:33:31,770 and we discretize the v's -- you know, 485 00:33:31,770 --> 00:33:33,730 we have a finite number of test functions. 486 00:33:33,730 --> 00:33:41,050 Well, I can't go entirely -- I'll come back to finite 487 00:33:41,050 --> 00:33:43,250 elements. 488 00:33:43,250 --> 00:33:45,815 Let me stay with this continuous problem, calculus 489 00:33:45,815 --> 00:33:47,550 of variations problem. 490 00:33:47,550 --> 00:33:55,490 v is v of x here, and it satisfies these boundary 491 00:33:55,490 --> 00:33:55,990 conditions. 492 00:33:55,990 --> 00:34:01,880 That's the only requirement that we need to think about here. 493 00:34:01,880 --> 00:34:02,950 What's the strong form? 494 00:34:05,780 --> 00:34:11,510 Also called the Euler-Lagrange equation. 495 00:34:14,100 --> 00:34:15,530 How do I get to that strong form? 496 00:34:15,530 --> 00:34:17,350 How did I get to it before? 497 00:34:17,350 --> 00:34:23,230 I would like to get this into something times v. Here 498 00:34:23,230 --> 00:34:25,570 I've got something times v but I've also 499 00:34:25,570 --> 00:34:27,380 got something times v prime. 500 00:34:31,560 --> 00:34:34,910 I wanted to get it like up here where 501 00:34:34,910 --> 00:34:37,560 it was times v. What do I do? 502 00:34:37,560 --> 00:34:40,630 You know what I do. 503 00:34:40,630 --> 00:34:43,420 There's only the one idea here. 504 00:34:43,420 --> 00:34:46,020 Integrate by parts. 505 00:34:46,020 --> 00:34:54,450 So I have the integral -- well, dF/du*v was no problem -- 506 00:34:54,450 --> 00:34:56,190 that had the v that I like. 507 00:34:59,130 --> 00:35:06,000 But it's this other guy that has a v prime and I want v. So, 508 00:35:06,000 --> 00:35:11,410 it doesn't take too much thinking. 509 00:35:11,410 --> 00:35:15,740 Integrate by parts, take the derivative off of v, 510 00:35:15,740 --> 00:35:19,240 get a minus sign, and put a derivative -- 511 00:35:19,240 --> 00:35:22,650 can I do it with a prime, but I'll do better below -- 512 00:35:22,650 --> 00:35:26,260 onto this, and then there's a boundary term, 513 00:35:26,260 --> 00:35:29,110 but the boundary term goes away because of the boundary 514 00:35:29,110 --> 00:35:32,620 condition. 515 00:35:32,620 --> 00:35:45,450 So now I have the v. Can I make it on this board? 516 00:35:45,450 --> 00:35:49,910 There's the dF*du multiplying v, and then there's the minus -- 517 00:35:49,910 --> 00:35:56,810 this is the derivative d by dx of dF / d u prime. 518 00:35:56,810 --> 00:36:02,680 Now all that is multiplying v and giving me 519 00:36:02,680 --> 00:36:04,500 the integral of zero. 520 00:36:04,500 --> 00:36:10,450 I promise to write that bigger now. 521 00:36:10,450 --> 00:36:13,180 But, again, the central point was 522 00:36:13,180 --> 00:36:18,740 to get the linear term times v. That's always the main point. 523 00:36:18,740 --> 00:36:20,940 Then what's the conclusion? 524 00:36:20,940 --> 00:36:24,780 What's the Euler-Lagrange equation? 525 00:36:24,780 --> 00:36:32,150 This integral is this quantity times v and v can be anything, 526 00:36:32,150 --> 00:36:34,680 and I've got to get zero. 527 00:36:34,680 --> 00:36:38,120 So, what's the equation? 528 00:36:38,120 --> 00:36:41,430 That stuff in brackets is zero. 529 00:36:41,430 --> 00:36:43,440 That's the Euler-Lagrange equation. 530 00:36:43,440 --> 00:36:48,050 Finally, let me just write it down here. 531 00:36:48,050 --> 00:36:52,020 Euler-Lagrange strong form. 532 00:36:55,740 --> 00:37:07,440 In this general problem it would be dF/du minus d by dx of dF / 533 00:37:07,440 --> 00:37:11,840 d u prime equals zero. 534 00:37:11,840 --> 00:37:14,250 Would you like me to put on what a -- 535 00:37:14,250 --> 00:37:17,940 if F happened to depend on u double prime, 536 00:37:17,940 --> 00:37:22,390 it would be a plus -- this would be what would happen -- 537 00:37:22,390 --> 00:37:29,640 just so you get the pattern -- equaling zero. 538 00:37:29,640 --> 00:37:33,430 So I've gone one step further by allowing 539 00:37:33,430 --> 00:37:38,730 f to depend on curvature, and writing down 540 00:37:38,730 --> 00:37:45,320 what the resulting term would be in the Euler-Lagrange equation. 541 00:37:45,320 --> 00:37:49,650 Could you figure out why it would be that? 542 00:37:49,650 --> 00:37:52,420 Where would this thing have come from? 543 00:37:52,420 --> 00:37:55,705 It would have come from -- there would have been a dF / d u 544 00:37:55,705 --> 00:38:01,690 double prime times v double prime in the weak form. 545 00:38:01,690 --> 00:38:06,020 Then I would have done two integrations by parts -- 546 00:38:06,020 --> 00:38:12,530 two minus signs making a plus, two derivatives moving off of v 547 00:38:12,530 --> 00:38:19,310 and onto the other thing and it would give me that. 548 00:38:19,310 --> 00:38:23,910 All times v, but now I've got everything times v, 549 00:38:23,910 --> 00:38:29,640 it's true for every v, so the quantity has to be zero. 550 00:38:29,640 --> 00:38:32,440 That's the Euler-Lagrange equation, 551 00:38:32,440 --> 00:38:34,690 strong form for 1D problem. 552 00:38:41,850 --> 00:38:42,620 Yes? 553 00:38:42,620 --> 00:38:45,810 AUDIENCE: [INAUDIBLE PHRASE]? 554 00:38:45,810 --> 00:38:47,040 PROFESSOR: Oh, you're right. 555 00:38:47,040 --> 00:38:48,750 You're right. 556 00:38:48,750 --> 00:38:53,900 If we had this situation then we would be up to -- 557 00:38:53,900 --> 00:38:57,880 this would typically be a fourth-order equation and we 558 00:38:57,880 --> 00:39:00,390 would have two boundary conditions at each end. 559 00:39:00,390 --> 00:39:02,450 Absolutely. 560 00:39:02,450 --> 00:39:06,310 When I slipped in this just to sort of show the pattern, 561 00:39:06,310 --> 00:39:09,180 I didn't account for the boundary conditions. 562 00:39:09,180 --> 00:39:14,490 That would be one level up as well. 563 00:39:14,490 --> 00:39:15,620 Exactly. 564 00:39:15,620 --> 00:39:19,560 And nature -- well, fortunately we don't get many equations 565 00:39:19,560 --> 00:39:23,220 of higher degree than four. 566 00:39:23,220 --> 00:39:28,050 This would be like the beam equation, the parallel to that 567 00:39:28,050 --> 00:39:32,790 would be a beam equation or a plate 568 00:39:32,790 --> 00:39:35,810 equation or a shell equation, God forbid. 569 00:39:35,810 --> 00:39:38,630 In shell theory they're incredibly 570 00:39:38,630 --> 00:39:41,500 complicated because they're on surfaces. 571 00:39:41,500 --> 00:39:44,870 But the pattern is always this. 572 00:39:47,790 --> 00:39:49,250 Well, and, of course, they're also 573 00:39:49,250 --> 00:39:52,060 complicated because they're in 2D. 574 00:39:52,060 --> 00:39:54,340 So maybe I should say a little bit. 575 00:39:54,340 --> 00:40:00,830 Could I maybe write -- I'm trying to do a lot today. 576 00:40:00,830 --> 00:40:05,260 Trying to do kind of the formal stuff today. 577 00:40:05,260 --> 00:40:07,420 So another step in the formal stuff 578 00:40:07,420 --> 00:40:12,520 would be to get into 2D, which I haven't done. 579 00:40:15,400 --> 00:40:18,260 What would be the famous 2D problem that 580 00:40:18,260 --> 00:40:20,430 leads to Laplace's equation? 581 00:40:20,430 --> 00:40:25,470 So 2D, what would I minimize, just to -- 582 00:40:25,470 --> 00:40:35,270 so P of u would be a double integral of du/dx squared -- 583 00:40:35,270 --> 00:40:41,830 maybe times a c -- du/dy squared. 584 00:40:41,830 --> 00:40:47,120 I would really like 1/2 on that just to make life good. 585 00:40:47,120 --> 00:40:51,580 Then minus a double integral of f times u. 586 00:40:51,580 --> 00:41:02,860 dx*dy, to emphasize these are double integrals. 587 00:41:02,860 --> 00:41:06,870 That's a very, very important problem. 588 00:41:06,870 --> 00:41:10,410 Many people have tried to study that. 589 00:41:10,410 --> 00:41:14,980 Euler and Lagrange would produce an equation for it. 590 00:41:14,980 --> 00:41:17,100 You might say OK, solve the equation, that 591 00:41:17,100 --> 00:41:18,350 finishes the problem. 592 00:41:18,350 --> 00:41:23,890 But mathematicians are always worried, is there a solution? 593 00:41:23,890 --> 00:41:25,840 Is there a minimum? 594 00:41:25,840 --> 00:41:30,230 So I'm dodging the bullet on that one. 595 00:41:30,230 --> 00:41:33,490 When I say minimize over all functions u, 596 00:41:33,490 --> 00:41:40,530 I could create problems where worse and worse and worse 597 00:41:40,530 --> 00:41:43,710 functions got closer and closer to a minimum 598 00:41:43,710 --> 00:41:47,150 and there was no limiting minimizer. 599 00:41:47,150 --> 00:41:49,610 I won't do that. 600 00:41:49,610 --> 00:41:52,860 This is a problem that works fine. 601 00:41:52,860 --> 00:41:54,280 What happens to it? 602 00:41:54,280 --> 00:41:59,340 Well, should we try to do the same weak form? 603 00:42:02,410 --> 00:42:07,610 We'd have a dP/du, and what do you think it would look like? 604 00:42:07,610 --> 00:42:12,260 It would have the integral, double integral. 605 00:42:12,260 --> 00:42:13,650 What would it have? 606 00:42:13,650 --> 00:42:21,100 It will have a c*du/dx and there will be a dv/dx. 607 00:42:21,100 --> 00:42:23,040 But then integration by parts will 608 00:42:23,040 --> 00:42:28,190 take that x derivative off of v and onto this, with a minus. 609 00:42:35,690 --> 00:42:39,430 I did it fast, did it way fast. 610 00:42:39,430 --> 00:42:43,980 Then out of this thing will come, if I look at the v term 611 00:42:43,980 --> 00:42:48,760 they'll be a dv/dy and I take that y derivative off 612 00:42:48,760 --> 00:42:51,290 of that and onto this. 613 00:42:51,290 --> 00:42:55,200 Oh, I can't use d anymore, I have to use partials. 614 00:42:55,200 --> 00:42:57,910 d by du of c*du/dy. 615 00:43:00,870 --> 00:43:05,060 Then the f is just sitting there. 616 00:43:08,050 --> 00:43:17,030 Oh, it's all multiplied by v. dx/dy I didn't put yet. 617 00:43:17,030 --> 00:43:32,770 Equals zero for all v. Again, this is the same thing 618 00:43:32,770 --> 00:43:38,330 that we had in ordinary calculus where it was just P prime of u, 619 00:43:38,330 --> 00:43:39,740 v equals zero. 620 00:43:43,940 --> 00:43:49,980 This is a level of sophistication up. 621 00:43:49,980 --> 00:43:53,820 It's producing differential equations, not 622 00:43:53,820 --> 00:43:56,480 scalar equations. 623 00:43:56,480 --> 00:43:59,180 So what's the deal? 624 00:43:59,180 --> 00:44:01,060 I've done the integration by parts, 625 00:44:01,060 --> 00:44:04,000 so I've got everything multiplying a v. 626 00:44:04,000 --> 00:44:15,020 So what's the strong form of this problem? 627 00:44:17,970 --> 00:44:25,550 Well, if this integral has to be zero for every v, 628 00:44:25,550 --> 00:44:30,910 then the conclusion is that this stuff in the brackets is zero. 629 00:44:30,910 --> 00:44:33,100 That's always the same. 630 00:44:33,100 --> 00:44:36,040 That's the strong form. 631 00:44:36,040 --> 00:44:38,590 So that's Laplace's equation or actually 632 00:44:38,590 --> 00:44:41,970 Poisson's equation because I have a right-hand side, f of x. 633 00:44:45,470 --> 00:44:47,360 Well, those are the mechanics. 634 00:44:47,360 --> 00:44:55,210 Now, what else comes into the calculus of variations? 635 00:44:55,210 --> 00:44:57,630 You've seen the pattern here. 636 00:44:57,630 --> 00:45:01,560 There's one important further possibility 637 00:45:01,560 --> 00:45:07,940 that we met last time, which was constraints. 638 00:45:07,940 --> 00:45:10,320 We're dealing here with a pure minimization. 639 00:45:13,370 --> 00:45:17,060 I didn't impose any side conditions 640 00:45:17,060 --> 00:45:21,740 on u except maybe the boundary condition. 641 00:45:21,740 --> 00:45:27,910 So let me give you an example which -- 642 00:45:27,910 --> 00:45:35,460 I'll just close with an example that I'm going to follow up, 643 00:45:35,460 --> 00:45:39,340 and it's going to have a constraint. 644 00:45:39,340 --> 00:45:44,960 So it'll look like the original problem, but, well, 645 00:45:44,960 --> 00:45:46,930 there will be two u's. 646 00:45:46,930 --> 00:45:49,510 So can I try to get it right here? 647 00:45:53,740 --> 00:46:04,580 My minimization, my unknown will have two components, u_1 648 00:46:04,580 --> 00:46:05,080 and u_2. 649 00:46:09,770 --> 00:46:13,980 And it'll be in 2D actually. 650 00:46:13,980 --> 00:46:21,860 What I'm going to produce here is called the Stokes problem. 651 00:46:21,860 --> 00:46:25,270 I'll study it next time, so if I run out of time, as I probably 652 00:46:25,270 --> 00:46:28,020 will, that's part of the plan. 653 00:46:31,010 --> 00:46:34,710 So it's Stokes and not Navier-Stokes. 654 00:46:34,710 --> 00:46:38,050 All I want to do is to write down a problem in which there 655 00:46:38,050 --> 00:46:40,120 is a constraint. 656 00:46:40,120 --> 00:46:48,730 So I write it as a minimization, say, dv_1/ d -- oh, 657 00:46:48,730 --> 00:46:50,830 probably all these guys are in here. 658 00:46:50,830 --> 00:47:00,990 dv_1/dx and dv_1/dy and dv_2/dx and dv_2/dy -- 659 00:47:00,990 --> 00:47:02,470 sorry about all this stuff. 660 00:47:05,360 --> 00:47:13,290 Probably an f_1*v -- oh, I've written v, 661 00:47:13,290 --> 00:47:16,150 because my mind is saying that the usual notation, 662 00:47:16,150 --> 00:47:20,050 I should be writing u because that would fit with 663 00:47:20,050 --> 00:47:22,120 today's lecture. 664 00:47:22,120 --> 00:47:25,400 It's a velocity and that's why many people call it v, 665 00:47:25,400 --> 00:47:30,450 and then they have to call the perturbation some w. 666 00:47:30,450 --> 00:47:35,240 f_1*u_1, f_2*u_2, all that stuff. 667 00:47:35,240 --> 00:47:36,960 No problem. 668 00:47:36,960 --> 00:47:39,820 That would lead to Laplace's equation, just the same, 669 00:47:39,820 --> 00:47:42,440 Poisson's equation. 670 00:47:42,440 --> 00:47:47,140 But I'm going to add a constraint. 671 00:47:47,140 --> 00:47:54,980 This [u 1, u 2] is a velocity, despite the letter. 672 00:47:54,980 --> 00:48:00,530 I want to make the material incompressible. 673 00:48:00,530 --> 00:48:06,740 I have flow here and it's like flow of water, 674 00:48:06,740 --> 00:48:09,580 probably incompressible. 675 00:48:09,580 --> 00:48:24,042 So incompressible means that dv_1/dx plus dv_2/dx is zero. 676 00:48:24,042 --> 00:48:25,000 So that's a constraint. 677 00:48:31,160 --> 00:48:32,550 How the heck do we deal with it? 678 00:48:35,790 --> 00:48:38,870 So I could do this minimization but with the constraint. 679 00:48:45,380 --> 00:48:50,260 So all this stuff I'm totally cool with now. 680 00:48:50,260 --> 00:48:53,310 That would just be calculus of variations, 681 00:48:53,310 --> 00:48:56,130 that would get me the dP/du, but I have 682 00:48:56,130 --> 00:48:57,490 to account for the constraint. 683 00:48:57,490 --> 00:49:01,230 So how do you account for a constraint? 684 00:49:01,230 --> 00:49:06,790 You build it into the problem with a Lagrange multiplier. 685 00:49:06,790 --> 00:49:11,140 So I multiply this thing by some Lagrange multiplier, 686 00:49:11,140 --> 00:49:14,360 and as I emphasized last time, Lagrange multipliers always 687 00:49:14,360 --> 00:49:16,940 turn out to mean something physically, 688 00:49:16,940 --> 00:49:18,980 and here it's the pressure. 689 00:49:18,980 --> 00:49:26,650 So it's natural to call the Lagrange multiplier p of x, y. 690 00:49:26,650 --> 00:49:34,320 I build that in, so I subtract the Lagrange multiplier 691 00:49:34,320 --> 00:49:38,820 times this thing that has to be zero. 692 00:49:38,820 --> 00:49:42,050 That gets in the problem. 693 00:49:42,050 --> 00:49:50,380 Now my function now depends on u and the pressure. 694 00:49:54,690 --> 00:50:00,420 I'm not going to push this to the very limit 695 00:50:00,420 --> 00:50:02,520 to find the strong form. 696 00:50:02,520 --> 00:50:04,940 But the strong form is the Stokes equations 697 00:50:04,940 --> 00:50:06,960 that we'll study. 698 00:50:06,960 --> 00:50:12,960 So we have a lot to do here to make this 699 00:50:12,960 --> 00:50:18,060 into practical calculations where we can compute something. 700 00:50:18,060 --> 00:50:22,560 And finite elements is a powerful way to do it. 701 00:50:22,560 --> 00:50:25,570 So we have to turn these continuous problems 702 00:50:25,570 --> 00:50:27,460 into discrete problems. 703 00:50:27,460 --> 00:50:32,490 And then, later, we have to turn this type of problem, which 704 00:50:32,490 --> 00:50:35,910 will be a saddle point problem because it's got this Lagrange 705 00:50:35,910 --> 00:50:40,010 multiplier in there, into a discrete problem. 706 00:50:40,010 --> 00:50:45,910 Let me just stop by putting the words saddle point there, 707 00:50:45,910 --> 00:50:52,690 and just as in the lecture this afternoon, saddle points 708 00:50:52,690 --> 00:50:57,190 appear as soon as you have constraints and Lagrange 709 00:50:57,190 --> 00:50:59,320 multipliers. 710 00:50:59,320 --> 00:51:00,750 Well, thanks for your patience. 711 00:51:00,750 --> 00:51:05,100 That's a lot of material that will quickly -- Now, 712 00:51:05,100 --> 00:51:11,260 those basic steps will be section 7.2 and will go up 713 00:51:11,260 --> 00:51:18,160 on the web quickly, just as soon as we get them revised. 714 00:51:18,160 --> 00:51:22,060 And I'm writing notes on your projects 715 00:51:22,060 --> 00:51:25,430 and I hope I'll have them ready for Friday. 716 00:51:25,430 --> 00:51:29,220 I'll aim for Friday because Monday is Patriot's Day 717 00:51:29,220 --> 00:51:32,160 and you have to run the marathon. 718 00:51:32,160 --> 00:51:33,440 So I'll see you Friday. 719 00:51:33,440 --> 00:51:33,940 Good. 720 00:51:33,940 --> 00:51:35,190 Thanks.