1
00:00:00,500 --> 00:00:01,990
PROFESSOR: OK.
2
00:00:01,990 --> 00:00:05,410
Here's an example that's
more or less for fun.
3
00:00:05,410 --> 00:00:08,200
Because you'll see
me try to do it.
4
00:00:08,200 --> 00:00:11,360
You can do it better.
5
00:00:11,360 --> 00:00:14,450
I call the problem
the tumbling blocks.
6
00:00:14,450 --> 00:00:18,090
Only in this example,
in my demonstration,
7
00:00:18,090 --> 00:00:20,270
it's going to be
a tumbling book.
8
00:00:20,270 --> 00:00:24,390
I'm going to take a
book, the sacred book,
9
00:00:24,390 --> 00:00:26,960
and throw it in the air.
10
00:00:26,960 --> 00:00:30,140
And I'll throw it
three different ways.
11
00:00:30,140 --> 00:00:38,560
And the question is, is the
spinning book stable or not?
12
00:00:38,560 --> 00:00:41,420
And let me tell
you the three ways
13
00:00:41,420 --> 00:00:45,610
and then give you the three
equations that came from Euler.
14
00:00:45,610 --> 00:00:47,950
So those are the
three equations.
15
00:00:47,950 --> 00:00:50,960
You see that they're not linear.
16
00:00:50,960 --> 00:00:55,270
And those are for
the angular momentum.
17
00:00:55,270 --> 00:00:57,590
So there's a little physics
behind the equations.
18
00:00:57,590 --> 00:01:00,630
But for us, those are
the three equations.
19
00:01:00,630 --> 00:01:07,750
So the first throw
will spin around
20
00:01:07,750 --> 00:01:10,730
the very short axis, just
the thickness of the book,
21
00:01:10,730 --> 00:01:12,810
maybe an inch.
22
00:01:12,810 --> 00:01:16,300
So when I toss that,
as I'll do now,
23
00:01:16,300 --> 00:01:22,960
you will see if I can toss
it not too nervously I hope.
24
00:01:22,960 --> 00:01:27,740
It came-- it was stable.
25
00:01:27,740 --> 00:01:33,820
The book came back to
me without wobbling.
26
00:01:33,820 --> 00:01:37,200
Of course, my nerves would
give it a little wobble,
27
00:01:37,200 --> 00:01:38,960
and that wobble would continue.
28
00:01:38,960 --> 00:01:41,970
It will be only
neutrally stable.
29
00:01:41,970 --> 00:01:44,490
The wobble doesn't disappear.
30
00:01:44,490 --> 00:01:47,990
But it doesn't
grow into a tumble.
31
00:01:47,990 --> 00:01:48,490
OK.
32
00:01:48,490 --> 00:01:53,920
So that's one axis,
the short axis.
33
00:01:53,920 --> 00:01:59,600
Then I'll throw it also
around the long axis, flipped
34
00:01:59,600 --> 00:02:00,780
like this.
35
00:02:00,780 --> 00:02:03,000
I think that will be stable too.
36
00:02:03,000 --> 00:02:06,410
And then, finally, on
the intermediate axis,
37
00:02:06,410 --> 00:02:08,470
is middle length axis.
38
00:02:08,470 --> 00:02:12,130
Notice the rubber band that's
holding the book together.
39
00:02:12,130 --> 00:02:15,210
Holding so the pages don't open.
40
00:02:15,210 --> 00:02:20,390
And this, we'll see, I
think, will be unstable.
41
00:02:20,390 --> 00:02:24,100
And similarly,
throwing a football,
42
00:02:24,100 --> 00:02:30,550
throwing other Frisbees,
whatever your throw.
43
00:02:30,550 --> 00:02:36,110
Any 3D object has got these
three axes: a short one,
44
00:02:36,110 --> 00:02:39,110
a medium one, and a long axis.
45
00:02:39,110 --> 00:02:43,570
And the equations
will tell us short
46
00:02:43,570 --> 00:02:47,990
and long axes should
give a stable turning.
47
00:02:47,990 --> 00:02:52,500
And the in between
axis is unstable.
48
00:02:52,500 --> 00:02:56,990
Well, how do we decide for
our differential equation
49
00:02:56,990 --> 00:03:02,510
whether the fixed
point, a fixed point,
50
00:03:02,510 --> 00:03:05,660
that's a critical
point, a steady state--
51
00:03:05,660 --> 00:03:07,850
we have to find
this steady state,
52
00:03:07,850 --> 00:03:12,080
and then for each steady
state we linearize.
53
00:03:12,080 --> 00:03:15,660
We find the derivatives
at that steady state.
54
00:03:15,660 --> 00:03:20,440
And that gives us a constant
matrix at that steady state.
55
00:03:20,440 --> 00:03:23,280
And then the
eigenvalue is decided.
56
00:03:23,280 --> 00:03:26,730
So first, find the
critical points.
57
00:03:26,730 --> 00:03:31,050
Second, find the derivatives
at the critical points.
58
00:03:31,050 --> 00:03:33,830
Third, for that
matrix of derivatives,
59
00:03:33,830 --> 00:03:38,250
find the eigenvalues
and decide stability.
60
00:03:38,250 --> 00:03:41,200
That's the sequence of steps.
61
00:03:41,200 --> 00:03:41,840
OK.
62
00:03:41,840 --> 00:03:44,630
The first time we've ever
done a three by three matrix.
63
00:03:44,630 --> 00:03:46,080
Maybe the last time.
64
00:03:46,080 --> 00:03:48,540
OK.
65
00:03:48,540 --> 00:03:51,430
Let me, before I
start-- before I
66
00:03:51,430 --> 00:03:57,970
find the critical points--
notice some nice properties.
67
00:03:57,970 --> 00:04:01,330
If I multiply this
equation by x, this one
68
00:04:01,330 --> 00:04:07,410
by y, this one by z, and
add, those will add to 0.
69
00:04:07,410 --> 00:04:10,910
When there's an x there,
a y there, and a z there,
70
00:04:10,910 --> 00:04:14,070
I get a 1 minus 2 and
a 1 they add to 0.
71
00:04:14,070 --> 00:04:17,269
So x times dx dt.
72
00:04:17,269 --> 00:04:21,750
y times dy dt. z
time dz dt adds to 0.
73
00:04:21,750 --> 00:04:23,570
That's an important fact.
74
00:04:23,570 --> 00:04:27,890
That's telling me that the
derivative of something is 0.
75
00:04:27,890 --> 00:04:30,090
That something
will be a constant.
76
00:04:30,090 --> 00:04:31,960
So I'm seeing here
the derivative
77
00:04:31,960 --> 00:04:38,685
of that whole business
would be the derivative
78
00:04:38,685 --> 00:04:41,760
of a half probably.
79
00:04:41,760 --> 00:04:45,520
x squared, because the
derivative of x squared
80
00:04:45,520 --> 00:04:48,760
will be with a half.
81
00:04:48,760 --> 00:04:51,860
The derivative will be x dx dt.
82
00:04:51,860 --> 00:05:00,630
And y squared and z squared
is the derivative is 0.
83
00:05:00,630 --> 00:05:05,680
The derivative of that
line is just this line.
84
00:05:05,680 --> 00:05:07,140
It's 0.
85
00:05:07,140 --> 00:05:08,280
So this is a constant.
86
00:05:13,780 --> 00:05:16,910
No doubt, that's
probably telling me
87
00:05:16,910 --> 00:05:21,710
that the total energy, the
kinetic energy, is constant.
88
00:05:21,710 --> 00:05:24,420
After I've tossed that
book up in the air,
89
00:05:24,420 --> 00:05:25,900
I'm not touching it.
90
00:05:25,900 --> 00:05:28,170
It's doing its thing.
91
00:05:28,170 --> 00:05:31,240
And it's not going to change
energy because nothing
92
00:05:31,240 --> 00:05:32,490
is happening to it.
93
00:05:32,490 --> 00:05:34,750
It's just out there.
94
00:05:34,750 --> 00:05:39,150
Now there are other-- so
that's a rather nice thing.
95
00:05:39,150 --> 00:05:40,170
This is a constant.
96
00:05:43,060 --> 00:05:47,560
Now there's another way.
97
00:05:47,560 --> 00:05:55,390
If I multiply this one by 2x,
and I multiply this one by y,
98
00:05:55,390 --> 00:05:59,800
and add just those
two, that cancels.
99
00:05:59,800 --> 00:06:05,740
So 2x dx dt-- 2x times
the first one-- and y
100
00:06:05,740 --> 00:06:08,100
times the second one gives 0.
101
00:06:08,100 --> 00:06:11,760
Again, I'm seeing
something is constant.
102
00:06:11,760 --> 00:06:14,600
The derivative of something,
and that something
103
00:06:14,600 --> 00:06:22,745
is x squared plus 1/2 y
squared is a constant.
104
00:06:25,640 --> 00:06:32,680
Another nice fact.
105
00:06:32,680 --> 00:06:35,300
Another quantity
that's conserved.
106
00:06:35,300 --> 00:06:38,480
And as I'm flying
around in space,
107
00:06:38,480 --> 00:06:43,200
this quantity x squared plus
1/2 y squared does not change.
108
00:06:43,200 --> 00:06:47,820
This sort of-- that
involved all of xyz.
109
00:06:47,820 --> 00:06:51,290
And of course that's the
equation of a sphere.
110
00:06:51,290 --> 00:06:58,020
So in energy space,
or in an xyz space,
111
00:06:58,020 --> 00:07:01,990
our solution is wandering
around a sphere.
112
00:07:01,990 --> 00:07:06,830
And this is the equation for,
I guess, it's an ellipse.
113
00:07:06,830 --> 00:07:08,790
So there's an ellipse
on that's sphere
114
00:07:08,790 --> 00:07:11,380
that it's actually
staying on that ellipse.
115
00:07:11,380 --> 00:07:13,720
And in fact there's
another ellipse
116
00:07:13,720 --> 00:07:20,550
because I could've multiplied
this one by 2z and this one
117
00:07:20,550 --> 00:07:22,470
by y and added.
118
00:07:22,470 --> 00:07:24,380
And then those
would have canceled.
119
00:07:24,380 --> 00:07:27,710
Minus 2 xyz plus 2xyz.
120
00:07:27,710 --> 00:07:31,270
So that also tells
me that it would
121
00:07:31,270 --> 00:07:41,730
be probably z squared plus 1/2
y squared equals a constant.
122
00:07:41,730 --> 00:07:44,670
That's another ellipse.
123
00:07:44,670 --> 00:07:46,380
z squared plus 1/2 y squared.
124
00:07:46,380 --> 00:07:47,810
You see this?
125
00:07:47,810 --> 00:07:50,730
If I take the
derivative of that,
126
00:07:50,730 --> 00:07:57,540
I have 2z times dz dt
plus y times dy dt.
127
00:07:57,540 --> 00:07:59,180
Adding give 0.
128
00:07:59,180 --> 00:08:00,540
The derivative is 0.
129
00:08:00,540 --> 00:08:02,110
The thing is a constant.
130
00:08:02,110 --> 00:08:03,870
But!
131
00:08:03,870 --> 00:08:05,880
But, but, but!
132
00:08:05,880 --> 00:08:10,650
If I subtract this
one from this one,
133
00:08:10,650 --> 00:08:12,850
take the difference
of these two.
134
00:08:12,850 --> 00:08:15,370
Suppose I take this
one minus this one.
135
00:08:15,370 --> 00:08:17,530
The 1/2 y squared will go.
136
00:08:17,530 --> 00:08:21,300
So that will tell me
that x squared minus z
137
00:08:21,300 --> 00:08:23,870
squared is a constant.
138
00:08:23,870 --> 00:08:26,490
Oh, boy!
139
00:08:26,490 --> 00:08:31,680
I haven't solved
my three equations.
140
00:08:31,680 --> 00:08:35,049
But I found out a whole
lot about the solution.
141
00:08:35,049 --> 00:08:39,549
The solution stays on the
sphere, wanders around somehow.
142
00:08:39,549 --> 00:08:43,039
It also at the same time
stays on that ellipse.
143
00:08:43,039 --> 00:08:45,010
And it stays on that ellipse.
144
00:08:45,010 --> 00:08:48,150
But this is not an
ellipse, not an ellipse.
145
00:08:48,150 --> 00:08:51,360
That's the equation
of a hyperbola.
146
00:08:51,360 --> 00:08:55,810
And that's why-- which, of
course, goes off to infinity.
147
00:08:55,810 --> 00:08:59,640
And that's why the-- well,
it goes off to infinity,
148
00:08:59,640 --> 00:09:01,800
but it has to stay
on the sphere.
149
00:09:01,800 --> 00:09:03,290
It wanders.
150
00:09:03,290 --> 00:09:08,080
This will be responsible
for the unstable motion.
151
00:09:08,080 --> 00:09:15,830
Professor [INAUDIBLE], who would
do this far better than me,
152
00:09:15,830 --> 00:09:19,800
his great lecture in 1803,
Differential Equations,
153
00:09:19,800 --> 00:09:21,150
was exactly this.
154
00:09:21,150 --> 00:09:24,470
The full hour to tell you
everything about the tumbling
155
00:09:24,470 --> 00:09:25,470
box.
156
00:09:25,470 --> 00:09:32,370
So I'm going to do
the demonstration
157
00:09:32,370 --> 00:09:39,380
and write down the main facts
and understand the stability,
158
00:09:39,380 --> 00:09:41,670
the discussion of stability.
159
00:09:41,670 --> 00:09:45,370
I'm ready to move on to the
discussion of stability.
160
00:09:45,370 --> 00:09:50,420
Again, here are my
three equations.
161
00:09:50,420 --> 00:09:52,250
We're up to three
equation, so we're
162
00:09:52,250 --> 00:09:55,010
going have a three
by three matrix.
163
00:09:55,010 --> 00:10:00,880
And first I have to find
out the critical points,
164
00:10:00,880 --> 00:10:04,080
the steady states
of this motion.
165
00:10:04,080 --> 00:10:07,550
How could I toss it so
that if I toss it perfectly
166
00:10:07,550 --> 00:10:10,640
it stays exactly as tossed?
167
00:10:10,640 --> 00:10:15,040
And the answer is,
around the axis.
168
00:10:15,040 --> 00:10:18,340
If I toss this perfectly,
with no nerves,
169
00:10:18,340 --> 00:10:21,620
it'll just spin exactly
as I'm throwing it.
170
00:10:21,620 --> 00:10:27,530
The x, y, and z will
all be constant.
171
00:10:27,530 --> 00:10:30,145
Now, when I toss
it on that axis.
172
00:10:36,660 --> 00:10:40,040
I'm looking for-- here
are my right hand side.
173
00:10:40,040 --> 00:10:47,500
YZ, minus 2XZ, and XY.
174
00:10:47,500 --> 00:10:50,320
And I wrote those
in capital letters
175
00:10:50,320 --> 00:10:54,560
because those are going
to be my steady states.
176
00:10:54,560 --> 00:11:00,610
Now I'm looking for are points
where nothing's happened.
177
00:11:00,610 --> 00:11:04,700
If those three right hand
sides of the equation are 0,
178
00:11:04,700 --> 00:11:06,380
I'm not going to move.
179
00:11:06,380 --> 00:11:09,350
xyz will stay where they are.
180
00:11:09,350 --> 00:11:13,340
So can you see solutions
of those three equations?
181
00:11:13,340 --> 00:11:16,040
Well, they're pretty
special equations.
182
00:11:16,040 --> 00:11:23,450
I get a solution when, for
example, solutions could be 1,
183
00:11:23,450 --> 00:11:26,230
0, 0/
184
00:11:26,230 --> 00:11:30,690
If two of the three--
if y and z are 0.
185
00:11:30,690 --> 00:11:34,400
y is 0, z is 0, y
and z are 0, I get 0.
186
00:11:34,400 --> 00:11:38,830
So that is a certainly
steady state.
187
00:11:38,830 --> 00:11:42,740
x equal 1, y and
z equal 0 and 0.
188
00:11:42,740 --> 00:11:49,310
And that steady state is
spinning around one axis.
189
00:11:49,310 --> 00:11:52,790
And, actually, I could have
also a minus 1 would also be.
190
00:11:52,790 --> 00:11:58,080
So I've found, actually, two
steady states with y and z 0.
191
00:11:58,080 --> 00:12:02,980
Then there'll be two
more with x and z 0.
192
00:12:02,980 --> 00:12:05,890
And this could be--
that'll be spinning
193
00:12:05,890 --> 00:12:07,870
around the middle axis.
194
00:12:07,870 --> 00:12:13,090
And then 0, 0, 1
or minus 1, that
195
00:12:13,090 --> 00:12:16,500
would be spinning around the
third axis, the long axis.
196
00:12:16,500 --> 00:12:18,840
So those are my steady states.
197
00:12:18,840 --> 00:12:21,130
And I guess, come
to think of it, 0,
198
00:12:21,130 --> 00:12:26,240
0, 0 would also
be a steady state.
199
00:12:26,240 --> 00:12:28,810
I think I found them all.
200
00:12:28,810 --> 00:12:30,180
These are the xy's.
201
00:12:30,180 --> 00:12:37,880
These are the x,
y, z steady states.
202
00:12:37,880 --> 00:12:39,360
OK.
203
00:12:39,360 --> 00:12:42,310
So now once you know the steady
states, that's usually fun,
204
00:12:42,310 --> 00:12:44,750
as it was here.
205
00:12:44,750 --> 00:12:50,800
Now the slightly less fun step
is find all the derivatives,
206
00:12:50,800 --> 00:12:54,970
find that Jacobian
matrix of derivative.
207
00:12:54,970 --> 00:12:58,190
So I've got three equations.
208
00:12:58,190 --> 00:13:01,350
Three unknowns, xyz.
209
00:13:01,350 --> 00:13:03,180
Three right hand sides.
210
00:13:03,180 --> 00:13:09,440
And I have to find-- I'm
going to have a three by three
211
00:13:09,440 --> 00:13:12,010
matrix of derivatives.
212
00:13:12,010 --> 00:13:14,050
This Jacobian matrix.
213
00:13:14,050 --> 00:13:19,680
So J for the Jacobian, the
matrix of first derivatives.
214
00:13:19,680 --> 00:13:24,300
So what goes into the
matrix of first derivative?
215
00:13:24,300 --> 00:13:26,710
Let me write Jacobian.
216
00:13:26,710 --> 00:13:30,380
It is named after Jacoby.
217
00:13:30,380 --> 00:13:32,270
It's the matrix of
first derivatives.
218
00:13:32,270 --> 00:13:35,650
On the top row are
the derivatives
219
00:13:35,650 --> 00:13:38,950
of the first function
with respect to x.
220
00:13:38,950 --> 00:13:42,400
Well, the derivative
with respect to x is 0.
221
00:13:42,400 --> 00:13:45,226
The derivative with
respect to y is z.
222
00:13:45,226 --> 00:13:49,910
The derivative with
respect to z is y.
223
00:13:49,910 --> 00:13:51,350
Those were partial derivatives.
224
00:13:54,150 --> 00:14:00,160
They tell me how much the
first unknown x moves.
225
00:14:00,160 --> 00:14:02,400
They tell me what's happening
with the first unknown
226
00:14:02,400 --> 00:14:07,940
x around the critical
point whichever it is.
227
00:14:07,940 --> 00:14:08,440
OK.
228
00:14:08,440 --> 00:14:13,860
What about the
partial derivatives
229
00:14:13,860 --> 00:14:15,390
from the second equation?
230
00:14:15,390 --> 00:14:19,040
it's partial derivatives
will go into this row.
231
00:14:19,040 --> 00:14:22,256
So x has a minus 2z.
232
00:14:22,256 --> 00:14:26,280
y derivative is 0.
233
00:14:26,280 --> 00:14:29,470
z derivative is minus 2x.
234
00:14:29,470 --> 00:14:33,600
And the third one, the
z derivative is 0 here.
235
00:14:33,600 --> 00:14:35,910
The y derivative in x.
236
00:14:35,910 --> 00:14:37,790
And the x derivative is y.
237
00:14:41,570 --> 00:14:45,950
I've found the 3 by 3
matrix with the nine
238
00:14:45,950 --> 00:14:48,800
partial first derivatives.
239
00:14:48,800 --> 00:14:49,540
OK.
240
00:14:49,540 --> 00:14:54,410
It's the eigenvalues of
that matrix at these points
241
00:14:54,410 --> 00:14:56,460
that decide stability.
242
00:14:56,460 --> 00:14:58,280
So I write that down.
243
00:14:58,280 --> 00:15:05,782
Eigenvalues of J at
the critical points x,
244
00:15:05,782 --> 00:15:09,230
y, z that's what I need.
245
00:15:09,230 --> 00:15:11,270
That's what decides stability.
246
00:15:11,270 --> 00:15:19,030
Let me just take the
first critical point.
247
00:15:19,030 --> 00:15:20,440
What is my matrix?
248
00:15:20,440 --> 00:15:24,490
I have to figure out what
is the matrix at that point?
249
00:15:24,490 --> 00:15:27,160
And I'll just take 1, 0, 0.
250
00:15:27,160 --> 00:15:28,600
1, 0, 0.
251
00:15:28,600 --> 00:15:37,050
If x is 1-- so I'm getting,
this is at the point x equal 1.
252
00:15:37,050 --> 00:15:41,090
y and z are 0.
253
00:15:41,090 --> 00:15:45,900
So if x is 1, then that
that's a minus 2 and a 1.
254
00:15:45,900 --> 00:15:47,830
And I think
everything else is 0.
255
00:15:52,070 --> 00:15:56,030
So it'll be the eigenvalues
of that matrix that
256
00:15:56,030 --> 00:16:03,800
decide the stability 1,
0, 0 of that fixed point.
257
00:16:03,800 --> 00:16:08,930
And remember, that's the
toss around the narrow axis.
258
00:16:08,930 --> 00:16:14,810
That's the toss
around the short axis.
259
00:16:14,810 --> 00:16:16,010
OK.
260
00:16:16,010 --> 00:16:19,110
What about the eigenvalues
of that matrix?
261
00:16:19,110 --> 00:16:24,420
Well, I can see here that
really it's three by three.
262
00:16:24,420 --> 00:16:26,930
But really, with
all those 0s, that
263
00:16:26,930 --> 00:16:29,700
gives me an eigenvalues of 0.
264
00:16:29,700 --> 00:16:33,290
So I'm going to have an
eigenvalue of 0 here.
265
00:16:33,290 --> 00:16:35,270
And then I'm going
to have eigenvalues
266
00:16:35,270 --> 00:16:40,310
from the part of that
matrix, which is two by two.
267
00:16:40,310 --> 00:16:43,560
So I'll have a
lambda equals 0 here.
268
00:16:43,560 --> 00:16:46,230
And two eigenvalues from here.
269
00:16:46,230 --> 00:16:51,290
And I look at that,
and what do I see?
270
00:16:51,290 --> 00:16:54,740
Now this is a two
by two problem.
271
00:16:54,740 --> 00:16:59,550
I see the trace is 0.
272
00:16:59,550 --> 00:17:00,420
0 plus 0.
273
00:17:00,420 --> 00:17:03,650
My eigenvalues are a
plus and minus pair
274
00:17:03,650 --> 00:17:05,859
because they add to 0.
275
00:17:05,859 --> 00:17:08,339
They multiply to
give the determinant.
276
00:17:08,339 --> 00:17:12,530
The determinant of
that matrix is 2.
277
00:17:12,530 --> 00:17:15,319
The determinant of
that matrix is 2.
278
00:17:15,319 --> 00:17:16,069
OK.
279
00:17:16,069 --> 00:17:18,470
So it has a positive
determinant.
280
00:17:18,470 --> 00:17:20,359
That's good for stability.
281
00:17:20,359 --> 00:17:22,680
But the trace is only 0.
282
00:17:22,680 --> 00:17:24,079
It's not quite negative.
283
00:17:24,079 --> 00:17:25,099
It's not positive.
284
00:17:25,099 --> 00:17:26,579
It's just at 0.
285
00:17:26,579 --> 00:17:30,500
So this is going to be a
case of neutral stability.
286
00:17:30,500 --> 00:17:36,440
The eigenvalues will be-- I'll
have a 0 eigenvalue from there.
287
00:17:36,440 --> 00:17:40,130
The eigenvalues from this
two by two will be-- there'll
288
00:17:40,130 --> 00:17:45,220
be a square root of
2 times i and a minus
289
00:17:45,220 --> 00:17:47,610
the square root of 2 times i.
290
00:17:47,610 --> 00:17:50,220
I think those are
the eigenvalues.
291
00:17:50,220 --> 00:17:54,760
And what I see there is
they're all imaginary.
292
00:17:54,760 --> 00:17:57,040
This is a pure oscillation.
293
00:17:57,040 --> 00:17:59,320
The wobbling keeps wobbling.
294
00:17:59,320 --> 00:18:00,570
Doesn't get worse.
295
00:18:00,570 --> 00:18:02,360
Doesn't go away.
296
00:18:02,360 --> 00:18:06,360
It's neutral stability
at this point.
297
00:18:06,360 --> 00:18:12,240
So neutral stability is what
we hopefully will see again.
298
00:18:12,240 --> 00:18:13,270
Yes.
299
00:18:13,270 --> 00:18:18,820
And I think, also, if I
flip on the long axis.
300
00:18:18,820 --> 00:18:19,320
Good.
301
00:18:19,320 --> 00:18:22,600
Did you see that
brilliant throw?
302
00:18:22,600 --> 00:18:24,300
It's neutral stability.
303
00:18:24,300 --> 00:18:30,310
It came back without
doing anything too bad.
304
00:18:30,310 --> 00:18:38,240
And I finally have to do the
axis that we're all intensely
305
00:18:38,240 --> 00:18:42,170
waiting for, the middle axis.
306
00:18:42,170 --> 00:18:45,960
And the middle axis is when
the book starts tumbling,
307
00:18:45,960 --> 00:18:48,590
and it's going to be a question
of whether I can catch it
308
00:18:48,590 --> 00:18:49,460
or not.
309
00:18:49,460 --> 00:18:50,610
May I try?
310
00:18:50,610 --> 00:18:55,440
And then may I find-- what am I
expecting on the neutral axis?
311
00:18:55,440 --> 00:18:57,920
I'm expecting instability.
312
00:18:57,920 --> 00:18:59,980
I think actually it
will be a saddle point.
313
00:18:59,980 --> 00:19:03,820
But there'll be a
positive eigenvalues.
314
00:19:03,820 --> 00:19:05,660
There will be a
positive eigenvalue.
315
00:19:05,660 --> 00:19:10,770
And it is responsible for the
tumbling, the wild tumbling
316
00:19:10,770 --> 00:19:12,280
that you will see.
317
00:19:12,280 --> 00:19:16,420
And it's connected
with the point staying
318
00:19:16,420 --> 00:19:20,670
on this hyperbola that
wonders away from-- so it's
319
00:19:20,670 --> 00:19:22,920
this one now that I'm doing.
320
00:19:22,920 --> 00:19:27,440
This guy is the-- I'll put
a box around-- a double box
321
00:19:27,440 --> 00:19:28,490
around it.
322
00:19:28,490 --> 00:19:34,590
That's the unstable one, which
I'm about to demonstrate.
323
00:19:34,590 --> 00:19:36,020
Ready?
324
00:19:36,020 --> 00:19:38,771
OK.
325
00:19:38,771 --> 00:19:39,270
Whoops.
326
00:19:39,270 --> 00:19:39,769
OK.
327
00:19:39,769 --> 00:19:42,130
It took two hands to catch it.
328
00:19:42,130 --> 00:19:44,820
Let me try it again.
329
00:19:44,820 --> 00:19:52,420
The point is it starts tumbling,
and it goes in all directions.
330
00:19:52,420 --> 00:19:56,860
It's like a football, a
really badly thrown football.
331
00:19:56,860 --> 00:20:05,010
It's like a football being
thrown that goes end to end.
332
00:20:05,010 --> 00:20:09,260
The whole flight breaks
up, and the ball is a mess.
333
00:20:09,260 --> 00:20:11,660
Catching it is ridiculous.
334
00:20:11,660 --> 00:20:14,590
And I'm doing it with a book.
335
00:20:14,590 --> 00:20:15,230
Yes.
336
00:20:15,230 --> 00:20:19,520
You saw that by
watching really closely.
337
00:20:19,520 --> 00:20:21,480
OK.
338
00:20:21,480 --> 00:20:23,140
Better if you do it.
339
00:20:23,140 --> 00:20:27,360
I'll end with the
eigenvalues at this point.
340
00:20:27,360 --> 00:20:29,400
So the eigenvalues
at that point--
341
00:20:29,400 --> 00:20:32,330
can I just erase my matrix?
342
00:20:32,330 --> 00:20:36,570
So this was a neutrally
stable one, a center
343
00:20:36,570 --> 00:20:38,810
in the language of stability.
344
00:20:38,810 --> 00:20:42,430
That's a center which you just
go around and round and round.
345
00:20:42,430 --> 00:20:47,640
But now I'm going to just take
x and z to be 0 and y to be 1.
346
00:20:47,640 --> 00:20:52,240
So can I erase that
matrix and take--
347
00:20:52,240 --> 00:20:57,890
If x and z are 0, and y is
1-- so I get a 1 down here.
348
00:20:57,890 --> 00:21:00,040
And I get a 1 up there.
349
00:21:00,040 --> 00:21:01,280
And nothing else.
350
00:21:01,280 --> 00:21:02,540
Everything else is 0.
351
00:21:05,810 --> 00:21:08,470
OK.
352
00:21:08,470 --> 00:21:10,240
That's my three by three matrix.
353
00:21:10,240 --> 00:21:12,300
What are its eigenvalues?
354
00:21:12,300 --> 00:21:16,240
What are the eigenvalues
of that three by three very
355
00:21:16,240 --> 00:21:18,010
special matrix?
356
00:21:18,010 --> 00:21:25,860
This is now the-- this was
the first derivative matrix,
357
00:21:25,860 --> 00:21:29,950
the Jacobian matrix, at
this point, corresponding
358
00:21:29,950 --> 00:21:31,670
to the middle axis.
359
00:21:31,670 --> 00:21:32,420
OK.
360
00:21:32,420 --> 00:21:40,810
Again, I'm seeing some 0s.
361
00:21:40,810 --> 00:21:47,150
I'll reduce this to that two
by two matrix and this matrix.
362
00:21:47,150 --> 00:21:52,190
Really, I have this two
by two matrix in the xz,
363
00:21:52,190 --> 00:21:54,590
and this one in the y.
364
00:21:54,590 --> 00:21:56,400
How about that guy?
365
00:21:56,400 --> 00:22:01,710
You recognize what we're
looking at with this matrix.
366
00:22:01,710 --> 00:22:08,830
So with that matrix, I can
tell you the eigenvalues.
367
00:22:08,830 --> 00:22:10,820
We can see the trace is 0.
368
00:22:10,820 --> 00:22:13,480
The eigenvalues add to 0.
369
00:22:13,480 --> 00:22:15,560
They multiply to
the determinant.
370
00:22:15,560 --> 00:22:18,870
And the determinant is minus 1.
371
00:22:18,870 --> 00:22:22,640
So the eigenvalues
here are 1 and minus 1.
372
00:22:22,640 --> 00:22:25,140
And then this guy gives 0.
373
00:22:25,140 --> 00:22:30,250
And it's that eigenvalue
of 1 that's unstable.
374
00:22:30,250 --> 00:22:32,980
That eigenvalue
of 1 is unstable.
375
00:22:32,980 --> 00:22:33,660
OK.
376
00:22:33,660 --> 00:22:38,270
So mathematics shows
what the experiment
377
00:22:38,270 --> 00:22:44,430
shows: an unstable
rotation tumbling
378
00:22:44,430 --> 00:22:47,150
around that middle axis.
379
00:22:47,150 --> 00:22:48,950
Thank you.