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So now let's get to the numerical results.

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And since we want to compare this with the analytical results, let's first calculate the energy corresponding

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to this term.

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So this will be two of a five oops.

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Sorry typo.

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Two over five times.

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I'm.

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Times Radius Square.

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And then because it's the energy, we also have to multiply with one half and all mega square.

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So the result will be zero point two and this will be something we are now trying to restore using numerical

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methods.

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So here we once again will need something like a number of points.

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And the idea that I'm using here is I'm going to create basically a cube consisting out of equidistant

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points.

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And then I'm going to check if the points that describe the cube are inside the sphere are nuts.

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And if they are not inside the sphere, then they will be dropped.

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And only if they are inside a sphere, they will be considered and will contribute to the integral.

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So here you have to think a bit in advance and think of a reasonable approach.

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So of course, you could try to do it in a similar way, as we did for the line where we have created

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some list or some array of points with these explicit commands.

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And this is probably possible, but it's much faster and much easier to just to use some loops.

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So considering loops is sometimes considered bad fashion, this is because using loops can be very,

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very slow.

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But I think here it's totally reasonable because the whole calculation will not take such a long time

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anyway.

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And also, even if we would just work with arrays, we would still have to check.

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Every element wants to see if the element is inside the sphere.

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Not so.

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I think it's totally fine to go ahead with loops.

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So what we are going to need first is a MBTI or an empty coordinate list.

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So this is where we will store our coordinates.

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So basically, we will create a list or an array as we did before by using ample little space.

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But this time we will not use line space, but we will go through loops and add the points manually,

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basically.

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So we are going to loop and you could use different type of loops, but I'm going to use here a for

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loop for X.

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And now we must provide an array or a list for the values over which we want to.

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Loops are right for X and MP Lyn Space and I go from minus R to R, which is actually the capital R

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here.

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The actual radius.

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And now I have to provide how many points we want to use, so I use these no points, of course.

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And then we do something here, but I don't only want to loop over the X coordinates.

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I also want to loop over the Y coordinates, which have the same limits and also over the Z coordinates,

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which also have the same limits.

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And now we must use an if statement.

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So if here is not a condition, the condition will be if the if the point is inside the sphere or not,

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we can check this by writing and p dot within towards the North.

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So basically the length of the vector x y z.

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So this would be something like, yeah, square root of x squared plus y square to square.

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So it's the distance of this point that we are currently considering in the loop with respect to the

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center of the sphere.

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And if this one is smaller or equal to the radius of the sphere down, it's inside and this will be

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good because then we are going to consider this point.

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So now we are going to write what happens then and what happens is we are adding this point to the coordinate

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list by writing quote list dot append and we are adding the point X, Y and Z.

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So let me run this, but actually both of these lines.

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And let me look at the output code list is a list of three coordinates.

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So X, Y Z and all of these points are inside of the sphere.

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So there are many of these points.

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So this is, of course, not the only thing that we want to have.

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I would say we would also add a counter to see how many points do we have because we will need this

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to calculate the total mass of the object later on.

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And so every time this is true, we increase the counter by one.

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And then also we will look at the actual contribution to the integral.

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So the contribution will be

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will be added up every time this is true.

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So we add up, so we increase it basically contribution as a contribution plus and now we are basically

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adding up the value our square.

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But please don't be confused here.

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This is again, this are here, which is the distance of the point that we are considering to the z

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axis.

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So this is really important.

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And P Dodds, Lynn Arc Dot know.

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Of X y coma zero.

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Because when you see X, this is our rotation axis.

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Then the distance will just be square root of x squared plus y square and the Z component doesn't play

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a role at all.

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And the equation we had are square, so we right here also squared.

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So here is oh yeah, like this.

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Now the arrows gone.

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And just to comment, this one here is actually just for plotting.

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It's not really needed.

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And of course, the same thing goes for this command here.

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So if you don't want to plot anything, then you don't need these two commands.

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And actually, it's pretty good to actually not use them because they are the most demanding things

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because it will always append these points to the list.

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But here we want to have a nice and beautiful result.

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So of course, I'm going to save them and I'm going to use them to plot the sphere later on so we can

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look at our results counter out of all the thirty times, thirty times.

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Thirty points, which are twenty seven thousand twelve thousand seven hundred twelve, are inside the

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sphere.

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And what I want to do next is I want to generate the coordinates list for the plot, and maybe you remember

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this from the crash course.

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We do not need to have the syntax as we have it right now, where we have for every point such a bracket

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of X, Y and Z.

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But we must have three individual arrays corresponding to all the X coordinates, all the Y coordinates

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and all the Z coordinates.

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So we have to restructure this array and we can do this by writing NP Dot transpose.

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So this leaves the data the same, but just reorder them and changes the brackets.

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So basically, if you think of it as a matrix, it transposes the matrix.

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So you see, now we have updated this, so we have overwritten the old coordinates list.

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And now the coordinates list looks like this.

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We have now a three individual arrays in this area.

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And here we have all the X coordinates.

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Here we have all the Y coordinates and here we have all the Z coordinates.

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So we can, for example, write this and then this will just be the array of all the X coordinates.

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And this is actually exactly what we are going to need next.

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We are going to make a three day plot by writing pelted ot axes.

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And as always, when we do three plots, we have to first enable the 3D option and we have to give it

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a name Pulte 3D.

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And now we can use this to make the plots.

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So we writes Pulte 3D.

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Can I use your eyes, get a 3D plot?

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Now we must provide the three individual lists x coordinate, y coordinate and do z coordinate.

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And I think this should work already.

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Yeah, so here is our sphere.

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So of course, we can now add labels for this.

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I will scroll up because we have done it already so we can just copy the code from here, for example.

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So we go here and we add to the coordinates, which in this case will be the actual Z coordinate.

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So like this, and since this looks like just blue blob, I want to make the individual points a bit

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smaller, which you can do by writing as is equal to some number.

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Oh yeah, that looks better.

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Or maybe even smaller.

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No, I wanted zero for one.

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Yes.

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This looks really good.

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So you see, now we started from an array or basically we looped over a cube.

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And we have considered only the points that are inside the sphere, whose radius is smaller than the

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radius of the sphere.

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And so all of the points that we are considering are these that are plotted here.

131
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And the contribution is already added up.

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00:10:42,090 --> 00:10:48,510
We have added up always the distance to the Z X squared, which is exactly what we need for the equation

133
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because the equation was, um, let me scroll up.

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We have this in the very beginning.

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00:10:56,250 --> 00:10:59,340
I think, yeah, here it is.

136
00:11:00,120 --> 00:11:07,860
So the equation was the rotational energy is I of a two omega square with IBM, and this is the contribution

137
00:11:07,860 --> 00:11:16,050
that we have just calculated, which means we have to now multiply this with the total mass and we have

138
00:11:16,050 --> 00:11:20,100
to then, you know, also consider these other terms for the rotational energy.

139
00:11:21,150 --> 00:11:23,400
So let's scroll down and do it.

140
00:11:24,870 --> 00:11:32,370
So as I just said, what we have to do is we have to multiply with the mass.

141
00:11:33,060 --> 00:11:40,650
And for this, we will use the total mass divided by the number of points that we have considered and

142
00:11:40,920 --> 00:11:43,290
then we have 40.

143
00:11:44,480 --> 00:11:46,380
You have to have the contribution.

144
00:11:49,010 --> 00:11:55,220
So this would be the eye, but simply want to have the energy to kinetic energy.

145
00:11:55,610 --> 00:12:00,980
We have to divide by two and have to multiply by omega square.

146
00:12:03,060 --> 00:12:08,690
And the result is 0.1 99, something which is basically.

147
00:12:09,910 --> 00:12:17,080
What we have from the analytical result, which was so difficult and so long to calculate, and here

148
00:12:17,080 --> 00:12:23,350
it was actually pretty simple, we just had to use three four loops to loop over a cube.

149
00:12:23,770 --> 00:12:28,120
And then we had to think about the condition, which was just a radius condition.

150
00:12:28,120 --> 00:12:34,360
And then we just add up the contributions and in the end normalized the contribution to get the correct

151
00:12:34,360 --> 00:12:34,870
results.