Why is the angular momentum conserved?

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We are always bombarded with this line "Law of conservation of angular momentum". For Example: Why does the earth speed up when it is near the sun (during winters in the northern hemisphere)? Or if you start spinning (like a ballet dancer) with your arms spread out, you can increase your angular velocity by just hugging yourself while spinning, why?

Or why is there a second rotor at the tail of a helicopter, I mean we all know what the big one is doing but what is that small one doing on the rear? And on and on and on... All these phenomena and many more have the same answer, law of conservation of angular momentum... But where is the intuition? Why is it conserved? And why am I even asking this question?

Let me first clear the meaning of the question.

It isn't about why angular momentum should be conserved; rather, it is about why any law should tell a System how to behave or evolve. After all, nature should not care about "a law". These kinds of misconceptions and confusion can become quite frustrating when you try to think about the conservation of angular momentum intuitively. Let me give you an example of how this law is used in reasoning conventionally. By using an animation given below:

As you can see in the video, a body is moving in a circular orbit around a source of central force (like a planet revolving around the sun). And as the body comes closer to the centre (say if you pull it in), its angular velocity starts increasing. Why does this happen? The famous and conventional answer is the Law Of Conservation Of Angular Momentum.

The value of angular momentum (L) for a body of mass "m" and velocity "v", moving at a perpendicular distance "r" (radius of the circular path) from the centre of the force is given as:

So according to the law of conservation of angular momentum, this quantity is a constant of motion. So as the radial distance from the centre decreases the value of velocity increases to keep the value of L constant. Basically, as radius goes down, velocity goes up and vice-versa. I used the same logic to generate the above animation.

So you see everything is explained. You may be satisfied with this explanation, but I am not. You see I wanna know how does the body which is revolving around the centre "know" that it has to maintain this value, it couldn't care less. How does all the atoms and molecules of the universe know that if I pull them inward they have to speed up? Is there a mysterious physics government which is making them do it?

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This confusion has the origin in the way the laws of physics are introduced to us in our basic physics classes. There seems to be a misconception that laws are like something encoded into fabric of cosmos which are guiding matter around, but in fact, it is the other way around. Let me show you what I mean by that, by using the same example we discussed above (a body moving around a source of a central force).

Fig.1) Body moving in circular path

Figure 1 shows the situation when the body is moving in a perfectly circular path around the centre. In a circular path, the direction of the velocity of the body (which is just tangent to the path) is always perpendicular to the direction of force (which is only the radial direction in case of a central force (towards the centre)). Because the velocity and force are perpendicular to each other, there is no component of velocity vector along the line of action of the force ( cos(90) = 0 ), and thus the force can only change the direction of the velocity vector, not increase/decrease its magnitude. And that is what happens in a circular motion, the magnitude of velocity remains constant.

Fig.2) Moving Towards the centre

Now, what will happen if I pull the body inwards?

Figure 2 shows the situation when you try to pull the body towards the centre. As you can see the angle between the velocity vector and the Force vector is no longer 90 degrees (in fact it is less than 90). This results in a velocity component (v cos(theta)) towards the centre, along the line of action of the force. Now, this attractive force can increase the velocity because of that component. And thus it has an effect of increasing angular velocity as it moves more towards the centre.

Fig.3) Moving away from the centre

What about moving away from the centre?

Well, this also has the same effect that the angle between the velocity vector and force is no longer 90 degrees (as shown in Figure 3), it is more than 90 degrees. As a result there exists a component of velocity along the line of action of attractive force but now it is in the opposite direction. This causes a decrease in the magnitude of the velocity and which has an effect of slowing down as you move away from the centre.

So that is why it is hard to hug yourself when you are spinning in comparison to when you not. You are trying to accelerate your arms when you try to hug yourself while spinning and that requires force.

So in conclusion, it is not the law which is telling the system to behave in a particular way, in fact, it is the dynamics of the system itself which are showing a pattern that the multiplication of certain quantities (mass, velocity and perpendicular distance from centre) remains constant over time.

Whatever we have discussed above has not happened "because" of the law of conservation of angular momentum, nothing ever happens "because" of physical laws, instead, physical laws are the patterns we discover in the dynamics of the reality.

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