# I In the list of "conservation of X due to symmetry Y" ...

1. Mar 30, 2017

### rumborak

That is, with Noether's theorem, it feels to me the conservation of angular momentum somewhat stands out because the measure is more of a derived one. That is, rotation of something is really more of an interplay between inertia of the constituent particles and the forces that hold them together. Isn't angular momentum at its core then more a measure of inertia? If that's the case, I feel when compared to the more "basic" pairs (time symmetry/energy conservation for example), angular momentum conservation feels oddly out of place.

Any insights?

2. Mar 30, 2017

### Orodruin

Staff Emeritus
No, you are thinking of the moment of inertia. Angular momentum is to moment of inertia as linear momentum is to mass. In generalised coordinates $q$, the equations of motion take the form
$$M_{ab} \dot q^b = -\partial_a V,$$
where $V$ is the potential energy of the system and $M_{ab}$ its inertia tensor. If $q$ is a position, then the inertia tensor is essentially the mass, the left-hand side the linear momentum and you recover Newton 2 with the right-hand side being the force. If $q$ is a rotation angle, then the inertia tensor is essentially the moment of inertia, the entire left-hand side is the angular momentum, and the right-hand side represents a torque.

3. Mar 30, 2017

### rumborak

Oof, I get the impression I need to read up more again about momentum

Does this correction touch my overall question though? In the sense that an object's rotation is maybe not as "inherent" as its energy content?

For example, take this thought experiment: You got two masses connected by a spring, rotating in free space. It's easy to define and calculate the angular momentum of that system. However, imagine you disengage the spring, in which case the two masses will just follow their inertial path. Now, of course mathematically, the angular momentum stays the same of the "two-mass system", but it seems more of an arbitrary choice to cluster the two masses together. In the end, it's just two unrelated masses flying their merry way, and it's your mathematical choice to cluster them and calculate a total angular momentum.
I feel that is very different from the energy of a single particle, which is "inherent" to it so to speak.

Maybe I'm attaching undue meaning to Noether's theorem. I so far understood it to indicate a deep connection between say time and energy, but maybe this is purely a mathematical connection, not so much a physical?

4. Mar 30, 2017

### Orodruin

Staff Emeritus
Why not? What do you feel is so different about it? It has exactly the same mathematical description as linear momentum in terms of generalised coordinates.

What makes you think a single particle cannot have angular momentum?

It is a deep connection between symmetries and conserved quantities. Energy is related to invariance under time translations, linear momentum to invariance under spatial translations, and angular momentum to invariance under spatial rotations. In other words:
• If a solution can be translated in time and still be a solution, its energy is conserved.
• If a solution can be translated in space and still be a solution, its linear momentum is conserved.
• If a solution can be rotated and still be a solution, its angular momentum is conserved.

This is a matter of how you draw the boundaries of your system and what symmetries that appear in your system based on this. If the particles are not interacting, the angular momentum of each particle is also conserved as the system becomes invariant under rotations of that particles position only. If they are interacting, the typical thing would be that the system is only invariant under rotations as long as you include both particles in your system. That total angular momentum is conserved is a consequence of Noether's theorem and the symmetry of the system - not an arbitrary clumping.

5. Mar 30, 2017

### Staff: Mentor

I don't see this. The laws of physics are the same yesterday and today, so energy is conserved. They are the same here and there, so momentum is conserved. They are the same in this direction as that direction, so angular momentum is conserved. Seems like it doesn't stand out at all to me.