How Do Geometrical and Physical Symmetry Differ in Objects?

[issacnewton]

Hello

I was reading some article on angular momentum.
And at some point, the author started talking about the symmetric objects and axis of symmetry. Now I am wondering if the author means the geometrical symmetry or the symmetry in physics. For an example, if we take a uniform rod of length L, then the center of mass lies at the geometric center and the geometric axis of symmetry is same as the physical axis of symmetry. But if the mass is not distributed uniform on the length of the rod, then the center of mass is not at the geometric center and geometric axis of symmetry is not same as the physical axis of symmetry. Similar argument can be applied to any other geometrical symmetric object which might have mass evenly/unevenly distributed. I am little confused here. Wikepedia's article on symmetry probably means geometric/mathematical symmetry. Can somebody please elaborate ?

 

[FactChecker]

But if the mass is not distributed uniform on the length of the rod, then the center of mass is not at the geometric center

If the mass is distributed symmetrically, then the center of mass will still be at the geometric center. The mass distribution does not have to be uniform.

and geometric axis of symmetry is not same as the physical axis of symmetry.

IMO, in this context, the terms are considered to have the same meaning.

PS. There is a very profound theory in advanced physics which associates mathematical fixed points under a mathematical transformation with conserved physical properties. (see https://en.wikipedia.org/wiki/Symmetry_(physics) ). The theory was developed by Emmy Noether (see https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math ).

 

[issacnewton]

Factchecker, there are situations where non uniform can lead to the same center of mass as the geometric center, I agree. But what happens when mass density for a rod increases linearly from left end to the right end ? Here we have the center of mass slightly to the right of the geometric center. So what will be considered the axis of symmetry for the consideration of the angular momentum ?

 

[fresh_42]

The author assumes homogeneous masses in the context of the article. However, if you assume not uniformly distributed masses, you will just get other, more complicated symmetries, e.g. a rotation on an ellipsoid instead of a circular. This would make the calculations much more difficult and distract from what his point is, namely to demonstrate some fundamental dependencies. So in the end you won't get rid of symmetries, which are equally mathematical aka geometric and physical. The equal meaning still holds, only the kind of symmetries will change - both: geometric as physical.

The reason for your question is, that you assume the geometric object to be uniform and homogeneous without questioning, however, doubt it for the physical world. The latter is a correct observation, but the former an assumption made only in your mind. You can still model a disk with an inhomogeneous distribution of matter, it's just not the one you can easily draw and the resulting new symmetries will be much more complicated as will be the mathematical model. However, an object rotating around a shifted center of mass still has symmetries.

 

[FactChecker]

Factchecker, there are situations where non uniform can lead to the same center of mass as the geometric center, I agree. But what happens when mass density for a rod increases linearly from left end to the right end ?

Then the mass is not distributed symetrically. If the mass is distributed symetrically (mass(x) = mass(-x)), then the center of mass is still at the geometric center (x=0). A uniform distribution of mass is not necessary.

Consider a bar-bell shape (or a hollow sphere). The mass is not uniform. There is more mass toward the ends (or outside). Yet the center of mass is at the geometric center.

 

[issacnewton]

So when the mass density of a rod increases linearly from left to right, would the axis of symmetry pass through the center of mass or the geometric center ?

 

[FactChecker]

So when the mass density of a rod increases linearly from left to right, would the axis of symmetry pass through the center of mass or the geometric center ?

I agree that they are different. But you are using the term "geometric symmetry" loosely. Symmetry of what? In a physics context, I would assume it is geometric symmetry of the mass distribution.

 

[issacnewton]

When we say geometric symmetry, we usually refer to the shape of the object only. So, while talking about physical objects, should we also take mass into the consideration ?

 

[FactChecker]

When we say geometric symmetry, we usually refer to the shape of the object only. So, while talking about physical objects, should we also take mass into the consideration ?

Of course.

 

[issacnewton]

Ok, it's making sense now. So one of the axis of symmetries for this uneven rod would pass through the center of mass and perpendicular to the rod. Correct ?

 

[fresh_42]

When we say geometric symmetry, we usually refer to the shape of the object only.

So the only question is, whether a geometric object can model the physical reality or not. If not, you simply will have to adjust the geometric model. You say that you distinguish between a ball with uniform mass distribution and others. So why don't you allow this for the geometric object? It is this different hidden assumption of yours which makes the difference, not the word as such. You can consider more complex geometric structures, you just do not allow them to be.

 

[issacnewton]

fresh_42, so geometric symmetry itself has no meaning in physics. We have to look at the mass distribution to check the symmetry. Is that right ?

 

[fresh_42]

Yes, but the vast majority of balls and rods have such a basic geometric symmetry, at least in a good approximation, which is why the principles taught in the article are important. If you plan the orbit of a satellite, then a Euclidean ball will be insufficient as a model and you will have to adjust the shape, or the metric, or consider a vector field instead or whatever. They are still geometric in a sense and do have symmetries, even if not as obvious ones.

 

[issacnewton]

OK, so far things are clear. Now in the article I referred to, the author is talking about the symmetric objects. So most probably, he is assuming that the mass is symmetrically distributed (not necessarily uniformly distributed). And then he goes on to talk about 1) Symmetric Objects Rotating about an Axis of Symmetry
2) Symmetric Objects Rotating about an Axis that is Not an Axis of Symmetry

So the rod I was talking about (mass density increasing from left to right), is that a symmetric object or a non symmetric object ?
If we refer to the center of mass of this rod, then I think it would be symmetric object. But if we refer to some other point, then the rod will be non symmetric object. But the author is talking in generic terms here.

 

[fresh_42]

So the rod I was talking about (mass density increasing from left to right), is that a symmetric object or a non symmetric object ?

It is symmetric along its longitude axis and asymmetric otherwise. But if there is a (usually much heavier) mass attach to one end, this can be neglected anyway.

If we refer to the center of mass of this rod, then I think it would be symmetric object.

Just that the barycenter isn't at the center of the line anymore, however, somewhere it is.

But if we refer to some other point, then the rod will be non symmetric object. But the author is talking in generic terms here.

Yes, his investigations are for a uniform distribution of masses of a body. The symmetries are still there, e.g. for discs along its length or with a shifted barycenter, they are just not the simple ones.

 

[issacnewton]

Ok, but for the rod I was talking about, I think there are two axes of symmetry. One is the longitude axis (axis along the length of the rod I guess) and another axis which passes through the center of mass (which is different from the midpoint of the rod) and is perpendicular to the rod. Correct ?

 

[fresh_42]

I don't see why it should be perpendicular, as it is still within the rod, but generally, yes.

 

[issacnewton]

There are two axes of symmetry here. One along the length of the rod and another is perpendicular to the rod but passes through the center of mass of the rod.

 

[fresh_42]

O.k., in this case: yes.

 

[issacnewton]

Yes, its making sense now.

 

[FactChecker]

In physics, mass is central. It is very rare that the outline boundary of an object is important while mass is ignored. In the context of physics, you are probably safe to assume that "symmetric" means a symmetric distribution of mass.

 

[issacnewton]

Yes, and in the context of electrostatics, symmetric will also mean both mass and charge. Right ?

 

[FactChecker]

Use your best judgement based on the context.