A differential geometry question from continuum mechanics

[wrobel]

Consider the flow of a continuous medium with a smooth velocity field $\boldsymbol v(x)$ in $\mathbb{R}^3\ni x$. The density of the medium is $\rho(x)$. Here $\mathbb{R}^3$ is equipped with the standard inner product $\delta_{ij}$.

By definition, the momentum of a material volume $D$ (where $D$ is a bounded domain in $\mathbb{R}^3$) is given by the formula:
$$\boldsymbol P=\int_D\boldsymbol v\mu, \qquad(1)$$
where $\mu=\rho\sqrt g dx^1\wedge dx^2\wedge dx^3$ is a differential form (the infinitesimal mass).

And what about formula (1) if we replace $\mathbb{R}^3$ with some other Riemannian manifold with non-zero curvature?
In this case, formula (1) becomes senseless, and I have no idea how to fix it—or if it is even possible.
Any opinions on this?

 

[QuarkyMeson]

On a curved manifold, velocities at different points belong to different tangent spaces, so they cannot be added directly. You need a chosen way to transport vectors or a symmetry of the manifold to define a total momentum.

So I don't think it's generally possible, but you might be able to rescue (1) in some cases.

Side note, even when a manifold has no symmetries and momentum is lost, ideal flows still have conserved quantities of a different kind, they're just now topological ones. This would be bleed into some of the braiding stuff I've been reading, but it's very tangential to this.

 

[Matterwave]

And what about formula (1) if we replace $\mathbb{R}^3$ with some other Riemannian manifold with non-zero curvature?
In this case, formula (1) becomes senseless, and I have no idea how to fix it—or if it is even possible.
Any opinions on this?

I am not aware of a way to in general fix this integral in the sense that there is a unique 1:1 correspondence between this integral and a "fixed version" that works in general Riemannian manifolds. And so, I might not have too many intelligent things to say about this. So take the following with a huge grain of salt.

One thing I might try, could be to turn the vector into a 1-form using the metric $g(v,\,\,)$ and then contracting with some Killing Field $\xi$ to perhaps obtain some conserved aggregate quantity. You'd get an integral like:

$$ I(\xi) = \int_D g(v, \xi)\mu$$

Your form of $\mu$ seems to me to work on a general manifold. This integral that I constructed is going to come out with just number though and not a vector.

Trying this integral from a transport-related view seems gnarly to me since the transport would be path dependent.

 

[wrobel]

I feel that if we want to generalize the equations of a continuous medium's motion to an arbitrary Riemannian manifold, then the only way is to start with the Lagrange–d'Alembert principle or with some other variational-like argument. The naive approach that I tried above and that is employed in most textbooks for $\mathbb{R}^3$ is, in general, hopeless.

 

[martinbn]

Why is 1 meaningless in the Riemannian case? Can you not take $\mu$ to be $\rho$ times a chosen volume form?

 

[wrobel]

Why is 1 meaningless in the Riemannian case?

On a curved manifold, velocities at different points belong to different tangent spaces, so they cannot be added directly.

 

[martinbn]

Oh, I see, it is the integral of the velocity.

 

[jbergman]

Well since you already added the restriction of a bounded domain $D$, then I don't see the restriction of adding the restriction of requiring the neighborhood to be parellelizable as problematic. Then, yes, you can define one using the vector fields basis.

It probably doesn't make sense anyways beyond that, physically.

 

[berkeman]

Thread closed temporarily for Moderation and AI posting check...

 

[berkeman]

After some cleanup, thread is reopened.

 

[Matterwave]

Well since you already added the restriction of a bounded domain $D$, then I don't see the restriction of adding the restriction of requiring the neighborhood to be parellelizable as problematic.

Can you define what it means for a neighborhood to be parallelizable?

 

[wrobel]

It probably doesn't make sense anyways beyond that, physically.

"Not making physical sense" and "not making geometrical sense" are in fact two sides of the same coin

 

[jbergman]

Can you define what it means for a neighborhood to be parallelizable?

It just means you can form a basis of linear independent non-zero vector fields such that at each point the vector fields form a full basis for the tangent space.

It's more general than a coordinate patch for example with the circle you have a canonical vector fields tangent to it but you can't cover it with a single patch.

 

[jbergman]

"Not making physical sense" and "not making geometrical sense" are in fact two sides of the same coin

I have to think about this more. The sphere is an interesting case.

 

[Matterwave]

It just means you can form a basis of linear independent non-zero vector fields such that at each point the vector fields form a full basis for the tangent space.

Ah got it, I was not familiar with this terminology.

It's more general than a coordinate patch for example with the circle you have a canonical vector fields tangent to it but you can't cover it with a single patch.

Hadn't realized this. Interesting point.

Given the basis exists on $D$, how does that define the integral in post #1? Make some arbitrary set of basis vector fields preferred, contract and do the integral component wise? This would seem quite arbitrary.

 

[jbergman]

Given the basis exists on $D$, how does that define the integral in post #1? Make some arbitrary set of basis vector fields preferred, contract and do the integral component wise? This would seem quite arbitrary.

Yes, I guess that's true. Now, I think you need some preferred basis of some kind.

 

[WWGD]

On a curved manifold, velocities at different points belong to different tangent spaces, so they cannot be added directly. You need a chosen way to transport vectors or a symmetry of the manifold to define a total momentum.

So I don't think it's generally possible, but you might be able to rescue (1) in some cases.

Side note, even when a manifold has no symmetries and momentum is lost, ideal flows still have conserved quantities of a different kind, they're just now topological ones. This would be bleed into some of the braiding stuff I've been reading, but it's very tangential to this.

Isn't a Connection precisely the instruction you need to " glue" tangent spaces at different points together; a choice of vector space isomorphism at said points?

 

[Matterwave]

Isn't a Connection precisely the instruction you need to " glue" tangent spaces at different points together; a choice of vector space isomorphism at said points?

Yes, but that choice is path dependent. Any two tangent spaces at $p_1,\, p_2$ can be made isomorphic via the connection if you specify a path between them. But this is a pairwise relation so good luck constructing the integral from the OP using that... Probably someone's thought of some way though...

This is what I meant in my final comment on post #3 (transport-related view).

 

[Matterwave]

Yes, I guess that's true. Now, I think you need some preferred basis of some kind.

Yeah, I think in general you need some vector field to "ground" you as you move around the manifold. In my post #3 I suggested using a Killing field. But of course that assumes a Killing field exists.

If no Killing field exists, you wouldn't get a conserved momentum anyways though. So that $P$ vector might not be very useful in such a space.