Galilean structure, inertial system and two bodies

[cliowa]

Say I have two bodies, idealized as points with mass, in Galilean spacetime $A^4$. When thinking about the 2-body problem (just two bodies with interaction forces in the entire universe), one usually goes from the 3-dim. to the 2-dimensional problem using some special idea. I read the following claim: For any initial conditions on those 2 bodies, one can find an inertial system such that the motion of those two bodies takes place in a fixed plane, for all time.
Why is that?

 

[StatusX]

It has to do with symmetry. The force on the particles should possesses whatever symmetries the configuration (positions and velocities) does. For example, if they are initially at rest, there is a rotational symmetry about the line connecting them, and so the force must lie on this line. Thus they will remain on the line for all time. Can you make the argument in your case?

 

[cliowa]

It has to do with symmetry. The force on the particles should possesses whatever symmetries the configuration (positions and velocities) does.

I don't think the force should really depend on the velocities (or the symmetries of the velocities, if you wish), we're dealing with simple interaction forces only. What makes you think that there is a dependence on the velocities?

For example, if they are initially at rest, there is a rotational symmetry about the line connecting them, and so the force must lie on this line.

Well, for the force to be along the line connecting the two particles their initial condition doesn't matter. That comes simply from the Galilean relativity principle, right? I am well aware of that one. I know that any force between the 2 particles can only depend on the distance of the particles, and must act along the line connecting them (there can be no time or velocity dependency).

Can you make the argument in your case?

No. Any help?
Thanks...Cliowa

 

[StatusX]

I'm talking about completely general forces. These can depend in an arbitrary way on the configuration of the system, ie, positions and velocities of all particles (for example, this would apply to the Lorentz force, which is velocity dependent). It might be true that the force depends only on position, or that it is directed along the line between the particles, but it isn't necessary to assume this for either my example or the thing you're meant to prove.

The only assumption is that whatever symmetries the configuration has, the forces must also have. Because if there was some transformation that left the system the same as it was but changed the forces, you'd have two identical configurations to which you're assigning different forces.

In my example, if the particles are at rest (or more generally, moving along the line connecting them), the system can be rotated around the line connecting them without changing anything. Unless the forces are also directed along this line, such a rotation would change them, which isn't allowed.

As a hint, the relevant symmetry to your problem will be reflection through the plane the particles are moving in.

 

[cliowa]

I'm talking about completely general forces. These can depend in an arbitrary way on the configuration of the system, ie, positions and velocities of all particles (for example, this would apply to the Lorentz force, which is velocity dependent).

Sorry for the misunderstanding, but I was talking in the context of classical mechanics in Galilean spacetime with the Galilean structure. I too am talking about completely general forces, and the constraints imposed by the principle of relativity do hold in galilean spacetime, as I said that those two particles are the only ones in the entire universe.

It might be true that the force depends only on position, or that it is directed along the line between the particles, but it isn't necessary to assume this for either my example or the thing you're meant to prove.

Now that surprises me. I can't imagine that that's the case, but why don't you just post your solution and we'll talk about it?

As a hint, the relevant symmetry to your problem will be reflection through the plane the particles are moving in.

Maybe I didn't make myself clear, but that's really the point of the whole thing: I don't know that there is a system in which the particles are moving in a plane. I first have to prove that this is really the case. Do you know what I mean?
Best regards...Cliowa

 

[StatusX]

Sorry for the misunderstanding, but I was talking in the context of classical mechanics in Galilean spacetime with the Galilean structure. I too am talking about completely general forces, and the constraints imposed by the principle of relativity do hold in galilean spacetime, as I said that those two particles are the only ones in the entire universe.

What's the misunderstanding?

Now that surprises me. I can't imagine that that's the case, but why don't you just post your solution and we'll talk about it?

I did. Look at the third paragraph.

Maybe I didn't make myself clear, but that's really the point of the whole thing: I don't know that there is a system in which the particles are moving in a plane. I first have to prove that this is really the case. Do you know what I mean?

You just need to find a frame where the initial positions and velocities lie in a plane, then the symmetry will dictate that the forces are such as to keep them there.

 

[cliowa]

What's the misunderstanding?

You started talking about velocity-dependent forces, which might arise in non-closed systems. Not a big deal, let's leave that aside.

You just need to find a frame where the initial positions and velocities lie in a plane, then the symmetry will dictate that the forces are such as to keep them there.

But that may well be impossible, right? Let's go to orthonormal coordinates and say one particle has a non-vanishing initial velocity in the direction of one basis vector, and the other one in the direction of a different basis vector, and let the particles be a certain (nonzero) distance apart. Now, where's that plane you're referring to? What's the frame i have to choose?

 

[StatusX]

Just take a rest frame of one of the particles.

 

[cliowa]

Just take a rest frame of one of the particles.

Alright, now I see. Thanks a lot and best regards...Cliowa