Confused about something

[Johnny5]

Suppose that I have two particles in vacuum, and there is only one interaction force between them, and that force is a central force.

 

Assume that at the beginning of time, the particles are a distance R apart, but are being pulled together by this one superforce S. How can I get the particles to orbit one another. Won't they just head right for the center of mass of the universe, collide, and then reflect or something in linear motion rather than circular motion?

 

Put another way, exactly how does a central force end up making things orbit the center of mass.

 

Regards

[ensonik]

Aren't orbits due to the interplay between multiple forces? Your example cites only one.

[jcarlson]

Another force besides the attraction between the two particles would be needed to accelarate one particle in a direction tangent to an ellipse around the other particle. Then theoretically, if the velocity of the orbiting object is the correct magnitude and direction in relation to the distance between the two objects at a given point, they should start orbiting another.

[Johnny5]

Another force besides the attraction between the two particles would be needed to accelarate one particle in a direction tangent to an ellipse around the other particle. Then theoretically, if the velocity of the orbiting object is the correct magnitude and direction in relation to the distance between the two objects at a given point, they should start orbiting another.

 

Yeah I thought so, so let me ask you this. Suppose I am a source which is going to "push" the thing in the direction which is tangent to the elliptical orbit this thing finally takes on. I would have to "push" perfectly to get the orbit to be elliptical. Not enough push would mean the things spiral together, and too much would mean they sail away from each other. So how is it that nature managed to get things perfect? Do you know?

 

Regards

[J.C.MacSwell]

Yeah I thought so' date=' so let me ask you this. Suppose I am a source which is going to "push" the thing in the direction which is tangent to the elliptical orbit this thing finally takes on. I would have to "push" perfectly to get the orbit to be elliptical. Not enough push would mean the things spiral together, and too much would mean they sail away from each other. So how is it that nature managed to get things perfect? Do you know?

 

Regards[/quote']

 

Isn't it the inverse squared laws which lead to stable orbits?

[Johnny5]

Isn't it the inverse squared laws which lead to stable orbits?

 

I've never actually proven this in enough detail. I've read other people's proofs but never attempted to construct my own, maybe now is a good time to do so.

 

When you say stable orbits, I presume you mean ellipses (circle also an ellipse). But also, inverse R^2 force can lead to hyperbolic orbits (not stable).

 

So first I guess you have to prove that only inverse R^2 forces can lead to conic sections, so that only inverse R^2 forces can lead to the stable ellipse orbits. I have Feynman's gravity lecture on tape, and he uses some very arcane properties of ellipses known to the ancient Greeks, but I couldn't even follow his entire proof, because he fudged in at least one area (used something without proof). This type of proof avoids using analytic geometry, and avoids using vector calculus, and so is the most intuitive type of proof, because you can visualize the motion so well.

 

Let me ask you something whose answer should be simple... what is the quickest way you know of to prove that only inverse square forces lead to ellipses?

 

Thank you very much again.

 

Regards