Newton's law of universal gravitation

[balta06]

Hi all,

Does Newton's law of universal gravitation depend on the geometry of the manifold? For example, what happens to formulas if we take the projective plane as our universe? How can one model "the mass" on the projective plane or on torus?

I am a math grad with very elementary physics knowledge. I know almost all mathematical objects used in the physics but don't know how you use or apply them. So, I need a book on these topics. Any recommendations?

Thanks.

 

[vanesch]

Hi all,

Does Newton's law of universal gravitation depend on the geometry of the manifold? For example, what happens to formulas if we take the projective plane as our universe? How can one model "the mass" on the projective plane or on torus?

I am a math grad with very elementary physics knowledge. I know almost all mathematical objects used in the physics but don't know how you use or apply them. So, I need a book on these topics. Any recommendations?

Thanks.

I would say "mathematical methods of classical mechanics" by Arnold, but maybe you're more sophisticated than that...

 

[nicksauce]

Um... In my opinion if you want to study gravitation in non-flat spaces, you should study general relativity, not try to figure out how Newton's law works in such spaces... Maybe I'm missing something here?

 

[vanesch]

Um... In my opinion if you want to study gravitation in non-flat spaces, you should study general relativity, not try to figure out how Newton's law works in such spaces... Maybe I'm missing something here?

I would think (but maybe I'm wrong) that you could study a Newton-like gravity in non-flat spaces (but not space-times!). What turns gravity into GR is the relativistic requirement of invariance under local lorentz transformations, I thought. But if you have no such requirement, and just go for a time x (curved) space manifold, I don't see why you can't have fun with Newton-like gravity. It's not the physics of our universe, but it is probably fun mathematical physics.

 

[balta06]

I would say "mathematical methods of classical mechanics" by Arnold, but maybe you're more sophisticated than that...

Thank you, I got the book.

nicksauce, I have insufficient info to reply your message. Let me state what my aim is:

I was trying to write a spacecraft software which calculates the necessary force required to move the spacecraft on a linear curve. I assumed that all planets in the universe (number of them is finite) is stationary and all the data required (masses of the planets and spacecraft ; and, the coordinates of them in three dimensional space) is known.
When I tried to generalize this software to arbitrary manifolds, I encountered the following problems: I had to calculate the distance between two points as the length of the geodesic connecting these two points. Hence, Newton’s law had to change, because, it is highly probable that Newton’s law is a result of an integration (of potential). Hence, it’s also confusing what we mean by ‘integration’ on an arbitrary manifold.

Are you suggesting that I should study general relativity?

I also have one more question. Is there a book discussing the spherical astronomy in a more general setting? For instance, suppose that we have a manifold M with an embedded submanifold N and a people X living on the surface of N. How does X determine the locations of objects?

I know that my questions are very general but I need a direction to start.

Thank you.

 

[vanesch]

Are you suggesting that I should study general relativity?

Well, that depends on what you want to do. Real gravity in the real universe seems to behave according to general relativity. Newtonian gravity has been experimentally falsified.

But if you want to toss around with mathematical theories about how you can generalise Newtonian mechanics to curved spaces (not space-times), which has nothing to do with our universe, but which might make for fun mathematics, then there's no point in looking into general relativity.

I also have one more question. Is there a book discussing the spherical astronomy in a more general setting? For instance, suppose that we have a manifold M with an embedded submanifold N and a people X living on the surface of N. How does X determine the locations of objects?

As you are building your own toy universe here, you are free to specify the laws of your toy universe.

I have the impression that you don't see a difference between mathematical physics, and theoretical physics. In mathematical physics, you are interested in the mathematical structures that go with certain laws and theories, and in order to explore that, you are free to change settings - knowing very well that this hasn't anything to do anymore with the "real universe", but changing the settings might help you understand better the mathematical structure of a certain theory. For instance, by changing the number of dimensions, or the metric, or something else, you can hope to get a better idea of what is "essential" and what not, in a certain structure. Your aim is not to "improve upon understanding real nature", your aim is to understand the mathematical structure of certain theories - whether they are correct or not.
In theoretical physics, people rather try to guess "how nature is". They try to guess deeper laws of nature. The verdict is the experiment.

So if you want to understand the natural phenomenon of gravity, then you are more like a theoretical physicist, and you want to study general relativity (and forget about Newton's law on manifolds). You might be interested in actual measurements and observations. If you are a mathematical physicist, and you want to learn more about the mathematical structure of Newton's law, then you forget about general relativity, and you go playing with Newton's law in different settings. You don't care about observations, you want to study a mathematical structure and its variations.