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easy
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Re: Newton's shell theorem with random density

January 8th, 2022, 9:34 am

Yes, that's a fair summary.
 
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Alan
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Re: Newton's shell theorem with random density

January 8th, 2022, 1:46 pm

So, take N=1 and unit masses and G=1. That is, there is a unit mass inside the shell at r < R, using the coordinates in Fig. 2 in the first link that trackstar posted. Here it is again. There is a single mass located randomly (uniformly) on the shell.

Then, the gravitational field strength at r is a random vector [$]\vec{E}[$] with random magnitude [$]E[$]. Then [$]E[$] has a probability density [$]p(E)[$] with support on the interval [$]\frac{1}{s_{max}^2}[$] to [$]\frac{1}{s_{min}^2}[$] where [$]s_{min} = R -r[$] and [$]s_{max} = R + r[$]. You should be able to work out this density analytically from the Fig. 2 coordinates. 

If there is more than one particle, so [$]N \ge 2[$], I would just use a Monte Carlo. Make [$]N[$] draws, distribute the N particles on the surface, calculate the field at the observation point, and repeat for say 10,000 trials to get the pdf and stats.
 
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Alan
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Re: Newton's shell theorem with random density

January 8th, 2022, 2:28 pm

I will add that it seems natural to split up the pdf [$]p(E)[$] into two pieces. For each random [$]\vec{E}[$], you can project it onto the radius line from the center of the sphere through the observation point. This projection will either be "inward pointing" or "outward pointing". So there an inward p, call it [$]p_+(E)[$] and an outward [$]p_{-}(E)[$]. By defining a new scalar [$]e = \pm E[$], where you take the + for inward projections and the - for outward projections, you get a new density [$]p(e)[$] with both positive and negative support. It's the mean of this latter density that should be zero, corresponding to the Shell Theorem.   
 
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Alan
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Re: Newton's shell theorem with random density

January 9th, 2022, 3:33 pm

@easy

For a Monte Carlo, there are lots of online references about drawing points so that they are uniformly randomly distributed on the surface of a sphere. For example: Sphere Point Picking (Wolfram Mathworld)

If you go that route and calculate some gravity field pdf's, it would be interesting to see some results. 
Please post any interesting pics. 

Analytically, besides N=1, perhaps some large N behavior of low moments could also be characterized.

Interesting problem!
 
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trackstar
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Re: Newton's shell theorem with random density

January 9th, 2022, 7:51 pm

Found a short paper on this - fun approach and there a nice summary below: "This paper is a gem, exhibiting the beauty of mathematical proof while at the same time revealing the mystery behind some deep classical results." 

Newton’s Shell Theorem via Archimedes’s Hat Box and Single-Variable Calculus - Peter McGrath, UPenn (2018)
George Pólya Awards 2019 

Right now, I am watching a game involving a prolate spheroid, which is being sent off in spiraling arcs down the field, with frequent multi-player collisions.

Model that! 
 
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Alan
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Re: Newton's shell theorem with random density

January 10th, 2022, 6:26 pm

Nice find!  

I did go looking for modelling that prolate spheroid game -- and found a robot field goal kicker! (youtube)

It would be fun if someday somebody starts a sports league called the "Anything goes League", where you can have robot players as long as they can make it onto the field on their own. There would still be lots of special rules like "no taking an ax to the robot".
 
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trackstar
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Re: Newton's shell theorem with random density

January 11th, 2022, 4:28 am

That is funny. I have thought that it would be interesting to have a game where some of the rules changed after each quarter. Either randomly, or as kind of a handicapping system to give a badly trailing team a chance. So many things you could alter - the number of downs, how many men allowed on the field, going offsides is ok, coin toss on close calls with yardage instead of bringing out the chains ... it would certainly keep the offensive and defensive coordinators on edge (as if they aren't already, this time of year). : D

On the mechanics of the sport: I found a nice write up here: The Physics of Football - How Stuff WorksSome of the games this weekend were spectacular.

And now, back to shells with random density. 
 
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bearish
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Re: Newton's shell theorem with random density

January 11th, 2022, 4:36 am

I think Bill Watterson has described this game. It’s Calvinball.