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(This is an edited version of the original question, since I'm starting a bounty)

I'm trying to find a function $y$ from given data. Reverse optimization, so to speak.

Say we have two (two-dimensional) bodies $A$ and $B$ of known (but arbitrary) shapes at known positions relative to each other. There are forces of attraction and repulsion between the two bodies, but I only know that the bodies are in stable equilibrium. If I understand "equilibrium" correctly, that means that the total force is zero and the total energy is minimized (or is that wrong to begin with?).

Let's assume we suspect that the force between two differential elements in the bodies (one element in one body, the other one in the other body) is given by a function $y({\bf r})$. ${\bf r}$ could be the coordinates of the elements $[x_1,y_1,x_2,y_2]$ or more complicated things, but let's assume it's simply the distance between the elements for now. The total force between the two bodies would then be: $$ \int_A \int_B y(r) \: dA \: dB$$

If have a couple thousand of such equilibrium configurations, with known body shapes and known body positions is there any way for me to find out what the function might look like? With actual, discrete data, and descriptions of the bodies -- can I get a numerical approximation somehow? E.g., is there a way to turn this into a system that can be least-square-approximated, for example?

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    What you're looking for isn't well defined (I believe) at the moment. When you say "those sums", I don't even see a sum in what you're saying, and perhaps you could give some more detail about the $w_i$ functions and how they interact with $y$.2012-04-06
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    Sorry, the $F_i$'s are the sums. It's a discrete problem, really, I'm just using continuous notation here. The $w_i(x)$ are really long columns of numbers, and for every $x$ the value of $y(x)$ is multiplied by $w_i(x)$. I then sum up all those products, so that each of the long columns gives me a number. I only know that this number is (approximately) minimal, and I'm trying to find a possible function $y(x)$. -- When you say "not well defined", do you mean it's over- or underconstrained?2012-04-06
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    The problem is under-specified. In general, you cannot minimize more than one objective at a time. Are you combining the $F_i$ in some way? The $F_i$ are linear functionals of $y$, so unless the $w_i$ have some property (such as $0 \in co \{ w_i \}$), it may be that all the $F_i$ are unbounded below.2012-04-06
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    "It's a discrete problem, really, I'm just using continuous notation here." Why would you want do that? And then not even explain it before someone asks?2012-04-06
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    @joriki, my apologies, the problem is much more complex than described here, and I'm not sure I completely grasp it myself. Discrete or continuous doesn't matter here, really -- if someone can simply explain why this won't work, or what additional information would be needed, that's helpful for my understanding too.2012-04-06
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    @copper.hat, I've been thinking about how I could combine the $F_i$ into a single criterion, but minimizing, say, their sum doesn't do justice to the fact that I *know* $y(x)$ exists, and every $F_i$ is minimized. I guess I need more information. What does $co\{w_i\}$ stand for?2012-04-06
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    You need to re-work your question, because right now all we can do is give suggestions as to how the question should be formulated, but without a context, this is just as efficient as guessing. I'm sorry.2012-04-06
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    @PatrickDaSilva -- alright, thank you anyway :) I'll rethink the problem and maybe post another question. (Should I delete this one?)2012-04-06
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    @sebastian_k : Just edit your question and we'll take a look then.2012-04-06
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    For the numerical approximation part, this sounds like a problem that will benefit from simulated annealing.2012-04-06
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    @EMS: That's a really helpful comment. Mind explaining a bit more and/or pointing me to a good intro? (And make it an answer so I can upvote you!)2012-04-06
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    @sebastian_k: The term stands for the convex hull of the vectors $w_i$. Your problem is linear in $y$, so any given $F_i$ can made as small as you want. It is possible that you could find a $y$ such that all $F_i$ are negative, in which case all $F_i$ are unbounded below. The convex hull condition was an off-the-cuff criterion for preventing this.2012-04-06
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    It's usually not the case that the force between two points is solely a function of their relative positions. Depending on what the physical function is, it will also depend on the mass/charge/magnetic moment/... at those points, which is something you haven't said is known.2012-04-09
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    @RahulNarain, let's say I know for sure that the force depends only on the distance (or any other known variables, be that as it may -- I'm not dealing with an actual physical problem here).2012-04-09
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    Another thing that comes to mind is that the functional form of the statement that the objects are in equilibrium (force is 0) should reduce to a statement about only the average force exerted by A on B (and vice versa) and the distance between the shapes centroids (or centers of mass if they have density). You should not actually have to integrate over all differential elements of both bodies.2012-04-09
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    @EMS That's an interesting aspect. Does that apply even if $y$ is a function of, say, the coordinates of the involved elements $[x_1,y_1,x_2,y_2]$, and maybe also of things like density and charge?2012-04-09
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    Think about it. The total force that A applies to B is an average of every point in A affecting every point in B. This can equally well be stated as the effect of an average point in A (i.e. the centroid/center of mass) for each point in B, averaged over all of B. This is why, to great approximation, calculating the force of Newtonian gravity between two bodies only cares about the distance between centers of mass, not the actual distance between every point. I'm sure this could be stated better using language of conservative vector fields.2012-04-09
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    @EMS, spherically symmetric rigid bodies can be replaced by point masses in the case of inverse-squared-law forces in three dimensions, but this is not necessarily true for other shapes or other force laws.2012-04-09
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    @EMS That sounds tempting, but how am I supposed to find the centroids with regard to the unknown force if I don't know what the force looks like to begin with? As Rahul pointed out, these aren't circular shapes in gravity.2012-04-09
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    @Rahul, it would be true for any shapes embedded in a conservative vector field, I think. But you're right that the force field here could be pathological. I have a friend for example who works on molecular sphere packing and I think the force calculations there are pretty hairy.2012-04-09
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    @sebastian_k: The question is very unclear. If things are in equilibrium, the total force on each body is zero. Are you trying to solve a mechanics problem? Why don't you describe a simple example of what you are trying to solve?2012-04-12

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