# Can there be forces between three things? [closed]

In all the physics I know forces are always between pairs of things. Gravity is between pairs of masses, so is electrostatic repulsion, so are the strong and weak forces as far as I understand them (which is not very far).

Are there any hypothetical forces that are defined between three or more things?

Maybe quarks are somehow an example since three of them come together to make a proton or a neutron. But I'm especially interested in large-scale forces.

Can there be one that's like gravity? For example, what if every three masses interact in a way that accelerates them in a direction within the plane they define and perpendicular to the direction of some centroid? If any two of the three masses are equal, this "force" would net zero by reading the triple in both directions, but if they are all different maybe not. What else in physics would that break?

Naively, this would be a way to explain a galaxy rotation curve. Split the galaxy into three pie slices, and reading clockwise give one slice an abundance of light elements and a deficit of heavy, the next slice a normal amount, and the next slice more heavy than light. Then if the atoms in the galaxy are treated as point masses they pull on each other in the same direction as the observed rotation with an acceleration determined by however we've defined the "force". Alternatively, let all the point masses be the same, and define the force so that it is larger when the distance between consecutive pairs in the triple is increasing, then it seems like the existing spiral structure gives the asymmetry needed. Basically as long as some triangles are more common than their mirror image, then the mirror triangles are more common when looking at the galaxy upside-down, so there can be a positive net acceleration on each of the three masses even when they are equal.

## closed as off-topic by sammy gerbil, Kyle Kanos, Jon Custer, Sebastian Riese, Chris♦May 28 '18 at 1:07

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• Would it be correct to say that the theory of emergent gravity requires 3 things: two entangled particles and the mass in-between them that actually has the gravitational field. – foolishmuse May 17 '18 at 20:34
• I don't know but I'm hoping there is a way of explaining this (proving it somehow implausible I imagine) that doesn't rely on entanglement or anything more complicated than what I've mentioned. – Dan Brumleve May 17 '18 at 20:42
• "In all the physics I know forces are always between pairs of things." Aren't you overlooking gravitational fields, electromagnetic fields, etc., that can express the effects of many more than just a pair of things? – D. Halsey May 17 '18 at 21:42
• I mean I also know you can define a force as a gradient on an energy landscape. But I think this is in some sense the same as adding up all the forces between pairs. I'm not really sure how a 3-way force would fit into that concept. – Dan Brumleve May 17 '18 at 21:44
• Isn't it the sum of the forces that generates the acceleration? So if there's just three objects, then each of the three accelerate due to the positions of the other two. – Kyle Kanos May 21 '18 at 10:10

Newton's third law is typically stated in terms of forces and so assumes that all forces come in pairs. However, if we substitute Newton's third law for the law of conservation of total momentum we can consider more general theories. For example, suppose there are three particles with positions $\vec{x}_{1,2,3}$ and the force on each particle is given by $$\vec{F}_1 = -k(\vec{x}_2-\vec{x}_3)$$ $$\vec{F}_2 = -k(\vec{x}_1 + \vec{x}_3 - 2\vec{x}_2)$$ $$\vec{F}_3 = -k(\vec{x}_2-\vec{x}_1)$$ You can check that these forces conserve the total momentum $\vec{p}_1+\vec{p}_2+\vec{p}_3$, but you can't really break them up into just forces between the pairs of particles. To calculate the force on 3 you have to know where particle 1 is, so it doesn't work without three particles.
You can construct a theory like this that conserves momentum, energy, and angular momentum by using a potential energy that is a function of only the scalar $W = (\vec{x}_1-\vec{x}_2)\cdot(\vec{x}_2-\vec{x}_3)$. My forces above come from the potential $V = kW$.