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I came across Saari's homographic conjecture in Three Body problem. I need more information on what exactly is a homographic solution and how is it different from a homothetic solution?

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2 Answers 2

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A homographic solution is one in which the configuration of all the masses is preserved for all time. One example of this are the relative equilibrium solutions. Each mass rotates at a fixed distance from the center of mass at constant angular velocity. For the 3BP these are the colinear solutions found by Euler and the equilateral triangle configurations found by Lagrange.

A homothetic solution is one in which the scale size changes while retaining the configuration and no rotation is permitted. An example of this would be a total symmetric collapse: start 3 equal masses from rest at the vertices of an equilateral triangle and they will retain the configuration as they collapse to the origin.

Both of these solution types are examples of central configurations. Saari talks about these things extensively in his book: "Collisions, Rings, and other Newtonian N-Body Problems"

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  • $\begingroup$ Thanks a lot. That was very helpful $\endgroup$
    – Injitea
    Jul 12, 2014 at 8:10
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JEM's answer is not exactly true. I am not an expert on the subject but I have also looked at many of the relevant papers (for eg. this one and this one) out of curiosity.

From these (also see good summary here) you can see that, in the context of central configurations and Saari's homographic conjecture, a "homographic configuration" of masses is one which is invariant under scaling and rotation.

Formally, the configuration is said to be homographic if at any time $t$ the coordinates of the masses in the configuration $q(t)$ satisfy: $$q(t)=\lambda(t)Q(t)q(t_0)$$ where $Q(t)$ is a rotation matrix, $\lambda(t)$ is a time-dependent scalar, and $t_0$ is some specific time in the orbit of the configuration.

JEM's answer refers to Saari's original conjecture and a special restrictive case of homographic solutions wherein there is rotation but no scaling--the $\lambda(t)$ drops out of the equation above and the masses rotate as a rigid body (with constant moment of inertia $I$).

In Saari's extended or homographic conjecture, the concern is with a wider range of homographic configurations that scale and rotate with time ($I$ is not constant but $UI$ is constant, where $U$ is the potential energy of the system).

Note, however, that this notion of homography is more restrictive than the usual definition of homography from projective geometry (see for eg. this, and this, and this MSE post). It seems to me a better way of referring to n-body orbital configurations that are invariant under scaling and rotation (as one of the lead researchers Moeckel suggests), is to call them self-similar configurations.

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  • $\begingroup$ Thank you so much ben. That gave a deeper understanding. Greatly appreciated. $\endgroup$
    – Injitea
    Aug 10, 2014 at 9:46

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