The equation for the ellipse only depends on the relative numerical values of the anchor positions so one can solve for the equilibrium position just by finding the lowest point on elllipse. There are no physical "laws" needed. Let's review the equation for the ellipse and those for formulas for the equilibrium found using Lagrange's undetermined multipliers.
What happens if we try to find the minimum value of y for the ellipse? To do that we need to take the derivative of the formula for it and set that equal to zero and solve for the unknown angle.
The cosine of the minimum angle, θmin, turns out to be equal to the eccentricity times the cosine of the angle of the line through the focal points of the ellipse, θ0, and we get the same coordinates as the other formulas gave for the equilibrium position.
So the path that the constraints impose the motion of the weight is all that we really need to solve this problem.
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