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I suspect this is an example of the spinning-egg problem, in which a prolate spheroid (such as an egg) spun on a table about one of its "short" axes will tend to "stand up" so that it's spinning about its long axis. A few explanations have been proposed for this phenomenon, most notably: H. K. Moffatt & Y. Shimomura, "Spinning eggs — a paradox ...

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This circle is the set of locations where the gravity of the sun and earth cancel out. Uhm. No. They are neither co-linear nor the same magnitude. To get a zero "force" at the Lagrange points you have to work in a rotating frame of reference, implying a centrifugal pseudo-force which closes the triangle. But the points on the circle you defined other ...

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At L4 and L5, the object would stay in the same relative position because it is in the same orbit around the Sun as Earth. Any other points on the circle would not be in the same orbit as Earth. They would be in orbits of differing inclination and so, in the course of the orbit of that object about the Sun, it would necessarily leave those points. The L4 and ...

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You need to imagine the two suns are rotating around the common center of mass. The simplest situation is if they are in circular orbits. In that case the point at the center of mass in the $L_1$ Lagrangian point, which is known to be unstable for just the reason you suspect. If you move the small mass toward one of the suns at very low velocity, it will ...

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Gravitional waves are a product of general relativity. In GR, gravity is due to curved space-time. Gravitational waves therefore are not how gravity affects things. What are gravitational waves then? They are what happens when a gravity "well" moves. Imagine the source of the above gravity "well" moving. The gravity well will move by gravitational waves. ...

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