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They presumably meant non-zero total momentum. If the total momentum were non-zero, then the center of mass will be moving in a straight line, and will not return to where it began, and therefore the orbit would not be periodic.

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Big picture the center of mass momentum is concerned and it does not matter if the system as a whole is moving up or down or to the right or the left. Like the Earth is basically orbiting the sun in an ellipse [1] so far as the sun is concerned and from the viewpoint of the solar system not care so much that it is moving around the galaxy unless we are interested that orbit being perturbed by other stars that we pass near over millions and milions of years.

[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets

Oh. That makes sense. Yes. The total momentum has to be zero.

All of the things in the solution can constantly have momentum.

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