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In the world of "absolute" speed values in the Newtonian mechanics, the moon must move faster (i.e., must have a higher speed) if he travels with the earth (i.e., has the same velocity direction) than when he travels "against" the earth (otherwise the orbit would not be an ellipse). So something must change its velocity which in turn needs some kind of force. Is this a problem for Newtonian mechanics already or does the gravitational force between earth and moon change the speed of the moon?

I hope my question is understandable to an advanced physicist's mind.

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You appear to be mistaken about something: Newtonian mechanics does not have a concept of absolute speed. All motion in Newtonian mechanics is measured relative to something else.

With that out of the way, if we measure the moon's motion from the reference frame of the sun (so that the sun appears to be stationary), we find that it does indeed travel faster when going with the Earth than going against. You are exactly correct that the force that causes this change in velocity is the gravitational attraction between the Earth and the moon.

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It is gravity (between the Earth and Moon) which causes the moon to speed up relative the sun when it is overtaking the Earth (and to slow down relative the sun when the Earth is overtaking it).

This is not a problem because forces are paired (Newton's 3rd law, right) and the Earth slows down relative the sun while the moon speeds up and vice versa; but it does so by smaller increments such that momentum remains conserved.

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