A Scalar Equation of Motion

In classical mechanics, this thread presents a scalar equation of motion, which can be applied in any reference frame (rotating or non-rotating) (inertial or non-inertial) without the necessity of introducing fictitious forces.

If we consider two particles A and B of mass Image may be NSFW.
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where Image may be NSFW.
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Clik here to view. are the accelerations of particles A and B, Image may be NSFW.
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Clik here to view. are the positions of particles A and B, and Image may be NSFW.
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This scalar equation of motion is invariant under transformations between reference frames.

In addition, this scalar equation of motion would be valid even if Newton's three laws of motion were false.