agroposo wrote:I didn't presume that the word series as used by Gopi meant a sum of elements, because nowhere in his paper he is summing the higher order forces, but after re-reading Gopi's paper, I've come to the conclusion that he is using the term series to indicate a set, as when he says in the paper's abstract "an infinite series of higher order rotational forces", meaning that the force F and its derivatives F', F'', F''', F(4), F(5), ..., constitute a set. If we consider all the derivatives, then we have an infinite set.

I am glad that agroposo caught the meaning as intended. I couldn't have put it that well myself.

agroposo wrote:If you're postulating that there are "infinite higher order forces", necessary to account for circular and elliptical motion, what would happen if at some point in the sequence, let's say n, the derivative F(n) is zero? Then all the subsequent derivatives would be zero, and your postulate will be contradicted.

1. I am not postulating it, I am deriving it. That is a big difference, as the method is the same as how velocity and acceleration are traditionally derived for circular motion, with the difference being that I see no logical reason to stop the derivative process at acceleration. If you can take the first and second derivatives, then you can take all of them, since they are all non-zero.

2. If F(n)=0, as suggested above, then that means that some Rw^n = 0. This implies that either R = 0 or w = 0, i.e. either the radius or the angular velocity has to go to zero. In that case you no longer have circular motion, so there is nothing to contradict. Not only the subsequent derivatives, but the preceding derivatives will vanish as well -- a peculiarity of circular motion that is unlike linear motion. It pulls the ground out from the entirety of the circular motion we are starting with.

Where Newton stopped was at the second derivative, acceleration, so that it could then be fit with Galileo's theory. Using geometry to stop the derivatives process is not simplification, it is plain wrong. You cannot chop an infinite series by fiat because it is convenient. If Newton has to be given his due, it will have to be in his derivation of the calculus, and the several mathematical techniques he applied in other places in the Principia.

hoi.polloi wrote: My guess is that Gopi accidentally mixed prosaic language with math language, therefore failing to communicate his idea in established understandings/agreements.

I admit I may have done a good bit of that. It is only with the three papers attached that I have done my best to keep the terminology strictly mathematical and peer-reviewable, so please do not take the wording I am using on the forum to be the literal mathematical phrase as used in the literature. I am only trying to get the meaning across here by trying to phrase it in different ways.