Classical Physics

   

Galileo's Solution for the 'Path of Quickest Descent'

Authors: Radhakrishnamurty padyala

Path of the quickest descent of a material particle from a point A to another point B at a lower lever, in a constant gravitational field, is a famous problem in mathematics. It was solved by Galileo around 1638. Galileo’s solution was that the quickest path is the arc of a circle with B as the lower end of the vertical diameter of the circle in the vertical plane. It is important to note that Galileo does not use the summation of time intervals of travel along the successive chords that connect A and B. He compares the times of travel between two paths from A to B. One path consists of the direct shortest path – the chord connecting A and B. The second path consists of two chords AC and CB. Galileo proves that the two chord path ACB is quicker than the single chord path. Then he compares the two chord path with the three chord path, AC, CD, DB and proves the three chord path ACDB is quicker than the two chord path ACB. He extends this procedure indefinitely to more and more chords and proves that the arc of the circle is the quickest path of travel from A to B. Later, John Bernoulli solves this problem and poses it as a challenge to peers to solve it. Many well known mathematicians that include Bernoulli’s elder brother Jacob Bernoulli, Newton, Pascal among others solved it. John Bernoulli’s solution was based on Fermat’s least time principle. To account for the path followed by a ray of light between two points, Fermat enunciated the principle of least time. According to this principle light takes the path of minimum time in going from the initial to the final point involving reflections or of refractions on its way. Bernoulli argued that if light follows Fermat’s principle in economizing the time of travel between two points, why not a material particle also follow that principle, so that nature economizes on the number of principles required to govern various processes? Arguing thus, he employed Fermat’s least time principle and arrived at a different solution from that of Galileo’s solution. This solution became very famous and gave rise to many other mathematical developments. In contrast to Galileo’s circular path, Bernoulli’s solution was a ‘brachistochrone’. We discussed Bernoulli’s solution in an article in this journal earlier. Bernoulli’s solution involves the summation of time intervals of travel along the successive chords of the brachistochrone. Galileo did not add the time intervals because time intervals along paths of different accelerations are not additive – they are additive if and only if they are along path of the same value of acceleration. Students must get a good grasp of this idea in order to appreciate Galileo’s solution. As Erlichson says, this study provides some very interesting information on Galileo’s geometrical methods.

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[v1] 2018-12-28 00:58:56

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