Analysis of motion of a rotating tube including a material point by Johann Bernoulli, Daniel Bernoulli, Clairaut, d’Alembert and Euler

One of the most attractive problems in 1740s was a motion of a rotating tube around a fixed point and that of a small body included in it. Representative mechanicians in those days, such as Daniel Bernoulli, Alexis-Claude Clairaut, Leonhard Euler and Jean le Rond d’Alembert, tried to solve this problem. This problem was later applied to astronomical mechanics. It was posed by Euler for Johann Bernoulli in his letter in 1741. According to Johann Bernoulli’s letter to Euler dated on March 15, 1742, this problem is to determine a motion of a rotating tube around a fixed point and that of a body included in it under the influence of the gravity. Sit tubus seu canalis (sive gravis sive gravitatis expers) mobilis circa axem fixum, in quo versetur globus, qui ob gravitatem in tubo sine frictione descendat (et quidem, quod sine dubio subintelligis, non rotando sed fluendo) simulque tubo motum inducat : quovis tempore• determinare situm tubi et globi in tubo, itemque utriusque celeritatem. Johann Bernoulli tried to solve this problem in his succeeding letter but he failed to solve it. Johann Bernoulli in reality solved a problem posed by J.S. König in autumn of 1743 when König studied with Clairaut and Maupertuis. It is as follows, Determiner la courbe, que decrit un corps renfermé dans un Tuyeau pendant que le tuyeau se meut uniformement autour d’un Centre sur un plan horizontal. 6 According to Johann Bernoulli’s letter to Euler dated on August 27, 1742, these two problems seem to be similar but for him, they were quite different. He could solve the problem posed by König but the problem posed by Euler was unsolvable for him. He admitted that there was a great disparity between them and summarized difference as follows. 1. König considers a case where the gravity does not exist (a motion on a horizontal plane) but Euler poses motion under the influence of gravity. 2.A tube rotates uniformly in the former case but it does not rotate uniformly in the latter case. 3. Motion of a body included in a tube depends on the tube in the former case but in the latter case, a tube is weightless and its motion depends on a heavy body. Johann Bernoulli published the resolution of this problem twice. First, he wrote it in French as a supplement in his letter to Euler dated on August 27, 1742 and after that, he rewrote and published it in the 4 volume of his complete works in Latin. Euler also posed this problem to Daniel Bernoulli and Clairaut. Daniel Bernoulli frequently mentioned it in a series of letters to Euler. Especially, in his letter dated on October 20, 1742, he discussed a motion of a heavy straight tube rotating around a fixed point on a horizontal plane and that of a material point included in it. Next, he solved this problem by using the principle of conservation of “le momentum du mouvement circulatoire”, which corresponds to the conservation of angular momentum for us, and published it in his memoir. Clairaut also gave an outline of this resolution in his letter to Euler and published it in his memoir. Euler himself published his resolution in his memoir. D’Alembert alone discussed this problem in his book independently. 16 In this article, we will examine Johann Bernoulli’s and Daniel Bernoulli’s methods to solve this problem in §.2 and 3. In §. 4, 5 and 6, we will summarize Clairaut’s, d’Alembert’s and Euler’s methods. Finally, we will compare and examine their five methods and discuss their positions in the 18 century mechanics, which was on the way to constitute the modern physics.