Bertrand's task

Bertrand's first task . Find the law of forces, depending only on the position of the moving point, and forcing it to describe the conic sections, whatever the initial conditions.

This problem was successfully solved by Darboux and Alphen ^[1] with the additional assumption that the force is central, and then it was possible to drop this condition ^[2] . It turned out that there are two such interactions - the law of world wideness and Hooke's law .

The assumption of centrality of force, however, could be made from general considerations of the symmetry of the problem.

The second task is Bertrand . Knowing that the force causing the planet to move around the Sun depends only on the distance and is such that it forces its point of application to describe a closed curve, whatever the initial conditions, if only the speed is less than a certain limit, find the law of this force.

The answer is short: the law of force can be either the law of Hooke or the law of the world wide.

The problem is solved by Bertrand himself ^[3] . The most complete solution is given in the Darboux note to the mechanic Depeiroh ^[4]

Koenigs G. proposed an even more general problem:

Koenigs task . Knowing that the force causing the planet to move around the Sun depends only on the distance and is such that it forces its point of application to describe an algebraic curve, whatever the initial conditions, to find the law of this force.

Surprisingly, the answer is the same: the law of force can be either the law of Hooke or the law of world wideness.

An exhaustive solution of the problem is given by Koenigs himself ^[5] . The idea of ​​the proof is reduced to the proof of the closedness of the analytic finite orbit ^[6] , which reduces the problem to the previous one.

The tasks of determining the type of forces when the body moves in orbits in the form of conic sections and the type of orbits according to a given law of forces are set and completely solved ^{[7] by} Isaac Newton in book I " Mathematical principles " using the synthetic method developed by him that combines geometric proofs of the main theorems of mathematical analysis and the theory of limits ^[8] with the theory of analytic series based on the Newton binomial theory ^[10] created by him ^{[9 ]} .

Section III ( On the movement of bodies along eccentric conic sections ) proves that movement along conic sections is possible only for the inverse square law ( Proposal XI - XIII ) or for the law of the first degree (Hooke, Proposition X ). Moreover, the first case corresponds to the direction of the force to the focus of the conic section, and the second to the geometric center of the ellipse. In Section II, it is tentatively proved that the motion of a body along a part of any smooth curve lying in a plane can be reduced to movement in the field of some central force with a attracting center on this plane ( Proposition VII, Corollaries 2 and 3 ).

In Section IX ( On the movement of bodies in moving orbits and on the movement of the apses ), it is proved using analytical series and passing from the orbit close to a circle to a circular orbit that a closed orbit can only be at a power of +1 (Hooke’s law, Example 2 ) or -2 (law of law, Example 3).

In the preface to the "Principals", the author of the translation and the editor of the first edition of the "Start" in Russian, the mechanic A.N. Krylov notes that the first translation into English was made in 1727, into French - only in 1759 by the Marquise du Chatelet , and Newton's work in modern European languages ​​became available only many decades after its first publication in 1686.