# Classical celestial mechanics, mathematical problems in

Problems arising in astronomy in connection with the study of the motion of heavenly bodies in a gravity field. The classical objects of study in celestial mechanics are the planets and satellites of the solar system. The motion of stars is the study of stellar astronomy (see Stellar astronomy, mathematical problems of). The motion of artificial heavenly bodies is the study of astrodynamics. Because the distances between the bodies of the solar system are large as compared with the dimensions of the bodies themselves, their translational and rotational motions can be studied separately; furthermore, in the study of the translational motion, all bodies of the solar system can be regarded as material points interacting according to Newton's law of attraction, leading to the so-called $N$-body problem. An idealized scheme of this problem is described by the majority of problems in classical celestial mechanics. Numerical integration is the only universal method of obtaining solutions to this problem for arbitrary initial values. However, numerical integration is unsuitable for clarifying the evolution of the system over a sufficiently large interval of time. Some particular classes of solutions are known which are suitable for any moment of time (e.g., the periodic solutions of Euler and Lagrange, [1]). The general properties of the three-body problem have received fuller treatment. Most of the results relating to the $N$-body problem have been obtained by methods of perturbation theory for planetary problems, i.e. when the mass of the $N-1$ bodies is small as compared to that of the central body. If one disregards the perturbations, then the equations of motion degenerate into the equations for the two-body problem and can be integrated in closed form. In this connection, the motion of planets is subject to Keppler's law. For sufficiently small masses one can obtain analytic continuations of certain important classes of solutions: the periodic and quasi-periodic ones. It is believed that the most general form of motion is quasi-periodic motion with incommensurable frequencies. These results enable one to approach the problem of stability of the solar system under the action of natural perturbations of the planets.

The most fully investigated problem is the restricted three-body problem. Here two bodies of finite mass move around the centre of inertia in elliptical orbits, while the third has negligibly-small mass. Also fairly fully investigated is Hill's problem, describing the motion of a satellite subjected to a perturbation from the direction of a distant, but massive, body. The restricted problem is of importance in the theory of motions of small planets, and Hill's problem is important in the theory of the motion of the Moon.

For the construction of a theory of motion of actual celestial bodies, effective computational methods have been developed, based on expansions in series of powers of a small parameter. These methods can be divided into two groups, depending on whether time enters only into the arguments of the trigonometric functions in the expressions for the perturbations, or whether these expressions contain time explicitly (so-called secular perturbations). The methods of the first group are considerably more complex and are used in those cases when the solution is required on intervals of time that are fairly large in comparison to the periods of the unperturbed motion (as, for example, in the theory of lunar motion). A characteristic feature of computational methods in celestial mechanics is the necessity of carrying out a vast amount of calculation.

The theory of motion of large planets reduces to the $N$-body problem for $N=10$, by which the Sun and its nine planets are meant. The equations of motion can be approximately integrated by means of series expansions (analytic methods) or by numerical integration. An ideal theory of motion of large planets must satisfy four conditions: 1) it must be founded on a single mathematical method; 2) the integration constants must be determined from a compatible system of astronomical constants; 3) the accuracy of the theory must conform to the requirements of modern cosmological experiments; and 4) the theory must be suitable for long time intervals. The theory in existence in the 1970s (and lying at the foundation of all astronomical experiments) does not satisfy the above requirements. It has been created by various authors using different methods and at a different time. The creation of a theory satisfying the modern requirements is inconceivable without a wide use of computers and perfect observations.

The theory of motion of satellites is, to a large extent, similar to that of large planets, although a singular feature of this problem is that the mass of the planet around which the satellite rotates is much smaller than that of the Sun, whose attraction substantially perturbs the motion of the satellites. For nearby satellites it is necessary also to take into account the fact that the central body is not spherical. In this connection, the problem of the two fixed centres becomes very important, since the potential of a non-spherical planet can be sufficiently well approximated by that of appropriately selected point masses.

#### References

[1] | G.N. Duboshin, "Celestial mechanics" , Moscow (1968) (In Russian) |

[2] | , Handbook of celestial mechanics and astronomy , Moscow (1971) (In Russian) |

#### Comments

One no longer believes that the most general form of motion is quasi-periodic with incommensurable frequencies. In fact, since H. Poincaré's paper [a1], people working in this area strongly believe that homoclinic motions (called "bi-assymptotique" by Poincaré) exist.

General references are [a2], [a3].

#### References

[a1] | H. Poincaré, "Sur les problèmes des trois corps et les équations de la dynamique" Acta Math. , 13 (1890) pp. 1–270 |

[a2] | V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) |

[a3] | H. Rüssmann, "Konvergente Reihenentwicklungen der Störungstheorie der Himmelmechanik" K. Jacobs (ed.) , Selecta Mathematica , 5 , Springer (1979) |

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