Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master


MMM1003 Celestical Mechanics
Hours: C+S+L
Applied Mathematics
Teaching Staff in Charge
Assoc.Prof. SZENKOVITS Ferenc, Ph.D.,
This course is an introduction to the Celestial mechanics which focuses on the dynamics of celestial bodies. The basic results concerning the classical models of the two-body and the many-body (including the three-body problems) are presented. Methods dedicated to general and particular perturbations are studied and applied to the basic models of the celestial mechanics. Home works include exercises of both practical and theoretical nature, and also writing computer programs.
By the graduation of this class, the students will get the following competences:
• Will be able to construct the mathematical model for different systems of celestial bodies.
• Will be able to deduce basic properties of the models.
• Will be able to apply the general and special perturbation methods.
• Will be able to perform numerical integration of the systems of ordinary differential equations involved.
1 The two-body problem
1.1 Newton’s laws of motion and of gravitation
1.2 The solution of the two-body problem
1.3 The elliptic orbit
1.4 The parabolic orbit
1.5 The hyperbolic orbit
1.6 The rectilinear orbit
1.7 Barycentric orbits
1.8 Classification of orbits with respect to the energy constant
1.9 The orbits in space
1.10 The f and g series
1.11 The use of recurrence relations
2. The many-body problem
2.1 The equation of motion in the many-body problem
2.2 The ten known integrals and their meanings
2.3 The force function
2.4 The virial theorem
2.5 Lagrange’s solutions of the three-body problem
2.6 The circular restricted three-body problem
2.7 The general three-body problem
2.8 Jacobian coordinates for the many-body problem
3. General perturbations
3.1 The nature of the problem
3.2 The equations of the relative motion
3.3 The disturbing function
3.4 The sphere of influence
3.5 The potential of a body of arbitrary shape
3.6 Potential at a point within a sphere
3.7 The method of the variation of the parameters
3.8 Lagrange’s equations of motion
3.9 Hamilton’s canonic equations
3.10 Derivation of Lagrange’s planetary equations from Hamilton’s canonic equations
4. Special perturbations
4.1 Factors in special perturbation problems
4.2 Cowell’s method
4.3 Encke’s method
4.4 The use of perturbational equations
4.5 Regularization methods
4.6 Numerical integration methods
1. ARNOLD, V. - KOZLOV, V.V. - NEISHTADT, A.: Mathematical Aspects of Classical and Celestial Mechanics. translated from the Russian by A. Iacob, Mir. Publishers, Moscow, 1988.
2. BOCCALETTI, D. - PUCACCO, G.: Theory of Orbits Volume 1: Integrable Systems and Non-Perturbative Methods. Springer-Verlag Berlin Heidelberg New York, 1998.
3. BOCCALETTI, D. - PUCACCO, G.: Theory of Orbits. Volume 2: Perturbative and Geometrical Methods. Springer-Verlag Berlin Heidelberg New York, 1999. BROWER, D. CLEMENCE, G.M.: Methods of Celestial Mechanics. Academic Press, New York, 1961 (trad. in l. rusa, Ed. Mir, Moscova, 1964)
4. DRÂMBA, C.: Elemente de mecanica cereasca. Ed. Tehnica, Bucuresti, 1958.
5. DUBOSIN, G. N.: Nebesnaya Mechanika. Osnovnie zadaci i metodi. Izd. Nauka, Moskva, 1963, 1968.
6. ÉRDI Bálint: Égi mechanika. Tankönyvkiadó, Budapest, 1992.
7. ÉRDI Bálint: A Napredszer dinamikája. ELTE Eötvös Kiadó, Budapest, 2001.
8. OPROIU, T. Et alii: Astronomie. Culegere de exercitii, probleme si programe de calcul. Univ. Babes-Bolyai din Cluj-Napoca, 1985, 1989.
9. ROY, A.E.: Orbital Motion. Third Edition, Adam Hilger, Bristol and Philadelphia, 1988.
Activity at the seminaries (40%).
Exam at the end of the term (60%).
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject