Mathematical Methods of Celestial Mechanics

Recommended Prerequisites

Linear Algebra and Analytic Geometry, Infinitesimal Analysis I, II and III, Differential Equations and Modelling.

Teaching Methods

A significant part of practical classes is expositive in nature, complemented by the presentation of specific examples and problem solving for applying the acquired knowledge. Personal and group work and discussion in the classroom are encouraged by the teacher. Students are also accompanied individually to discuss their work and to identify any difficulties.

Learning Outcomes

This course aims to enable the students to understand the fundamental principles of celestial mechanics. Initially the study of the two-body problem is carried out and later the one of the N-body problem. Students are expected to be able to determine the evolution of the orbits of planets, asteroids and comets and to consult astronomical databases.

Many mathematical methods were first developed in order to understand the N-body problem. An attempt is made to present some of these methods, selecting those that have a more applied point of view.

The main skills to be developed are: analysis and synthesis skills; ability to formulate and solve problems; ability to work in groups; critical reasoning; autonomous learning ability; initiative and entrepreneurial spirit; ability to use the modern tools of celestial mechanics and to apply the acquired knowledge to concrete problems of the dynamics of celestial bodies.

Work Placement(s)

No

Syllabus

The evolution of celestial mechanics. Brief reference to the interdisciplinary nature of modern celestial mechanics.

Central motion. Law of areas. Binet’s formulas. Characterization of the motion plane. Nodal reference system. Description of the motion of planets in the Solar System: Kepler's laws.

Newtonian mechanics. Kepler's problem. Integration of the mechanical system. Orbit description: eccentricity; true anomaly; celestial reference systems. Calculation of the classical orbital elements from the observations. Relation between position and time: ephemeris, eccentric anomaly.

Two-body problem. Reduction: motion around the center of mass; relative motion.

N-body problem. Hamiltonian system. Integrals. Stability. Singularities: collision; total collapse; regularization.

Three-body problem. Restricted problem. Libration points. Stability.

Euler-Lagrange equations. Figure eight of the three-body problem.

Perturbation theory. Precession of the Earth.

Head Lecturer(s)

Margarida Maria Lopes da Silva Camarinha

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
Mini Tests: 30.0%
Frequency: 70.0%

Bibliography

Victor G. Szebehely, Hans Mark (1998). Adventures in celestial mechanics. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York.

F. P. J. Rimrott (1989). Introductory attitude dynamics. Mechanical Engineering Series. Berlin etc: Springer-Verlag.

Archie E. Roy (1978). Orbital motion. Halsted Press. John Wiley & Sons, New York.

Harry Pollard (1966). Mathematical introduction to celestial mechanics. Prentice-Hall Inc., Englewood Cliffs, N.J..

Sérgio B. Volchan (2007). Uma introdução à mecânica celeste. Publicações Matemáticas do IMPA. Rio de Janeiro: IMPA.

Carl D. Murray, Stanley F. Dermott (1999). Solar system dynamics. Cambridge University Press, Cambridge.

Alessandro Morbidelli (2002). Modern celestial mechanics. Aspects of solar system dynamics. Advances in Astronomy and Astrophysics 5. London: Taylor & Francis.

 Kenneth R. Meyer, Daniel C. Offin (2017). Introduction to Hamiltonian dynamical systems and the N-body problem. Applied Mathematical Sciences 90. Cham: Springer.