PhD Course in Computer Science, Mathematics and Physics


An investigation with two bodies but no crime: the Kepler problem

Basic Course


Departamento de Matemática, Universidade de Lisboa
Board Contact
2 for attendance + 2 for validation
March 25-29, 2019
Lessons / Hours
8h total

Training goals
The Kepler problem is a simple but fundamental topic in Celestial Mechanics, which consists in the study of the motions of a non attracting body in the gravitational field of a second one. The aim of these lectures is:

  • to provide an elementary but (possibly) not so widely known proof of the elliptic form of the orbits following Hamilton’s velocity circles theorem and an ingenious construction of Feynman (and, if time allows, also a nice proof by Lagrange);
  • to discuss how to determine the position of a planet along its orbit at a given time by solving the so called Kepler’s equation;
  • to illustrate some of its deeper geometrical aspects


  • The surprising connection of the Kepler’s equation with the linear oscillator (whose orbits have, by the way, an interesting topological structure, which we will address in a little detour) through the Levi-Civita regularization, a particular case of the Kasner-Arnol’d Theorem;
  • (if time allows and depending on the audience one of the two following):
    • how the presence of an hidden SO(4) symmetry accounts for a mysterious constant of motion, the so-called Runge –Lenz vector (following Moser)
    • how the space of velocities corresponding to a negative energy can be endowed a Riemannian structure with the velocity circles as geodesics (following Milnor)


On the geometry of Kepler Problem, J. Milnor, The American Mathematical Monthly, 1983

Feynman’s Lost Lecture. The motion of the Planets around the Sun, D.L. Goodstein and J.R. Goodstein 1996

Introducción a la Mecánica Celeste, R. Ortega and A. Ureña, 2010

Classical Mechanics with Calculus of Variations and Optimal Control. An intuitive introduction, Mark Levi, 2014

Huygens and Barrow, Newton and Hooke, V. I. Arnold, 2001

Notes on Dynamical Systems, J. Moser and E.J.Zehnder, Courant lecture notes, AMS, 2005

Discussion of a list of assignments/exercises related to the subject of the course
Basic theory of ODEs.