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Kepler's problem and spaces of constant curvature

Kepler's problem (around 1706) concerns the motion of a planet around an immovable planet in the presence of the gravitational force. Based on empirical evidence, Kepler stated that the moon moves on an elliptical path around the earth (known as Kepler's first law, and secondly he stated that the position vector of the moving planet sweeps out equal areas in equal time intervals as the planet orbits around the stationary planet (known as Kepler's second law). It was not until Newton's gravitational laws were stated in 1687 (in Principia Mathematica) that Kepler's observations could be verified mathematically.

This lecture is principally motivated by another remarkable discovery that the solutions of Kepler's problem are intimately related to the geometry of spaces of constant curvature. These discoveries go back to V.A. Fock, who in 1935 linked the motion of the hydrogen atom to non-Euclidean Geometry. Then J. Moser, in 1970, showed that Kepler's problem restricted to the manifold of constant negative energy is equivalent to the geodesic flow on the sphere. Finally, Y. Osipov showed in 1977 that Kepler's problem is equivalent to the geodesic flow on the Euclidean space on the energy level zero, and is equivalent to the geodesic flow on the hyperboloid when restricted to the manifold of a constant positive energy.

My lecture will address these "enigmatic" connections between planetary motion and space forms. I will show in this lecture that an n-dimensional Kepler's system is equivalent to the geodesic flow on space forms (spaces of constant curvature) precisely under the same conditions as in the paragraph above. I will also show that the famous integrals of motion for the problem of Kepler, the angular momentum and the Runge-Lenz vector, correspond to a particular moment map associated to the geodesic flow on the appropriate space form.


Start Date:  2018-06-20
Start Time:   15:00
Speaker:  Velimir Jurdjevic (Univ. Toronto, Canada)
Institution:  Univ. Toronto, Canada
Place:  Room 2.4, DMat
Biography:  Professor Velimir Jurdjevic is one of the founders of geometric control theory. His pioneering work with H. J. Sussmann was deemed to be among the most influential papers in this subject area, and his book, Geometric Control Theory, revealed the geometric origins of the subject and uncovered important connections to physics and geometry. It remains a major reference on non-linear control. Jurdjevic's expertise also extends to differential geometry, mechanics and integrable systems. His publications cover a wide range of topics including stability theory, Hamiltonian systems on Lie groups, and integrable systems. His most recent book Optimal Control and Geometry: Integrable systems published by Cambridge Press in 2017, a synthesis of symplectic geometry, the calculus of variations and control theory, provides a foundation for many problems in applied mathematics and strongly reflects his current research interests. Professor Jurdjevic has spent most of his professional career at the University of Toronto.
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