Abstract
One of the highlights of classical mechanics is the mathematical derivation of the three experimentally observed Kepler laws of planetary motion from Newton’s laws of motion and of gravitation. Newton published his theory of gravitation in 1687 in the Principia Mathematica [1]. After two short introductions, one with definitions and the other with axioms (the laws of motion), Newton discussed the Kepler laws in the first three sections of Book 1 (in just 40 pages, without ever mentioning the name of Kepler!) Kepler’s second law (motion is planar and equal areas are swept out in equal times) is an easy consequence of the conservation of angular momentum L = r × p, and holds in greater generality for any central force field. All this is explained well by Newton in Propositions 1 and 2. Kepler’s first law (planetary orbits are ellipses with the center of the force field at a focus) however is specific for the attractive 1/r force field. Using Euclidean geometry Newton derives in Proposition 11 that the Kepler laws can only hold for an attractive 1/r force field. The reverse statement that an attractive 1/r force field leads to elliptical orbits Newton concludes in Corollary 1 of Proposition 13. Tacitly he assumes for this argument that the equation of motion F = ma has a unique solution for given initial position and initial velocity. Theorems about existence and uniqueness of solutions of such a differential equation have only been formulated and mathematically rigorously been proven in the 19th century. However there can be little doubt that Newton did grasp these properties of his equation F = ma [2]. Somewhat later in 1710 Jakob Hermann and Johan Bernoulli gave a direct proof of Kepler’s first law, which is still the standard proof for modern text books on classical mechanics [3]. One writes the position vector r in the plane of motion in polar coordinates r and θ. The trick is to transform the equation of motion ma = −kr/r with variable the time t into a second order differential
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