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Exercises on the Kepler ellipses through a fixed point in space, after Otto Laporte

Gert Heckman

TL;DR

This paper revisits Otto Laporte's geometric exercises on Kepler ellipses through a fixed point, recasting them in a modern geometric framework built on conservation laws. It shows that fixing the energy $H<0$ yields a common major axis $2a$ and period for all ellipses, while the foci trace a circle around the fixed point and the region swept by all such ellipses is bounded by a companion ellipse; furthermore, the envelope of directrices forms another ellipse with clearly described foci and eccentricity, revealing rich geometric structure behind Kepler's laws. The discussion centers on the Lenz (Runge–Lenz) vector $\mathbf{K}$, its conservation, and its role in expressing the focus positions and eccentricities, tying classical dynamics to geometric constructions. The work also situates Laporte's contributions within a historical arc connected to the Kowalevski top and quantum integrability, highlighting the enduring relevance of geometric perspectives in classical and integrable systems.

Abstract

This article has a twofold purpose. On the one hand I would like to draw attention to some nice exercises on the Kepler laws, due to Otto Laporte from 1970. Our discussion here has a more geometric flavour than the original analytic approach of Laporte. On the other hand it serves as an addendum to a paper of mine from 1998 on the quantum integrability of the Kovalevsky top. Later I learned that this integrability result had been obtained already long before by Laporte in 1933.

Exercises on the Kepler ellipses through a fixed point in space, after Otto Laporte

TL;DR

This paper revisits Otto Laporte's geometric exercises on Kepler ellipses through a fixed point, recasting them in a modern geometric framework built on conservation laws. It shows that fixing the energy yields a common major axis and period for all ellipses, while the foci trace a circle around the fixed point and the region swept by all such ellipses is bounded by a companion ellipse; furthermore, the envelope of directrices forms another ellipse with clearly described foci and eccentricity, revealing rich geometric structure behind Kepler's laws. The discussion centers on the Lenz (Runge–Lenz) vector , its conservation, and its role in expressing the focus positions and eccentricities, tying classical dynamics to geometric constructions. The work also situates Laporte's contributions within a historical arc connected to the Kowalevski top and quantum integrability, highlighting the enduring relevance of geometric perspectives in classical and integrable systems.

Abstract

This article has a twofold purpose. On the one hand I would like to draw attention to some nice exercises on the Kepler laws, due to Otto Laporte from 1970. Our discussion here has a more geometric flavour than the original analytic approach of Laporte. On the other hand it serves as an addendum to a paper of mine from 1998 on the quantum integrability of the Kovalevsky top. Later I learned that this integrability result had been obtained already long before by Laporte in 1933.

Paper Structure

This paper contains 5 sections, 3 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The familiar conservation laws
  3. A geometric focus-focus proof
  4. All Kepler ellipses through a fixed point
  5. Final remarks

Key Result

Theorem 3.1

The point $\mathbf{t}$ is equal to $\mathbf{K}/(\mu H)$ with the so called Lenz vector. The Lenz vector $\mathbf{K}$ and therefore also the vector $\mathbf{t}$ are conserved quantities for the Kepler problem.

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2