The Foundation of the Sommerfeld Transformation

The analysis of the one-dimensional journal bearing leads to an interesting integral that is continuous but has an analytic singularity involving the inverse tangent at π/2. This difficulty was resolved by a clever and non-intuitive transformation attributed to Sommerfeld. In this technical brief we show that the transformation has its origin in the geometry of the ellipse and Kepler’s equation that is based upon his observations of the planets in the Solar system. The derivation of the transformation is a problem or exercise in Sommerfeld’s monograph, Mechanics. The transformation is the relation between the two angles that characterize the ellipse, the closed orbit of a body in a central inverse square force field. The angle measured about the focus is the true anomaly (angle) and the angle measured about the center is the eccentric anomaly (angle). We establish the analogy between the orbital radius in terms of the eccentric anomaly and the film thickness of the journal bearing in terms of its central angle.