Pedal coordinates, dark Kepler, and other force problems

Abstract

Abstract. We will make the case that pedal coordinates (instead of polar or Cartesian coordinates) are more natural settings in which to study force problems of classical mechanics in the plane. We will show that the trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. This will allow us to generalize Newton theorem of revolving orbits to include nonlocal transforms of curves. Finally, we apply developed methods to solve the "dark Kepler problem", i.e. central force problem where in addition to the central body, gravitational influences of dark matter and dark energy are assumed.

Pedal coordinates and free double linkage

“…Using pedal coordinates we are also able to show a link between trajectories of a free double linkage and orbits of a test particle inside a Schwarzschild Black hole (see Section 4). We have also found that the free double linkage provides a particular solution to the "Dark Kepler problem" (discussed in [7]). Although certainly just a mathematical coincidence, it is amazing that a connection between such dramatically different problems exists at all.…”

“…Obviously, the pedal coordinates do not care about the rotation around the pedal point and about the curve's parametrization, but it is actually not easy to tell in general the nature of ambiguity associated to a pedal equation -in fact, it differs from equation to equation. (For more information see [7]. )…”

“…There is some activity in the area [1], [2], [3], [4], [5], [6] but not as much as it deserves. As proved in [7], pedal coordinates are particularly well suited for studying force problems within a plane. Specifically, certain class of central and Lorentz-like force problems can be easily solved in pedal coordinates algebraically (see Theorem 2).…”

“…The purpose of this paper is to showcase these qualities on a non-trivial example that is outside the scope of papers [7,8] -demonstrating that pedal coordinates are far more versatile. Though the example itself is not without interest, the emphasis is laid on the solution's method.…”

Notes on pedal and contrapedal curves of fronts in the Euclidean plane

“…These coordinates are also useful for solving certain type of force problems in classical mechanics and celestial mechanics. Blaschke has recently showed that the trajectory of a test particle under the influence of central and Lorentz‐like forces can be translated into pedal coordinates at once without solving any differential equation and have also applied developed methods to solve a dark Kepler problem.…”