# Pedal coordinates, Dark Kepler and other force problems

## TL;DR

This paper advocates using pedal coordinates for classical mechanics force problems, enabling direct trajectory analysis and generalization of Newton's revolving orbit theorem, with applications to dark matter-influenced Kepler problems.

## Contribution

It introduces pedal coordinates as a natural framework for force problems and extends Newton's theorem to include nonlocal curve transforms, applied to dark matter scenarios.

## Key findings

- Pedal coordinates simplify force problem analysis without differential equations.
- Generalization of Newton's revolving orbit theorem to nonlocal transforms.
- Application to dark matter-influenced Kepler problems.

## Abstract

We will make the case that \textit{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.