# Steiner triangular drop dynamics

## TL;DR

This paper introduces the Steiner triangle as a minimal geometric model for liquid droplet dynamics, revealing complex behaviors such as bouncing, rocking, and quasiperiodic motions, and capturing symmetries of realistic models.

## Contribution

It presents a novel minimal model based on Steiner's circumellipse for analyzing droplet dynamics, with detailed dynamical system analysis and identification of invariant manifolds.

## Key findings

- Identified bouncing and rocking periodic motions.
- Discovered nested quasiperiodic motions around equilibrium.
- Model captures symmetries of more realistic droplet models.

## Abstract

Steiner's circumellipse is the unique geometric regularization of any triangle to a circumscribed ellipse with the same centroid, a regularization that motivates our introduction of the Steiner triangle as a minimal model for liquid droplet dynamics. The Steiner drop is a deforming triangle with one side making sliding contact against a planar basal support. The center of mass of the triangle is governed by Newton's law. The resulting dynamical system lives in a four dimensional phase space and exhibits a rich one-parameter family of dynamics. Two invariant manifolds are identified with "bouncing" and "rocking" periodic motions; these intersect at the stable equilibrium and are surrounded by nested quasiperiodic motions. We study the inherently interesting dynamics and also find that this model, however minimal, can capture space-time symmetries of more realistic continuum drop models.