The motion of an unbalanced circular foil in the field of a point source

TL;DR

This paper models the complex motion of an unbalanced circular disk in a flow created by a point source, revealing Hamiltonian dynamics, bifurcations, and nonintegrability that explain natural leaf behavior.

Contribution

It provides an idealized mathematical model of an unbalanced disk in a flow, demonstrating Hamiltonian and nonintegrable dynamics relevant to natural phenomena.

Findings

Equations of motion are Hamiltonian for fixed sources.

Balanced disks have integrable equations of motion.

Unbalanced disks exhibit nonintegrable, complex behavior.

Abstract

Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of joint motion of the unbalanced circular disk and the point source. It is shown that in the case of a fixed source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.