# Optimal dynamics of a spherical squirmer in Eulerian description

## TL;DR

This paper formulates the optimization of a spherical microsquirmer's propulsion cycle in a viscous fluid as a noncanonical Hamiltonian system, providing analytical insights into its dynamics and previously obtained numerical results.

## Contribution

It introduces a Hamiltonian framework for the squirmer's surface velocity optimization and derives an integrable three-mode approximation of the system.

## Key findings

- The system is quadratically nonlinear but integrable in a three-mode approximation.
- The Hamiltonian formulation offers a new theoretical perspective on microsquirmer dynamics.
- Analytical results support and interpret previous numerical findings.

## Abstract

The problem of optimization of a cycle of tangential deformations of the surface of a spherical object (microsquirmer) self-propelling in a viscous fluid at low Reynolds numbers is represented in a noncanonical Hamiltonian form. The evolution system of equations for the coefficients of expansion of the surface velocity in the associated Legendre polynomials $P^1_n(\cos\theta)$ is obtained. The system is quadratically nonlinear, but it is integrable in the three-mode approximation. This allows a theoretical interpretation of numerical results previously obtained for this problem.