Optimal dynamics of a spherical squirmer in Eulerian description

The problem of optimization of a cycle of tangential deformations of the surface of a spherical object (micro-squirmer) 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.