Hamiltonian formulation for extended active particle models

Determine whether a Hamiltonian formulation exists for generalized models of active particle motion beyond unidirectional steady laminar straight duct flows, specifically including (i) general ellipsoidal particle shapes with arbitrary axis ratios, (ii) time-dependent duct flows, (iii) models incorporating particle–wall collisions, (iv) models with finite-size particles that interact with the flow, and (v) flows through curved duct geometries. The goal is to ascertain if the Hamiltonian structure identified for infinitesimal, non-inertial, self-propelled spheroidal particles in steady straight ducts persists under these extensions.

Background

This paper establishes a Hamiltonian formulation for the motion of infinitesimal, non-inertial, self-propelled spherical, prolate spheroidal, and oblate spheroidal particles suspended in unidirectional steady laminar flow through straight ducts of arbitrary cross-section. The authors derive explicit constants of motion, potentials, and demonstrate confinement of orbits within basins determined by these potentials.

In the conclusions, the authors outline several natural extensions of the model—such as general ellipsoidal shapes, time-dependent flows, particle–wall interactions, finite-size effects, and curved duct geometries—and explicitly state uncertainty about whether these extended settings retain a Hamiltonian structure. Resolving this question would clarify the scope and robustness of the Hamiltonian framework for active particle dynamics in more realistic fluid environments.