Hamiltonian structure of passive defect dynamics in two-dimensional nematic liquid crystals in unbounded domains

Abstract Hamiltonian structures for the passive dynamics of topological defects in nematic liquid crystals in two-dimensional systems are presented. Following the work by Miyoshi et al. (Miyoshi et al. 2024 Proc. R. Soc. A 480, 20240405. (doi:10.1098/rspa.2024.0405)), which demonstrated that the regularized Frank free energy associated with topological defects is equivalent to the Kirchhoff–Routh path function, the harmonic conjugate of the regularized Frank energy is investigated to elucidate the defect dynamics. Since defect motion is driven by the gradient of the Frank energy, the application of the Cauchy–Riemann equations shows that the harmonic conjugate of the regularized Frank energy serves as a Hamiltonian governing the defect dynamics. This Hamiltonian is expressed as a weighted sum of arguments between defect positions. Several conserved quantities, as well as the conditions for self-similar defect dynamics, are identified based on this Hamiltonian. The Hamiltonian structure is also applicable for estimating defect motion and defect–defect collisions within a limited field of view of the alignment angles.