Phase-space structure and nonlinear dynamics of a charged particle on a helicoidal manifold under a magnetic field

We analyze the classical dynamics of a charged particle constrained to a helicoidally embedded Riemannian manifold in ${\mathbb{R}}^3$ under a uniform magnetic field in the ambient space. The induced metric $d{s}^2=d{u}^2+(1+w^2{u}^2)d{v}^2$ and the pulled-back symmetric gauge yield an exact reduction to a one-dimensional nonlinear Hamiltonian system. The resulting effective potential couples geometry and magnetic field, producing transitions between bounded and unbounded motion and a reorganization of phase-space topology. In the asymptotic regime, the dynamics reduces to a harmonic oscillator with ${\omega }_{\mathrm{e}\mathrm{f}\mathrm{f}}={\omega }_c\mathrm{/}2$ and $\mathrm{\ell }=\sqrt{2}{\mathrm{\ell }}_{\mathcal{B}}$. The system admits a Landau-type semiclassical spectrum and exhibits a geometry–magnetic control parameter $\mathrm{\Lambda }=q\mathcal{B}+\mathrm{\hslash }{k}_{v}w$ governing a chirality transition.