The origin and nature of spiral arms in barred galaxies are still under debate. Among others, the theory of invariant manifolds provides a framework capable of explaining spiral and ringed structures in these systems. We investigated the dynamical origin and kinematic signatures of spiral structure in an N-body simulation of an isolated barred galaxy to assess whether invariant manifold theory provides a consistent dynamical framework to disentangle the disc particle populations and identify those that genuinely build, trace, and sustain the spiral arms. For each simulation snapshot, we measured the bar and spiral properties and reconstructed the corresponding gravitational potential using. We AGAMA computed the Jacobi energy of disc particles and classified them relative to the energies of the equilibrium points, thereby isolating manifold-compatible orbits. We analysed their spatial distribution and velocity structure to characterise spiral-related streaming motions. The Jacobi constant provides a physically motivated dynamical separator that reveals three distinct kinematic populations: (i) low-energy particles on nearly circular orbits populating most of the disc (sim60%), (ii) high-energy particles associated with banana orbits (łesssim10%), and (iii) manifold-compatible particles (sim30--40%) originating near the bar and following transit orbits along the spiral arms. Only the manifold-compatible population generates the prominent outward-migrating ridge observed in the R-v_̌arphi plane and reproduces the characteristic spiral streaming pattern. In contrast, the low-energy population exhibits a global quasi-circular motion with small perturbations induced by the self-gravity of the spiral structure. Our results demonstrate that the spiral arms are dynamically traced by the manifold-compatible population, which forms the backbone of the structure and drives effective radial transport. The bulk of low-energy disc particles respond to the spiral perturbation similarly to the traditional density wave picture, enhancing the density contrast caused by the invariant-manifold compatible particles. In this framework, barred spiral arms emerge as material structures sustained by manifold-guided transport, with the surrounding disc behaving as a system of material density waves.
Soler-Terricabras et al. (Tue,) studied this question.