A Dynamical-Time Framework for the Dynamics of Charged Particles

Abstract

We present a dynamical framework for modeling the motion of point-like charged particles, with or without mass, in general external electromagnetic fields. A key feature of this formulation is the treatment of time coordinate as a dynamical variable. The framework applies to the relativistic regime while consistently admitting a nonrelativistic limit. We also introduce a representation of particle trajectories in velocity space, which provides clear insight into the nature and asymptotic behavior of the dynamics. As an application, we compare the motion of massive and massless particles in a constant electromagnetic field and find that, for identical field configurations, their asymptotic behavior is independent of both mass and initial conditions. Finally, we explore the computational advantages of the dynamical-time formulation over the conventional uniform-time approach in two case studies: a uniform electromagnetic field, and an elliptically polarized wave propagating along a uniform magnetic field. In both scenarios, the proposed scheme exhibits improvements in accuracy and computational efficiency.

Figures (4)

• Figure 1: Dynamics of massless and massive charged particles in a constant electromagnetic field. Parameters: particle charge $q=1$; electric field $\omega_E=0.212$; magnetic field $\omega_B=1$; relative angle between the fields $\alpha = 3\pi/8$. Mass values: $m=0$ (red), $m=0.8$ (blue). Initial conditions: energy $\mathcal{E}_0=1$, velocity angles $(\varphi_0, \psi_0) = (\pi/2, 0)$, position $\mathbf{x}_0 = \mathbf{0}$. (a) Projection of spatial trajectories onto the $z=0$ plane. (b) Particle trajectories on the velocity sphere; the massive particle explores the interior, whereas the massless one remains confined to the surface. The asymptotic behavior is governed by the fixed points, where the orbits spiral inward or outward. (c) Spatial trajectories represented in the compactified $\xi$--coordinates. (d) Evolution of the mass-shell constraint $\chi(t)$ and of the conserved quantity $|Q_0(t) - Q_0(0)|$ as functions of the time coordinate $t$ (the remaining integrals of motion display similar behavior). Smaller deviations from zero indicate better numerical accuracy.
• Figure 2: Numerical comparison between the UTF (black) and the DTF (orange) for the massless particle in a constant electromagnetic field. The parameters and initial conditions are the same as in Fig. \ref{['fig:massless_vs_massive']}. Integration was performed using the RK4 method with $\lambda \in [0,1000]$ for the UTF and $\lambda \in [0,70]$ for the DTF, in both cases with $\Delta\lambda = 10^{-3}$ (all plots were mirrored for negative $\lambda$ values to highlight the symmetry of the solutions). (a) Evolution of the temporal coordinate $t$ as a function of $\lambda$. Although $t$ behaves similarly in both frameworks for small $\lambda$, the DTF exhibits exponential growth at large $\lambda$. (b) Evolution of the mass-shell constraint $\chi(t)$ versus $t$ using the RK4 integrator. (c) Adaptive step size of the RKDP integrator as a function of $t$, comparing the UTF (black $\diamond$) and DTF (orange $\circ$). The UTF step size varies over several orders of magnitude, whereas the DTF step size exhibits significantly smaller fluctuations, remaining around $0.01$. (d) Evolution of $\chi(t)$ versus $t$ using the adaptive RKDP integrator; in this case, the slight advantage of the UTF over the DTF becomes negligible.
• Figure 3: Dynamics of massless and massive charged particles in an elliptically polarized electromagnetic wave with a uniform axial magnetic field. Mass values: $m=0$ (red), $m=1$ (blue). Parameters: $q=1$, $a_1=1$, $a_2=1.6180$, $\omega=0.3236$, $\delta=\pi/4$, $b_0=1.6180$. Initial conditions: massless particle, $\mathbf{p}_0 = (1,0,0)$; massive particle, $\mathbf{p}_0 = \mathbf{0}$; for both cases, $\mathcal{E}_0 = 1$ and $\mathbf{x}_0 = \mathbf{0}$. (a) Spatial trajectories. (b) Projection of spatial trajectories onto the $z=0$ plane, showing periodic orbits for $b_0 = 1.6180$ (upper panel) and quasiperiodic ones for $b'_0 = 1.6280$ (lower panel). (c) Particle trajectories on the velocity sphere; the massive particle explores the interior, whereas the massless one remains confined to the surface; periodic orbits for $b_0 = 1.6180$ (upper panel) and quasiperiodic ones for $b'_0 = 1.6280$ (lower panel).
• Figure 4: Numerical comparison between the UTF (black) and the DTF (orange or green) for the massive particle in an elliptically polarized electromagnetic wave with a uniform axial magnetic field. The parameters and initial conditions are the same as in Fig. \ref{['fig:Massive_Massles_EMW']}. (a) Evolution of the temporal coordinate $t$ as a function of the parameter $\lambda$. (b) Evolution of the mass-shell constraint $\chi(t)$ versus $t$, comparing UTF with $\Delta\lambda=10^{-2}$ (black), DTF with $\Delta\lambda=10^{-2}$ (green), and DTF with $\Delta\lambda=5\times10^{-3}$ (orange); smaller deviations from zero indicate better numerical accuracy. (c) Adaptive step size of the RKDP integrator as a function of $t$, comparing the UTF (black $\diamond$) and DTF (orange $\circ$). In both cases, the step size oscillates with similar behavior, but the amplitude is markedly smaller for DTF.