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Hamiltonian guiding-center trajectory calculations in Boozer coordinates

This paper presents a Boozer-coordinate energetic-particle trajectory tracker (BEPT) for axisymmetric tokamak equilibria that utilizes Hamiltonian guiding-center equations, Fourier-spline field interpolation, and a fourth-order Runge-Kutta integrator to achieve high accuracy in field reconstruction and conservation of kinetic energy and toroidal canonical momentum.

Original authors: Diankun Li, Limin Yu, Zhengjie Ma, Yunming Zhang, Tong Wu

Published 2026-09-16

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CC BY 4.0

Original authors: Diankun Li, Limin Yu, Zhengjie Ma, Yunming Zhang, Tong Wu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of a fusion reactor, a special kind of plasma is held in place by powerful magnetic fields, much like a bead on a wire. This plasma contains not only the hot fuel needed for fusion but also high-speed particles born from the reaction itself, known as energetic particles. These particles are crucial because they carry immense energy that can help heat the plasma further, acting as a self-sustaining engine for the reactor. However, their paths are not simple; they drift and bounce around in complex ways, and if the magnetic cage is not perfectly shaped, these particles can escape. When they escape, they carry away valuable heat and can strike the reactor walls, potentially damaging the machine. To build a safe and efficient reactor, scientists need to predict exactly where these particles will go, but calculating their paths is a difficult mathematical challenge because the magnetic fields are intricate and the particles move at incredible speeds.

To solve this, researchers at the East China University of Science and Technology have developed a new computer tool designed to track these fast-moving particles with high precision. They focused on a specific type of magnetic environment found in doughnut-shaped reactors called tokamaks. In these machines, the magnetic field lines twist around the plasma, creating a complex three-dimensional maze. The team created a program that uses a special set of coordinates, named after a physicist who helped define them, to describe this magnetic maze. These coordinates allow the equations that govern the particle motion to be written in a much cleaner, more compact form than traditional methods. By using this streamlined approach, the researchers could build a simulation that follows the particles over long periods without the computer errors that usually accumulate and ruin the results.

The team tested their new tool, which they named BEPT, against a real-world magnetic configuration from a large tokamak experiment. They started by taking a detailed map of the magnetic field and converting it into a smooth, continuous mathematical description. This step was vital because the particles move so fast that the computer needs to know the exact shape of the magnetic field at every tiny point along their path, not just at the rough grid points where the original data was measured. By using a method that blends different mathematical techniques to fill in the gaps between data points, they created a field map so smooth that the difference between their version and the original data was smaller than one part in a billion. This level of smoothness is necessary because the particles drift based on how the magnetic field changes from one point to the next; if the map is jagged or rough, the calculated path will be wrong.

Once the magnetic map was ready, the researchers launched thousands of simulated particles into the virtual reactor to see how they would behave. They watched to see if the particles would stay trapped in the center or if they would drift out. The simulation successfully reproduced the two main types of paths these particles take: those that zip all the way around the reactor without stopping, and those that get caught in a magnetic trap, bouncing back and forth like a ball in a bowl while slowly drifting sideways. The computer tracked these paths for a very long time, equivalent to thousands of trips around the reactor, and checked if the fundamental laws of physics were being respected. In a perfect simulation, the total energy of a particle and a specific measure of its position should never change. The researchers found that their tool kept these values constant with an error so small it was barely detectable, proving that the simulation was stable and reliable over long durations.

To be absolutely sure their results were correct, the team compared their new tool against an established, trusted computer code that uses a completely different mathematical approach to solve the same problem. It is like two different navigators using different maps to find the same destination. Despite the different methods, both tools produced nearly identical paths for the particles. The time it took for particles to complete their loops and the width of their wandering paths matched up with differences of less than one percent. This agreement confirmed that the new tool works correctly and that the special coordinates they used are a valid and powerful way to study these particles. The researchers noted that while their tool is excellent for tracking particles in a steady, unchanging magnetic field, it does not yet include the effects of collisions between particles or sudden changes in the magnetic field caused by external forces.

The work represents a significant step forward in the ability to model fusion plasmas with high accuracy. By proving that a smooth, mathematically elegant description of the magnetic field can be combined with a robust tracking method, the team has provided a reliable foundation for future studies. This tool will help scientists better understand how energetic particles move and interact with waves in the plasma, which is essential for designing reactors that can hold their heat long enough to generate clean energy. The success of this simulation suggests that with the right mathematical framework, the complex chaos of a fusion reactor can be tamed and understood, bringing the dream of limitless fusion power one step closer to reality.

Technical Summary: Hamiltonian Guiding-Center Trajectory Calculations in Boozer Coordinates

Problem Statement
Energetic particles, such as fusion-born alpha particles and fast ions, are critical to tokamak performance but are susceptible to losses due to magnetic geometry and perturbations. Accurate numerical trajectory tools are essential for studying their confinement, neoclassical transport, and wave-particle interactions. While existing orbit-following codes utilize cylindrical coordinates or straight-field-line (PEST) coordinates, the authors note that Boozer coordinates offer a distinct advantage for Hamiltonian guiding-center dynamics. In Boozer coordinates, the covariant and contravariant representations of the magnetic field yield a compact form for the equations of motion, and the toroidal canonical momentum possesses a simple invariant structure for static axisymmetric equilibria. However, a complete workflow connecting standard equilibrium reconstructions (like EFIT) to differentiable Boozer-coordinate field representations for long-time integration has not been fully established in a single, verified framework.

Methodology
The authors developed a Boozer-coordinate energetic-particle trajectory tracker (BEPT) for axisymmetric tokamak equilibria. The methodology consists of three main components:

Hamiltonian Formulation: The guiding-center equations of motion are derived in Boozer flux coordinates $(\psi, \theta, \zeta)$ . The magnetic field is represented in the form $\mathbf{B}_0 = q(\psi)\nabla\psi\times\nabla\theta - \nabla\psi\times\nabla\zeta = I(\psi)\nabla\theta + g(\psi)\nabla\zeta$ . The equations of motion are projected onto this basis, incorporating the modified magnetic field $\mathbf{B}^*$ and the magnetic moment $\mu$ . The system is normalized using the magnetic-axis field, major radius, and a reference energetic-particle energy.
Field Reconstruction: To enable numerical integration, the authors implemented a Fourier-spline reconstruction pipeline. Starting from EFIT equilibrium data, they transformed the coordinates to straight-field-line Boozer coordinates. The poloidal dependence of equilibrium quantities (field strength, current functions, Jacobian) is expanded in a Fourier series, while the radial dependence is represented by cubic splines. This $C^2$ -continuous reconstruction ensures that the spatial derivatives required for drift calculations are smooth and accurate.
Numerical Integration: The normalized equations are integrated using a fourth-order Runge-Kutta (RK4) scheme. Field quantities and their derivatives are re-evaluated at each substep of the integrator to maintain accuracy.

Key Contributions
The paper makes three specific contributions to the field:

Formulation: It formulates the Boozer-coordinate guiding-center equations in a normalized form suitable for direct trajectory integration.
Pipeline Development: It establishes a complete pipeline from EFIT equilibria to differentiable Boozer-coordinate field quantities, including the coordinate transformation and the Fourier-spline reconstruction method.
Verification: It provides a comprehensive verification of the trajectory calculation through four distinct tests: equilibrium reconstruction accuracy, orbit topology validation, invariant conservation, and benchmarking against an established code.

Results
The BEPT code was tested using an axisymmetric equilibrium from the DIII-D tokamak (shot #159243).

Reconstruction Accuracy: The reconstructed magnetic field strength and Jacobian data agreed with the original equilibrium data with relative errors of order $O(10^{-9})$ . The spline fits successfully passed through discrete grid points while providing the smooth derivatives necessary for the drift equations.
Orbit Topology: The solver correctly reproduced standard orbit classes, including passing particles (completing poloidal transits) and trapped particles (forming banana orbits with magnetic mirror reversals). Three-dimensional trajectories showed expected toroidal precession.
Conservation Laws: Over long integration times ( $\tau = 10^4$ ), the normalized kinetic energy and toroidal canonical momentum were conserved with relative errors of approximately $O(10^{-8})$ and $O(10^{-6})$ , respectively. No secular growth was observed, indicating that the error is dominated by spatial interpolation rather than temporal discretization.
Benchmarking: A direct comparison with KAEC, a well-established code using PEST coordinates, showed consistent results. For representative passing and trapped particles, the transit and bounce frequencies agreed within $0.74\%$ and $0.26\%$ , respectively. The radial excursions and toroidal precession frequencies also showed close agreement, with deviations attributed to the sensitivity of secular drifts to equilibrium representation details.

Significance and Scope
The authors conclude that the BEPT solver successfully combines Hamiltonian guiding-center theory with a smooth Boozer-coordinate representation and explicit Runge-Kutta integration to produce accurate long-time trajectories in axisymmetric settings. The work serves primarily as a verification study rather than a complete fast-ion transport simulation. The authors explicitly state that the current implementation does not include collisions, magnetic ripple, non-axisymmetric perturbations, time-dependent fields, or self-consistent coupling to Alfvénic modes. The significance of the work lies in establishing a robust, verified workflow for Boozer-coordinate dynamics, which the authors plan to extend in future work to include non-axisymmetric configurations and DESC equilibrium data.

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