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Charged particle motion in a strong magnetic field: The first order expansion

Ugo Boscain, Wadim Gerner

TL;DR

The paper provides a mathematically rigorous derivation of the first-order expansion for the motion of a charged particle in a strong magnetic field, using only the assumption of a strong field and without pre-imposing gyroradius or length-scale structure. It delivers a full trajectory expansion with explicit leading parallel motion and $O(1/\omega)$ perpendicular corrections, then rewrites the perpendicular part as curvature- and grad-$B$-drifts in the guiding-centre framework. By carefully comparing with the physics literature, the authors show that their results reproduce the standard drift terms and establish the a posteriori validity of common guiding-centre approximations, even near bounce points in magnetic mirrors. The work clarifies how mathematical and physical approaches align, emphasizes the minimality of mathematical assumptions, and provides a robust foundation for applications in magnetic confinement, neoclassical transport, and stellarator physics.

Abstract

We provide a mathematically rigorous derivation of the first order expansion of the motion of a charged particle in a strong magnetic field. In contrast to the derivations that can be found in the physics literature we solely assume throughout that the magnetic field is strong. In particular we do not need to make any structural assumptions on the particle motion, such as the gyroradius being small in comparison to the magnetic length scale. Instead, some of the additional assumptions which are usually made in the physics literature turn out to be an a posteriori consequence in our approach. Our approach further justifies the utilisation of the guiding centre approximation at "bounce points" within magnetic mirrors, a situation which violates the usual assumptions which are made in the physics literature when deriving the guiding centre approximation.

Charged particle motion in a strong magnetic field: The first order expansion

TL;DR

The paper provides a mathematically rigorous derivation of the first-order expansion for the motion of a charged particle in a strong magnetic field, using only the assumption of a strong field and without pre-imposing gyroradius or length-scale structure. It delivers a full trajectory expansion with explicit leading parallel motion and perpendicular corrections, then rewrites the perpendicular part as curvature- and grad--drifts in the guiding-centre framework. By carefully comparing with the physics literature, the authors show that their results reproduce the standard drift terms and establish the a posteriori validity of common guiding-centre approximations, even near bounce points in magnetic mirrors. The work clarifies how mathematical and physical approaches align, emphasizes the minimality of mathematical assumptions, and provides a robust foundation for applications in magnetic confinement, neoclassical transport, and stellarator physics.

Abstract

We provide a mathematically rigorous derivation of the first order expansion of the motion of a charged particle in a strong magnetic field. In contrast to the derivations that can be found in the physics literature we solely assume throughout that the magnetic field is strong. In particular we do not need to make any structural assumptions on the particle motion, such as the gyroradius being small in comparison to the magnetic length scale. Instead, some of the additional assumptions which are usually made in the physics literature turn out to be an a posteriori consequence in our approach. Our approach further justifies the utilisation of the guiding centre approximation at "bounce points" within magnetic mirrors, a situation which violates the usual assumptions which are made in the physics literature when deriving the guiding centre approximation.
Paper Structure (18 sections, 4 theorems, 75 equations, 1 figure)

This paper contains 18 sections, 4 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of mathematical results
  3. A useful identity
  4. Full particle trajectory
  5. Guiding centre expansion
  6. Physics literature approach
  7. Some definitions
  8. Smallness assumptions
  9. Structural assumptions
  10. The guiding centre equations of motion
  11. Comparison of the mathematical and physics literature results
  12. Reconciling the mathematical and physical approach
  13. Comparing the smallness assumptions
  14. Mathematical structural consequences vs. physical structural assumptions
  15. Final remarks
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $B\in C^2_{\operatorname{loc}}(\mathbb{R}^3,\mathbb{R}^3)$ be a no-where vanishing div-free vector field and let $x_0,v_0\in \mathbb{R}^3$. Let further $x_{\omega}$ denote the (unique) solution to (S1E2) with initial conditions $x_0,v_0$. Then there exists some $x\in C^3_{\operatorname{loc}}(\ma In addition, $\dot{x}(t)= h(t)b(x(t))$ for a suitable $h\in C^2_{\operatorname{loc}}(\mathbb{R})$ f

Figures (1)

  • Figure 1: The real particle position is depicted in red. The "dominant" guiding centre motion is depicted in blue. The gyromotion is depicted in black. As we follow the trajectory, the gyro motion rotates at some angle $\phi$.

Theorems & Definitions (9)

  • Theorem 1.1: BG25
  • Theorem 1.2: First order expansion (full particle trajectory)
  • Remark 1.3
  • Definition 1.4: Guiding centre and gyromotion
  • Theorem 1.5
  • Lemma 2.1
  • proof : Proof of \ref{['5L4']}
  • proof : Proof of \ref{['S2T2']}
  • proof : Proof of \ref{['S2T4']}