Q = d + k + 2. This post builds the actual geometry for a magnetic field, shows
its geodesics are the orbits every physicist already knows, and measures the law.
The structure
A charged particle in the plane, tracking one extra number: the magnetic flux $\varphi = \int \mathbf A\cdot d\boldsymbol\ell$ it has swept, where $\nabla\times\mathbf A = B\,\hat z$. The state is $(x,y,\varphi)$, and the allowed moves are “step in the plane, and let the flux follow”:
\[X_1 = \partial_x + A_x\,\partial_\varphi,\qquad X_2 = \partial_y + A_y\,\partial_\varphi.\]You may drive along $X_1$ and $X_2$; you may not drive along $\partial_\varphi$ directly. But you reach it anyway, because the bracket of the two allowed moves is
\[[X_1, X_2] \;=\; (\partial_x A_y - \partial_y A_x)\,\partial_\varphi \;=\; B(x,y)\,\partial_\varphi.\]Wherever $B\neq0$, one bracket reaches the forbidden flux direction: the structure is contact, growth vector $(2,3)$, and — as Part 1 promised — $Q = d + 2 = 4$.
Larmor motion is Heisenberg — exactly
Now solve for the geodesics. The sub-Riemannian geodesics of this structure are unit-speed curves whose velocity direction $\theta$ turns at a rate set by the field:
\[\dot x = \cos\theta,\qquad \dot y = \sin\theta,\qquad \dot\theta = B(x,y)\,w,\]for a conserved momentum $w$. A curve of constant curvature $B\,w$ — a circle. This is precisely the Larmor orbit of a charged particle in a magnetic field: the shortest paths of the magnetic contact geometry are the trajectories the Lorentz force already draws. And the flux swept as the particle goes around is the area it encloses — the third coordinate.
For a uniform field $B_0$ in the symmetric gauge $\mathbf A = \tfrac{B_0}{2}(-y, x)$, write this out and it is, letter for letter, the Heisenberg group: the frame is the Heisenberg frame, the geodesics close at
\[t_c = \frac{2\pi}{B_0\,|w|}\qquad(\text{the Larmor period}),\]and the caustic — the set where a family of orbits refocuses — is the Heisenberg group’s central axis. The abstract flat model of sub-Riemannian geometry and the first system in every plasma-physics course are the same object — a statement about this planar lift; the three-dimensional field’s lift is quasi-contact with a different flat model (Part 4). (Details in Appendix D4; the code reproduces $t_c$ to one part in $10^{11}$ and confirms it is gauge-independent.)
The flux is area-like — so it is weight two
Why is $\varphi$ a weight-2 coordinate and not weight 1? Because you cannot get flux by going somewhere — only by going around. A loop of spatial size $r$ encloses area $\sim r^2$ and sweeps flux $\varphi \sim B\,r^2$. So as you shrink the available path length $r$, the reach in $x$ and $y$ shrinks like $r$, but the reach in $\varphi$ shrinks like $r^2$ — twice as fast. On a log–log plot of reach versus path length, the spatial coordinates have slope 1 and the flux has slope 2. The figure shows the real measurement.
research/preferred-directions (scripts/run_p2_law.py confirms
$k=0..3$; an equivalent 2D check lives in the sibling caustics-to-groups program).
The null jump, and what it means
Read the figure as a story about a magnetic null. Almost everywhere the field is ordinary, the flux is weight 2, and $Q = d + 2$. But approach a point where $B$ vanishes to order $k$ and the flux line steepens: the coordinate becomes weight $k+2$, and $Q$ jumps to $d + k + 2$. In 2D, $Q$ goes $4 \to 5$ at a simple null; in 3D (Part 4) it goes $5 \to 6$.
So the homogeneous dimension is a curvature-degeneracy meter. It sits at its floor value everywhere the field is healthy, and rises by exactly the vanishing order on the measure-zero set where the field fails. A sub-Riemannian invariant, read off how far a small ball reaches in each direction, locates the magnetic nulls — the candidate reconnection sites (a null alone does not reconnect anything: that takes localized non-ideal evolution, and 3D reconnection can also happen with no null — Pontin & Priest 2022) — and reads how degenerate each one is. Part 4 turns that into a working estimator and checks it against the standard null finder — a consistency check on known nulls, not a blind-detection claim.
What the growth vector does not yet tell you is the shape of the field near the null — how it curves, which way it leans. That information is not in the ball’s reach exponents; it is in the shape of the caustic. Reading the field’s gradient off that caustic is Part 3.
Glossary
- Magnetic contact structure — the sub-Riemannian structure on (position, flux) with frame $X_i = \partial_i + A_i\partial_\varphi$; contact wherever $B\neq0$.
- Larmor orbit — the circular trajectory of a charged particle in a magnetic field; here, a sub-Riemannian geodesic.
- Flux coordinate $\varphi$ — the swept magnetic flux $\int\mathbf A\cdot d\boldsymbol\ell$; the forbidden, area-like direction.
- Weight of a coordinate — the power of path length $r$ at which it fills in; the slope on the log–log reach plot. Spatial coordinates: 1; flux: $k+2$.
- Reach — how far a coordinate ranges over geodesics of a given length; a high quantile of its absolute value.
- Null jump — the increase of $Q$ from $d+2$ to $d+k+2$ at a magnetic null of order $k$.
References
- L. D. Landau & E. M. Lifshitz. The Classical Theory of Fields — Larmor motion.
- A. Bellaïche (1996). “The tangent space in sub-Riemannian geometry.” In Sub-Riemannian Geometry, Progr. Math. 144, Birkhäuser. (Ball–Box, weights — at equiregular points; the null, where the growth vector jumps, is a singular point, and what the exponents mean there is spelled out in Appendix D3’s fine print, with F. Jean’s monograph and Ghezzi–Jean’s non-equiregular volume paper as the governing references.)
- A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press. (Heisenberg, Martinet.)
- D. I. Pontin & E. R. Priest (2022). “Magnetic reconnection: MHD theory and modelling.” Living Rev. Solar Phys. 19, 1. doi:10.1007/s41116-022-00032-9. (Why the nulls are only candidate reconnection sites.)