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Appendix D4 — The Magnetic Geodesic Flow and Its Conjugate Locus

Where the identity "Larmor motion is Heisenberg" is proved. The Pontryagin Maximum Principle turns the magnetic contact structure into a flow whose curves are unit-speed circles of curvature B·w — the Larmor orbits — and for a uniform field the conjugate locus is Heisenberg's, reproduced by the code to eleven digits.

By Igor Moiseev · 20 August 2026
The Geometry of Forbidden Directions
  1. The Geometry of Forbidden Directions: A Research Program
  2. The Magnetic Contact Geometry: Larmor Motion Is Heisenberg
  3. Reading the Field Gradient from the Caustic
  4. Finding Magnetic Nulls with a Growth Vector
  5. The Null Gallery: Grounding the Estimator in the Real Sun
  6. The Transition State: Reading a Null Collision from One Point
  7. The Litmus Tests: What Survived Our Own Review
  8. Jupiter's Buried Skeleton: The Polar Patch as a Combination of Separatrices
Appendices — Theory Background
  1. D1. Connections, Holonomy, and Curvature
  2. D2. The Selection Rule: Why Anisotropic Transport Isn't Sub-Riemannian
  3. D3. The Law Q = d + k + 2, Derived
  4. D4. The Magnetic Geodesic Flow and Its Conjugate Locus ← you are here
  5. D5. The Nilpotent-Deviation Gradient Formula
What this appendix covers
The geodesic flow behind Part 2 and Part 3: the derivation that the sub-Riemannian geodesics of the magnetic contact structure are Larmor orbits, the uniform-field conjugate locus (Heisenberg's), and how the caustic is computed and shown gauge-invariant.

The Hamiltonian

The magnetic contact structure has orthonormal horizontal frame $X_1 = \partial_x + A_x\partial_\varphi$, $X_2 = \partial_y + A_y\partial_\varphi$ (Appendix D1). The sub-Riemannian geodesics are the projections of the flow of the Hamiltonian $H = \tfrac12(h_1^2 + h_2^2)$ on the cotangent bundle, where $h_i = \langle p, X_i\rangle$. Writing $w = p_\varphi$ for the momentum conjugate to the flux — conserved, because neither $\mathbf A$ nor $B$ depends on $\varphi$ — the Pontryagin Maximum Principle gives, for unit-speed geodesics ($h_1^2+h_2^2=1$, so $(h_1,h_2)=(\cos\theta,\sin\theta)$),

\[\dot x = \cos\theta,\qquad \dot y = \sin\theta,\qquad \dot\varphi = A_x\cos\theta + A_y\sin\theta,\qquad \dot\theta = B(x,y)\,w .\]

Larmor motion is the geodesic flow

Read the last equation: the velocity direction $\theta$ turns at rate $B\,w$. A unit-speed curve whose direction turns at a constant rate is a circle of radius $r_L = 1/(B\lvert w\rvert)$ — exactly the Larmor orbit of a charged particle of the corresponding charge-to-mass ratio in the field $B$. The sub-Riemannian shortest paths are not an analogy for cyclotron motion; they are cyclotron motion, with the sub-Riemannian momentum $w$ playing the role set by the particle’s speed and charge. The flux $\varphi$ swept is the enclosed area — the third coordinate.

The uniform-field conjugate locus is Heisenberg’s

For a uniform field $B_0$, the orbit is a circle traversed at unit speed; it closes, and the whole family of orbits from a point refocuses, after one period:

\[t_c \;=\; \frac{2\pi}{B_0\,|w|}\qquad(\text{the Larmor / cyclotron period}).\]

In the symmetric gauge $\mathbf A = \tfrac{B_0}{2}(-y, x)$ the frame is, term for term, the Heisenberg frame; the conjugate locus is the Heisenberg group’s central axis; and the whole structure is the flat model of 3D contact sub-Riemannian geometry. The abstract Heisenberg group and the first system in a plasma course are the same object, and the identification is exact, not approximate.

The code reproduces $t_c = 2\pi/(B_0\lvert w\rvert)$ to one part in $10^{11}$ across a range of $B_0$ and $w$, and the deviation for a varying field (Part 3) is measured against this exact baseline.

Computing the conjugate locus for a general field

For a non-uniform field there is no closed form, so the caustic is found numerically. The conjugate time along a geodesic is the first positive time at which the Jacobian determinant of the exponential map vanishes:

\[J(t) = \det\!\big[\,\partial_{\theta_0}\gamma \;\big|\; \partial_w\gamma \;\big|\; \dot\gamma\,\big] = 0 .\]

The base geodesic and its perturbations in $(\theta_0, w)$ are integrated together in one batch, $J(t)$ is formed along the trajectory, and its first sign change is bracketed. This is the routine that produces the nilpotent-deviation $\delta$ of Part 3.

(A scope note the criterion needs: this Jacobian test finds conjugate points of normal geodesics. It is exhaustive here only because wherever $B \neq 0$ the structure is contact, and contact structures have no nontrivial abnormal geodesics; approaching a null, $B \to 0$, contact fails and abnormals appear — not hypothetically: the horizontal lift of a nondegenerate zero curve is Montgomery’s strictly abnormal minimizer (1995) — which is one more reason everything near nulls in this series is handled through the tangent cone rather than this routine.)

Gauge invariance of the caustic

A gauge change $\mathbf A \to \mathbf A + \nabla\chi$ shifts the flux coordinate $\varphi \to \varphi + \chi(x,y)$ — a diffeomorphism of the target space. A diffeomorphism cannot move the set where a map’s Jacobian drops rank, so the conjugate time is gauge-invariant. The code confirms it: the symmetric gauge $\tfrac12(-y,x)$ and the Landau gauge $(0,x)$ give identical conjugate times to eight decimals. This is the check that certifies $\delta$ measures geometry rather than a gauge artefact — the same gauge discipline flagged in Appendix D1.

References

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