One side · one boundary · one reflected frame

Carry water around
the twist.

A positive water depth is globally meaningful on a Möbius strip. Clockwise vorticity is not: after one trip around the strip, a local orientation has reversed. This prototype evolves shallow water while enforcing that distinction at the seam.

WHAT WORKS NOWGeometry-correct shallow-water baseline

Curved metric, Levi–Civita source terms, conservative mass flux, reflective physical boundary, Möbius seam parity, twisted vorticity, two advected material loops, and the single boundary circulation.

WHAT THE KELVIN AUDIT REVEALSNot yet a variational integrator

The RK2 finite-volume update uses dissipative Rusanov flux. Material-loop circulation is therefore a measured error, not an invariant imposed by the algorithm. That makes this a useful baseline for the later Clebsch/Vakonomic method.

EMBEDDED STRIPdepth is color · water is not displaced along a nonexistent global normal

positive negative patch loop one-sided loop

VIEW

FIELD

LOOPS

time0.000

mass drift0

energy drift0

minimum depth—

patch-loop Γ—

patch-loop ΔΓ—

enclosed ∫ω—

Stokes defect—

one-sided Γ—

one-sided ΔΓ—

boundary Γ—

boundary ΔΓ—

vorticity RMS—

seam constraintexact

Initial experiment

Gravity gwave restoring force2.40 Mean depth Hpositive density on the strip1.00 Initial amplitudeheight or velocity strength0.18 Steps per framesimulation speed, not the CFL step2 Transport the local frameone lap reflects e₂; two laps restore it0.00 laps

KELVIN CIRCULATION AUDIT

\Gamma(t)=\oint_{\gamma_t}u^\flat,\qquad \dot\Gamma=0

gold

bounds a material patch; the readout independently compares Γ with the enclosed vorticity two-form ∫ω

cyan

winds once around the Möbius strip and has no globally oriented interior

The curves move with the computed velocity. In the exact inviscid equations both circulations remain fixed; here their drift measures the baseline solver’s numerical dissipation.

O(2) HOLONOMY AUDIT

Q_\gamma\in O(2),\qquad\det Q_\gamma=-1

The cyan arrow is the longitudinal frame vector. The orange arrow is transverse. Follow them for one lap: the orange vector returns reversed, so a global $J$ cannot be chosen.

Kelvin’s theorem, made operational

Follow the loop.
Integrate the covector.

A fixed curve samples whatever fluid happens to pass through it. Kelvin’s theorem instead follows the same particles, which is why both colored curves deform as the simulation advances.

MATERIAL LOOP

\frac{d}{dt}\oint_{\gamma_t}u^\flat=\oint_{\gamma_t}d\!\left(\tfrac12|u|^2-gh\right)=0

Pressure and gravity change the local velocity through an exact differential. A closed loop cannot detect that exact part.

01
Gold bounds a patch.
It is contractible. In an oriented local chart, Stokes gives $\oint_{\partial S_t}u^\flat=\int_{S_t}du^\flat$ .
02
Cyan detects topology.
It is one-sided. Its lift starts on sheet A and ends on sheet B, so there is no ordinary oriented patch whose boundary is this loop.
03
ΔΓ grades the integrator.
The continuum theorem predicts zero. Rusanov dissipation and interpolation create the displayed drift; a Kelvin–Noether method should reduce it substantially.

What changed in the equations?

The connection enters locally.
The twist enters globally.

The embedded parameterization has diagonal metric $g=E(s,r)\,ds^2+dr^2$ . Local curvature produces Christoffel terms; the reflected seam produces the global orientation holonomy.

01 · EMBEDDED METRIC

No flat-coordinate shortcut.

E=(R+r\cos\tfrac{s}{2})^2+\tfrac{r^2}{4}

The finite-volume cell measure is $\sqrt E\,ds\,dr$ . The solver includes the associated Levi–Civita geometric source.

02 · MASS

Depth is an ordinary density.

\partial_t(\sqrt E h)+\partial_i(\sqrt E h u^i)=0

The seam pulse is one physical bump even though it appears in two corners of the fundamental rectangle.

03 · MOMENTUM

Covariant acceleration.

\partial_t(hu^i)+\nabla_j\!\left(hu^iu^j+\tfrac12gh^2g^{ij}\right)=0

Pressure is intrinsic. It does not require choosing a normal side of the Möbius strip.

04 · TWISTED VORTICITY

Rotation changes sign after one lap.

d u^\flat=\zeta\,\rho_g,\qquad \zeta\in\Gamma(o(M))

The two-form $du^\flat$ is intrinsic; the displayed scalar ζ depends on a local orientation.

WHY THE DOUBLE COVER HELPS

Solve on a cylinder; constrain the two sheets.

The oriented double cover has two coordinate sheets. Ordinary fields copy with a transverse reflection. Twisted fields copy with an additional minus sign. The “double cover” view makes those rules visible without hiding them in an anti-diagonal matrix block.

quantity	bundle/type	deck transformation
depth h	ordinary scalar density	h₂(s,r) = h₁(s,−r)
velocity u	tangent vector	(uˢ,uʳ)₂ = (uˢ,−uʳ)₁(s,−r)
vorticity ζ	orientation-line scalar	ζ₂(s,r) = −ζ₁(s,−r)
streamfunction ψ	orientation-line scalar	ψ₂(s,r) = −ψ₁(s,−r)

THE NEXT IMPLEMENTATION STEP

Replace the baseline update, not the geometry.

The spatial ingredients above can feed a vakonomic semidirect-product solver. The low-rank state then needs Clebsch label pairs, the advected depth, its conjugate potential, and the one harmonic circulation coordinate.

1
Build equivariant scalar, twisted-scalar, one-form, and half-density bases on the double cover.
2
Assemble the Koopman advection map using the O(2) edge transports and curved metric.
3
Derive the depth-weighted sharp map and semidirect-product momentum map.
4
Integrate mass, Kelvin–Noether circulation, energy, and even potential-vorticity Casimirs together.