Model and methods
A smooth directional Finsler shell
The laboratory explores a smooth directional Finsler geometry on a prescribed Euclidean background.
The model
The model defines a directional length function. Its geodesic equations determine the resulting trajectories. A smooth radial shell localizes the directional effect.
Let x denote position and y a nonzero tangent vector. The Euclidean radius is r = |x|. The Euclidean norm is α = |y|.
s = (x · y) / (rα)
F(x, y) = α[1 + A(r)q(s)]
Negative s denotes inward motion. Positive s denotes outward motion. Zero denotes a tangent direction to a sphere about the origin.
q(s) = exp(−1/s2) for s < 0
q(s) = 0 for s ≥ 0
Every derivative of q vanishes at zero. Thus the directional transition is smooth. Reversal of a vector can change its length.
The prescribed shell
The amplitude is A(r) = εχ(r/ℓ). The length scale ℓ sets the shell size. The dimensionless amplitude ε sets its strength.
ρ = r/ℓ
χ(ρ) = exp[1 − 1 / (1 − (ρ − 2)2)] for 1 < ρ < 3
χ(ρ) = 0 otherwise
The profile reaches one at ρ = 2. It vanishes smoothly at both shell boundaries. The geometry equals Euclidean geometry near the origin and outside the shell.
The proof gives strong convexity for nonnegative amplitudes A < 1.9176430337…. The laboratory restricts ε to [0, 1]. This range preserves a positive definite fundamental tensor in every direction.
What follows from the equations
- Exact closest approach. For impact parameter b > 0, the minimum radius is rmin = b.
- A positive excess angle. The accumulated polar angle exceeds π when the incoming ray crosses an active shell interval.
- A straight outgoing branch. The directional correction vanishes on that branch.
- No radial reflection. An incoming radial geodesic crosses the shell and reaches the Euclidean central region.
- Complete geodesics. Uniform norm bounds and smooth equations permit continuation through every finite affine parameter.
For constant positive Finsler speed vF, radial motion obeys the following relation.
dr/dτ = −vF / [1 + A(r)/e] < 0 for r > 0
A finite smooth amplitude cannot produce reflection at a positive radius. The affine parameter describes curve progress and carries no physical clock interpretation.
Verification and reproducibility
The research package compares scalar quadrature with direct Cartesian Euler–Lagrange integration. Their agreement supplies a numerical diagnostic. The analytical proofs establish the structural results.
The browser model source exposes the interactive calculation. The source archive contains the manuscript and reproduction code.
Scientific scope
The model defines spatial lengths on prescribed Euclidean space. A fixed annular profile changes those lengths by direction. Its geodesic action determines trajectories in this positive definite geometry.
General (α, β)-metrics supply the established mathematical setting. The research note develops this specific shell and its trajectory equations. Its conclusions apply under the stated assumptions.
Primary sources
- Yu and Zhu · On a new class of Finsler metrics (2011). General (α, β)-metrics and convexity criteria. Author manuscript.
- Mo and Zhou · The curvatures of spherically symmetric Finsler metrics in ℝn (2014). The broader spherical metric calculus.