HCRP project 2009
Caustic on spheroid
Pictures of Caustics on ellipsoid
Office: SciCtr 434

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The exponential map on a spheroid x2/4+y2+z2=1. The primary caustic (the first root of the Jacobi field f starting at the initial point) is drawn in yellow, the secondary caustic (the second root of f(g(t)) orange, the ternary red. One clearly sees the 4 cusps as the still unproven Jacobi's last geometric statement claims. 100 geodesics of the 6000 computed geodesics have been drawn. The picture was computed by solving the geodesic equations g''k = -Gijk g'i g'j (where G is the connection using Einstein notation) in conjunction with the Gauss-Jacobi equation f'= - K(g(t)) f (where K is the curvature of the surface ) numerically with Mathematica. Since special needs are required (identifications of the map, assuring that we stay on the energy surface, checking whether the Jacobi field f reaches zero), the differential equations were "hand" integrated using Runge-Kutta and not using built-in DSolve routines. 6000 geodesics g(t) were computed on the ellipsoid and drawn in the spherical coordinate plane with (theta,phi) coordinates.

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The same picture but with the four primary caustics computed with 8000 geodesics. Only the caustics are shown. It is natural to ask whether the caustic C, the union of all primary caustics, the set of points C={ exp_p(v) | det D exp_p(v) = 0 } has a closure with interior or whether it is a Cantor set. Sturm-Liouville theory could help to answer the question. The Jacobi field operators L = d/dt^2 + V(t) = d/dt^2 + K(g(t)) are almost periodic Schroedinger operator because the geodesic flow on the ellipsoid is integrable. We need to know about the structure of roots L u = 0, but analytic expressions for the roots is already difficult if the potential V(t) is known in closed form.

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The non-rotationally symmetric ellipsoid x2/1.12 + y2/1.062 + z2 = 1 has caustics close to the antipode in the sphere case. We see again the primary, secondary and ternary caustic. 4000 geodesics were computed, 50 of them shown.

Questions and comments to knill@math.harvard.edu
Oliver Knill | Department of Mathematics | Harvard University