Variational rigidity for alternating symplectic billiards

The task is to classify smooth positively curved pairs of planar convex curves forwhich every finite physical orbit of the alternating symplecticbilliard has non-positive fixed-endpoint second variation. Such pairsare the independently translated homothetic ellipses, and all oftheir nontrivial finite Hessians are negative definite. The proofcombines an exact finite-block averaging identity with a commonarea-preserving normalization and independent scaling. The sameidentity bounds the orbit length at which a positive second variationoccurs outside the classified family. An exact polar reduction and aSchur-complement identity transfer the theorem to Minkowski billiards,giving a classification of totally integrable metric--table pairs forstrongly convex gauges. In the self-Minkowski case, total integrabilityforces the gauge to come from an inner product. No symmetry orpositive-curvature hypothesis is imposed in this case and curvature isobtained from calibration of continuous invariant graphs.