Jacobian determinant as a deformation field in static billiards

We develop a deformation-based framework for analyzing static billiard systems by mapping the spatial distribution of the Jacobian determinant in noncanonical angular coordinates, treating it as continuous deformation field. Although these systems are conservative, the determinant is not identically equal to unity, generating structured domains of local phase–space expansion ( det J > 1 ) and contraction ( det J < 1 ). We show numerically that these domains balance globally, providing a geometric manifestation of area preservation in noncanonical variables. The curves defined by det J = 1 act as a deformation-based skeleton of the phase space that intersect unstable periodic points and correlate with invariant manifolds. We prove analytically that period-two orbits restore exact unit determinant under composition and show that global conservation forces the composed Jacobian determinant of any higher-period orbit to strictly equal unity. The Jacobian determinant thus reveals an additional geometric layer in phase–space organization and offers a complementary perspective on conservative billiard dynamics.