Jacobian determinant as a deformation field in static billiards

TL;DR

This paper introduces a deformation-based framework using the Jacobian determinant in noncanonical coordinates to analyze static billiard systems, revealing geometric structures and phase-space organization.

Contribution

It develops a novel approach to study billiard dynamics through Jacobian determinants, showing their role in phase-space structure and invariants, with analytical and numerical insights.

Findings

Jacobian determinant varies in noncanonical coordinates, indicating local phase-space expansion and contraction.

Curves where det J=1 act as deformation boundaries linked to invariant manifolds.

Period-two orbits restore unit determinant, higher-period orbits show angular modulation.

Abstract

We develop a deformation-based framework for analyzing static billiard systems through the Jacobian determinant computed in noncanonical angular coordinates. Although these systems are conservative, the determinant is not identically equal to unity, generating structured domains of local phase-space expansion and contraction. 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 deformation boundaries that intersect unstable periodic points and correlate with invariant manifolds. We prove analytically that period-two orbits restore exact unit determinant under composition, while higher-period orbits exhibit angular modulation consistent with reversibility. The Jacobian determinant thus reveals an additional geometric layer in phase-space organization and offers a complementary perspective on conservative billiard dynamics.