Jacobian determinant as a deformation field in static billiards

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.

Figures (12)

• Figure 1: Geometry of the boundary for the elliptical-oval billiard with different values of the control parameters $\epsilon$, $e$, $p$ and $q$. In d) the trajectory of a single particle is shown for the first 20 collisions with $(\theta_0,\alpha_0)=(1.5,1.5)$.
• Figure 2: Phase space for the elliptical-oval billiard considering different sets of control parameters $\epsilon$, $e$, $p$ and $q$.
• Figure 3: Phase space and determinants of the Jacobian for the static elliptic-oval billiard with $e=0.3$,$\epsilon=0.1$,$p=1$ and $q=2$. The color scale represents the values of the determinant $\det J$, as given by Eq. (\ref{['detJ']}). Panel (a) shows a continuous range from $-2$ to $2$, whereas panel (b) highlights only two regions: red for $\det J>1$ and blue for $\det J<1$.
• Figure 4: Phase spaces and determinants of the Jacobian for the static elliptic-oval billiard with different values of $e$, $\epsilon$, $p$ and $q$. Determinant values below the unity are represented in blue, while those above unity are represented in red.
• Figure 5: Phase spaces and determinants of the Jacobian for the static elliptic-oval billiard with different values of $e=\epsilon$ and $p=q$, showing the multiplication of the colored regions and their subsequent deformation and merging.