Elliptic billiard with harmonic potential: Classical description

Abstract

The classical dynamics of the isotropic two-dimensional harmonic oscillator confined by an elliptic hard wall is discussed. The interplay between the harmonic potential with circular symmetry and the boundary with elliptical symmetry does not spoil the separability in elliptic coordinates; however, it generates non-trivial energy and momentum dependencies in the billiard. We analyze the equi-momentum surfaces in the parameters space and classify the kinds of motion the particle can have in the billiard. The winding numbers and periods of the rotational and librational trajectories are analytically calculated and numerically verified. A remarkable finding is the possibility of having degenerate rotational trajectories with the same energy but different second constant of motion and different caustics and periods. The conditions to get these degenerate trajectories are analyzed. Similarly, we show that obtaining two different rotational trajectories with the same period and second constant of motion but different energy is possible.

Figures (9)

• Figure 1: Geometry of the elliptic billiard with harmonic potential. For a given focal distance $f,$ the boundary is defined by the radial elliptic coordinate $\xi=\xi_{0}$ or, alternatively, by the parameter $\gamma_{0}=\left( b/f\right) ^{2}=\sinh^{2}\xi_{0}.$
• Figure 2: a) Surfaces $u_{\pm}\left( \gamma,\beta\right)$ for $\gamma \in\left[ -1,\gamma_{0}\right] ,$$\beta\in\left[ 0,2\right] ,$ and $u\in\left[ 0,\gamma_{0}\right]$ with $\gamma_{0}=1.5.$ Red curve is the branch line Eq. (\ref{['curveu']}). The surface is doubled-valued in the region defined by Eq. (\ref{['E0']}). (b) Surfaces $v_{\pm}\left( \gamma,\beta\right)$ in the interval $v\in\left[ 0,1\right]$. Red curve is the branch line Eq. (\ref{['curvev']}). The surface is doubled-valued in the region defined by Eqs. (\ref{['vd']}).
• Figure 3: (a) Regions on the plane $\left( \gamma,\beta\right)$ corresponding to different types of trajectories in the billiard with $\gamma_{0}=1.5.$ (b) Representative periodic trajectories and their caustics (red dashed lines). Orbits $\left( 3,7\right)$ for $\mathcal{L}$- and $\mathcal{R}$-motions.
• Figure 4: Poincaré phase mappings $\left( \xi,p_{\xi}\right)$ and $\left( \eta,p_{\eta}\right)$ of the billiard with $\gamma_{0}=1.5$ for several values of the energy parameter $\beta=\{0,0.2,0.5,0.8,1.1\}.$ The iso-$\gamma$ lines are contour lines of the surfaces $\gamma\left( \xi,p_{\xi}; \beta \right)$ and $\gamma\left( \eta,p_{\eta}; \beta \right)$ obtained from Eqs. (\ref{['pe']}) and (\ref{['pn']}), respectively.
• Figure 5: (a) Winding number surface $w\left( \gamma,\beta\right)$ with $\gamma_{0}=1.5$ showing curves iso-$\beta$. The cutoff condition $w_{\mathrm{c}}\left( \beta\right)$ is given by Eq. (\ref{['wc']}). (b) Iso-$w$ level curves of $w(\gamma,\beta)$$=\left\{ 0,0.025,0.05,...,0.5\right\}$. The winding number cannot be greater than 1/2. (c) Detail of the $\mathcal{R}_{1}$ region. For $\beta>\beta_{e}$, the level curves have two roots of $\gamma,$ which correspond to degenerate orbits in the billiard. For $\gamma=0.4$ we have $\gamma_{p}=0.1525$ and $\gamma_{q}=0.3284$.