Geometrical Optics — Poisson Variety Interactive

Poisson Variety Learning Hub

From ray optics to the singular variety $XY-Z^2=0$

This page visualizes the reduced Lie–Poisson variables $X=q^2$, $Y=p^2$, and $Z=q\cdot p$. The Casimir $C=XY-Z^2$ cuts the affine Poisson variety into leaves: $C>0$ gives regular hyperboloids, while $C=0$ gives a singular cone with a vertex at the origin.

Reduced optics model

In an axisymmetric paraxial medium, the optical Hamiltonian may be written as $H^2=n(X)^2-Y$. Radial turning points occur where $Z=0$, so the constants $(H_0,C_0)$ reduce the problem to the single equation $F(X)=0$.

$$\{X,Y\}=4Z,\qquad \{X,Z\}=2X,\qquad \{Y,Z\}=-2Y$$ $$C=XY-Z^2,\qquad H^2=n(X)^2-Y$$ $$F(X)=X\bigl(n(X)^2-H_0^2\bigr)-C_0$$

How to use

Drag the knobs or use arrow keys while focused. Choose the Casimir level $C_0$, the energy $H_0$, and the parabolic GRIN parameter $\alpha$. The big 3D panel shows the Poisson leaf $C=C_0$; the 2D panels show the turning-point polynomial and the simulated radius $r(z)$.

Poisson variety and GRIN ray simulator

3D reduced space $(X,Y,Z)$: regular leaves $C>0$, singular cone $C=0$, energy surface, and trajectory. Drag to rotate.

Turning-point polynomial $F(X)$; roots are radial extrema.

Numerical ray radius $r(z)=\sqrt{X(z)}$ in the parabolic GRIN model.

Rotary controls

Casimir level
C₀=XY−Z²

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0 gives singular cone

Hamiltonian
H₀

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propagating if H₀>0

GRIN gradient
α

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$n^2=n_0^2-\alpha X$

central index
n₀

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index on the axis

initial phase
φ₀

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starting point on level curve

animation speed
v

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visual speed only

The cyan curve is $C=C_0\cap H=H_0$. Its intersections with the plane $Z=0$ are the positive roots of $F(X)$.

Diagnostics

current regime

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Poisson variety level

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discriminant $\Delta$

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positive roots of $F(X)$

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turning radii

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current point $(X,Y,Z)$

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check $XY-Z^2$

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check $H^2=n^2-Y$

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Legend

Energy–Casimir curve $C\cap H$

Animated trajectory segment

Singular cone $C=0$

Energy parabola $Y=n_0^2-\alpha X-H_0^2$

Turning plane $Z=0$

Current reduced ray state

Control–Geometry Summary

Algebraically, the coordinate ring $\mathbb R[X,Y,Z]$ has a Lie–Poisson bracket. Since $C=XY-Z^2$ is a Casimir, the ideal $(C-c)$ is a Poisson ideal, so each level set is a Poisson subvariety.

Object	Equation	Geometric meaning	Optical interpretation
Affine Poisson variety	`Spec R[X,Y,Z]` with `{X,Y}=4Z`	The reduced Lie–Poisson space `sp(2,R)*`.	Coordinates are `X=q²`, `Y=p²`, `Z=q·p`.
Casimir level	`C=XY−Z²=c`	A Poisson subvariety because `C` Poisson-commutes with every function.	Fixes the transverse angular-momentum-like invariant.
Regular leaf	`C>0`	A smooth two-dimensional hyperboloid branch in the physically accessible region.	Generic skew rays; radial motion may be guided depending on $F(X)$.
Singular Poisson variety	`C=0`, i.e. `XY−Z²=0`	A quadratic cone singular at the origin.	Meridional/radial rays. The vertex is the degenerate state `q=p=0`.
Turning-point criterion	`F(X)=X(n(X)²−H₀²)−C₀`	Positive simple roots are intersections with the plane `Z=0`.	Two positive roots mean guided radial oscillation; a double root is critical.

For the parabolic GRIN model $n^2(X)=n_0^2-\alpha X$, the polynomial is $F(X)=-\alpha X^2+(n_0^2-H_0^2)X-C_0$, so the discriminant $\Delta=(n_0^2-H_0^2)^2-4\alpha C_0$ controls the transition between guided, critical, and unguided regimes.