Classical Spin — Poisson Leaf Interactive

Poisson Geometry Learning Hub

Classical spin precession with conservation-law knobs

The two main rotary knobs set the conserved quantities: the Casimir leaf $C = |\vec{S}|²$ and the energy $H = \vec{B}\cdot \vec{S}$. The orbit is the circle cut out by the energy plane on the sphere.

Physical model

The classical spin Poisson bracket is $\{Sᵢ,Sⱼ\}=εᵢⱼₖSₖ$. With $H(\vec{S})=\vec{B}\cdot \vec{S}$, the equation of motion is $d\vec S/dt=\{\vec{S}, H\}=\vec{B} \times \vec{S}$. Hence $\vec{S}$ rotates around $\vec{B}$ while preserving both $|\vec{S}|²$ and $\vec{B}\cdot \vec{S}$.

$$C=|\vec{S}|²=\text{constant}, \quad H=\vec{B}\cdot \vec{S}=\text{constant},$$ $$\vec{S}(t)=H \vec{b} + ρ(\hat e₁ \cos(φ₀+ωt)+\hat e₂ \sin(φ₀+ωt)),$$ $$\quad ρ=\sqrt{(C−H²)}, \quad\vec{b}=\vec{B}/|\vec{B}|$$

How to use

Drag a knob, use the mouse wheel over it, or focus it and press arrow keys. Drag the main 3D panel to rotate the camera. The default values match the direction of the Julia example’s magnetic field.

Spin visualization

3D Poisson leaf: sphere C=|S|², reference sphere C=1, vector field dS/dt=B×S, energy plane B·S=H, trajectory, S(t), and B. Drag to rotate.

Conservation-law map: allowed region |H| ≤ √C

Spin components over one precession cycle

Rotary controls

Casimir leaf
C = |S|²

–

sphere radius √C

energy
H = B·S

–

allowed by |H| < √C

field tilt
β

–

angle from S₃ axis

field azimuth
α

–

direction in S₁S₂ plane

initial phase
φ₀

–

starting point on orbit

animation speed
ω

–

visual speed only

Read the geometry from back to front: blue sphere = Casimir leaf, purple disk = Hamiltonian level set, bright cyan ring = their intersection.

Diagnostics

current orbit type

–

Casimir C = |S|²

–

leaf radius R = √C

–

energy H = B·S

–

energy level η = H/√C

–

orbit radius in energy plane ρ

–

current spin S(t)

–

magnetic field direction b

–

Legend

Intersection circle C∩H (full orbit)

Animated path traced on the intersection

Magnetic field direction B

Vector field dS/dt = B×S

Current spin vector S(t)

Filled energy plane B·S=H

Control–Geometry Summary

The motion is easiest to read geometrically: the Casimir fixes a sphere, the energy fixes a plane, and the spin traces their intersection circle.

Control	Mathematical meaning	Visual effect	Conservation-law interpretation
Casimir `C`	`C=\|S\|²`	Changes the radius of the Poisson leaf sphere from `√C`.	Choosing `C` means choosing a coadjoint orbit of `SO(3)`.
Energy `H`	`H=B·S`; the normalized energy level is reported as `η=H/√C`.	Moves the energy plane parallel to itself. Near `\|H\|=√C`, the circular orbit shrinks toward an equilibrium point.	The energy knob is dynamically bounded by `\|H\|≤0.98√C`, so the plane always intersects the Casimir sphere.
Field tilt `β` and azimuth `α`	Set the unit vector `b=B/\|B\|`.	Rotates the axis of precession and the normal direction of the energy plane.	They change the Hamiltonian function, but after choosing them, `H=B·S` remains conserved along each trajectory.
Initial phase `φ₀`	Chooses the initial point on the same intersection circle.	Slides the current marker around the same orbit without changing `C` or `H`.	It changes initial condition but not the selected conservation-law levels.
Speed `ω`	Sets visual angular speed in the browser.	Makes the animation faster or slower.	It does not alter the conserved quantities or the orbit shape.

Rule of thumb: C controls the sphere size, H moves the energy plane, the output η=H/√C tells how close the plane is to the poles, and β, α rotate the whole construction in spin space.