Euler's polhode for solid body rotation - under construction
Modify the solid body
largest edge length aa, 0.6 ... 3:
middle edge length bb, 0.3 ... 2:
smallest edge length cc, 0 ... 1:
angular momentum, min-direction L1: 0 ... 1
angular momentum, middle-direction L2: 0 ... 1
angular momentum, max-direction L3: 0 ... 1
Integration of Euler's ODE with L=(L1,L2,L3), W=(L1/D1,L2/D2,L3/D3),
components in body coordinates, D1,D2,D3: principal moments of inertia.
Angular velocity determines the rotation which has constant
Dotted other solutions on angular momentum sphere!
The angular velocity draws the herpolhode in observer space.
Euler's polhode is the time dependent angular velocity curve W(t) in the coordinate system
of the rotating object. One chooses the eigen frame (e1,e2,e3) of the tensor of inertia to have a simple relation
between angular velocity W=(W1,W2,W3), angular momentum L=(L1,L2,L3) and the principal moments of inertia D1,D2,D3:
L1 = D1*W1, L2 = D2*W2, L3 = D3*W3. The polhode is the solution of Eulers ODE: L' = L x W. Once the angular velocity
(W1,W2,W3) is known in the body frame (e1, e2, e3), the rotation in observer space is given by
(e1, e2, e3)'(t) = (W1*e1 + W2*e2 + W3*e3)(t) x (e1, e2, e3)(t).
Euler's ODE guarantees conservation of angular momentum L in this rotation and also conservation of
L*L and of kinetic energy E = 0.5L*W. The polhode is therefore the intersection of a sphere and an ellipsoid
(in angular momentum coordinates, otherwise two ellipsoids). A more detailed explanation of how W(t) controls the movement
of the object is given here:
From angular velocity to rotational motion.
The definition of the tensor of inertia and the derivation of Euler's ODE is explained here:
Free rotational motion of rigid bodies.