We consider trajectories in the configuration space of a freely rotating rigid body, the center-of-mass of which
is held fixed. A point in this space corresponds to a *angular position* which is most conveniently described by
a triplet of Euler-Rodrigues parameters.
Initial conditions for trajectories specify angular position **r**_{0} and angular velocity **ω**_{0}.
During a short span of time Δ*t* a trajectory can be approximated by a *parabolic trajectory*
which can be defined in a natural manner by supplementing the initial conditions with a single constant
vector **α** of *angular accelertation*.
A working method for computing trajectories for given
initial conditions is by means of an algorithm which gives just this angular acceleration as a function of initial
angular position, initial angular velocity, and time step. This defines an integrator which can be used for building
up a time-discrete trajectory.
Here we study a specific algorithm of this kind which is given by equation (52) in
Ulrich Mutze: *Rigidely connected overlapping spherical particles: a versatile grain model*, Granular Matter (2006)
or equation (59) in article8 .
We have **α**=Δ**ω**/Δ*t* where the quantities on
the r.h.s. are those in equation (52).
The animation shows that this formula defines trajectories which approximately
satisfy the principle of least action.
We are now in a position to explain what precisely is shown in this animation.

Each pixel of each frame corresponds to a finite parabolic trajectory and the pixel color corresponds to the value of the action integral along this trajectory. Violet colors correspond to the largest values, red colors to the lowest ones ('photon energy color coding'). The dark bands mark numerical values just like Fraunhofer lines mark frequencies in the solar spectrum. They form lines of equal action. They give a good indication where the action takes a minimum.

All parabolic paths under consideration are variations of an approximate physical trajectory ** p**
which is taken as originating from rather arbitrary initial conditions

These are the data adopted for the computation: The moments of inertia are (1.0, 1.4142, 3).
The initial angular position has Euler-Rodrigues parameters (1,1,1) with respect to the axes to which these
momenta of inertia refer. The initial angular velocity is (0,5,10) and Δ*t* is 0.0002.
For these data, the integrator (52) gives the angular acceleration as **α** = (-6.9033, -0.0207, 0.0073).
The absolute value |**α**| of **α** sets the scale for the representation of d**α**:
the frames cover all values da the components of
which are less than 10% of |**α**| in absolute value. UM 2006-06-26, last change 2008-05-01