Rigid body motion 45I 2 ω ̇ 2 −ω 3 ω 1 (I 3 −I 1 ) =τ 2
I 3 ω ̇ 3 −ω 1 ω 2 (I 1 −I 2 ) =τ 3. (1.11.35)These are known as therigid body equations of motion. Given the solutions of these
equations of motion forωi(t), the three Euler angles, denoted (φ,ψ,θ), are then given
as solutions of the differential equations
ω 1 =φ ̇sinθsinψ+θ ̇cosψ
ω 2 =φ ̇sinθcosψ−θ ̇sinψ
ω 3 =φ ̇cosθ+ψ. ̇ (1.11.36)The complexity of the rigid body equations of motion and the relationship be-
tween the angular velocity and the Euler angles renders the solutionof the equations
of motion a nontrivial problem. (In a numerical scheme, for example, there are singu-
larities when the trigonometric functions approach 0.) For this reason, it is preferable
to work in terms of a new set of variables known asquaternions. As the name sug-
gests, a quaternion is a set offourvariables that replaces the three Euler angles. Since
there are only three rotational degrees of freedom, the four quaternions cannot be
independent.
In order to illustrate the idea of the quaternion, let us consider theanalogous prob-
lem in a smaller number of dimensions (where we might call the variables “binarions”
or “ternarions” depending on the number of angles being replaced). Consider again a
rigid diatomic moving in thexyplane. The Lagrangian for the system is given by eqn.
(1.11.9). Introduce a unit vector
q= (q 1 ,q 2 )≡(cosθ,sinθ). (1.11.37)Clearly,q·q=q^21 +q^22 = cos^2 θ+ sin^2 θ= 1. Note also that
q ̇= ( ̇q 1 ,q ̇ 2 ) = (−(sinθ)θ, ̇(cosθ)θ ̇) (1.11.38)so that
L=μd^2 q ̇^2 −U(q), (1.11.39)
whereU(q) indicates that the potential depends onqsincer=dq. The present
formulation is completely equivalent to the original formulation in terms of the angle
θ. However, suppose we now treatq 1 andq 2 directly as the dynamical variables. If we
wish to do this, we need to ensure that the conditionq^21 +q^22 = 1 is obeyed, which
could be achieved by treating this condition as a constraint (in Section 3.12, we shall
see how to formulate the problem so as to avoid the need for an explicit constraint
on the components ofq). In this case,qwould be an example of a “binarion.” The
“binarion” structure is rather trivial and seems to bring us right back to the original
problem we sought to avoid by formulating the motion of a rigid diatomicin terms
of the angleθat the outset! We must wait until Section 3.12 to see how to sidestep
the constraint. For a diatomic in three dimensions, the rigid-body equations of motion
would be reformulated using three variables (q 1 ,q 2 ,q 3 ) satisfyingq 12 +q 22 +q^23 = 1 so
that they are equivalent to (sinθcosφ,sinθsinφ,cosθ).