A differential equation involving the cross product

Question

In the course of work in atomic physics, I've run into a differential equation of the form $$\frac{d}{dt}\vec{X}(t) = \vec{\Omega}(t) \times \vec{X}(t)$$ where $\vec{\Omega}(t)$ is given and we wish to solve for $\vec{X}(t)$. Is there a name for this type of equation? Can you please provide references on the structure of solutions to this equation?

Answer 1

In classical dynamics, this kind of equation is associated with the kinematic transport theorem (not to be confused with the continuum mechanical transport theorem).

The general form is an identity that describes the rate of change of a vector quantity $\vec{X}(t)$ in time as measured in an inertial reference frame $A$—let's call that $\frac{d\vec{X}(t)_A}{dt}$—as compared to the rate of change of the quantity in a rotating reference frame $B$, $\frac{d\vec{X}(t)_B}{dt}$. The general form is:

$$\frac{d\vec{X}(t)_A}{dt} = \frac{d\vec{X}(t)_B}{dt} + \vec{\Omega}(t)\times\vec{X}(t)$$

where $\vec{\Omega}(t)$ is the angular velocity of frame $B$ with respect to $A$. When the quantity you are considering is not changing in the rotating frame, it simplifies to the form you have above. Most textbooks on dynamics contain a discussion of this equation—see, for example, see Chapter 18 of Ruina's text.