Sometimes I have to revise #physics things, like "Why does the Larmor frame work?"
<br />Suppose you have circular motion at two angular frequencies A and B:
<br />x(t) = R(At)a + R(Bt)b
<br />Now define the average C=(A+B)/2 and difference D=(A-B)/2 and "un-rotate" at rate C:
<br />R(-Ct)x = R(Dt)a + R(-Dt)b
<br />Looks hopeful, the rotation rates are now equal and opposite.  We can use R(x)+R(-x)=2cos(x)I and R(x)-R(-x)=2sin(x)R(90°) but first have to rearrange to:
<br />R(-Ct)x = (R(Dt)+R(-Dt))(a+b)/2 + (R(Dt)-R(-Dt))(a-b)/2
<br /> = cos(Dt)(a+b) + sin(Dt)R(90°)(a-b),
<br />where the first step uses (p+q)(r+s)+(p-q)(r-s)=2pq+2rs.  The second line is two-dimensional simple harmonic motion at angular frequency D, in a frame rotating at rate C, as required.