In vector form the coupled ODEs are
dA/dt = A×B
dB/dt = B×e3
where A = (Ax, Ay, Az), B = (Bx, By, Bz) are 3D real vectors and e3 = (0, 0, 1).
6 coupled non-linear ODEs In rectangular components of A and B are
dAx/dt = Ay*Bz - Az*By
dAy/dt = Az*Bx - Ax*Bz
dAz/dt = Ax*By - Ay*Bx
dBx/dt = By
dBy/dt = -Bx
dBz/dt = 0
I want to find bounded real-valued functions Ax(t), Ay(t), Az(t), Bx(t), By(t), Bz(t) of real variable t.
But how can i find the exact solution A(t) from dA/dt = MA when elements of the anti-symmetric matrix M are sinusoidal functions of t?
Since Magnus expansion would give approximate solution due to truncation. So does it mean that exact solution of my problem is not possible?
You can notice that Bz=const and that Bx*Bx+By*By=constant. Also that Ax^2+Ay^2+Az^2=const. Having solved the equations for (Bx,By,Bz)=(a*cos(t)+b*sin(t),-b*cos(t)+a*sin(t),Bz), the conserved quantity for A means that its equations describe a motion on a sphere, of fixed radius, in ``A'' space, that preserves the sphere. So the group theory of SO(3) is useful here: you write out the equations for (Ax, Ay, Az) and find how the matrix on the right hand side, which is known, since the Bx, By, Bz are, now, known, can be written in terms of the generators of SO(3).
The vector B simply precesses about the e3 axis with constant angular velocity of 1. Its length does not change. The vector A in turn precesses about the vector B with constant angular velocity |B|.
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