$r(t)$ - Distance between Star and Planet
$f(t)$ - Angular Coordinate of Planet in its orbit
$\varphi (t)$ - Angular Coordinate of Star’s rotation
$\psi (t)$ - Angular Coordinate of Planet’s rotation
Kinetic Energy
$\mathcal{T}=\frac{\left(m_p m_s\right) \left(r(t)^2 f'(t)^2+r'(t)^2\right)}{2 \left(m_p+m_s\right)}+\frac{1}{2} \mathcal{I}_p \psi '(t)^2+\frac{1}{2} \mathcal{I}_s \varphi '(t)^2$
Gravitational Potential
$\mathcal{P}=-\frac{\left(3 \text{Gm}_s\right) \left(\frac{1}{2} T_p \cos (2 (f(t)-\psi (t)+\pi ))+Q_p\right)}{2 r(t)^3}-\frac{\text{Gm}_s m_p}{r(t)}$
Magnetic Potential
$\mathcal{U}= \frac{- \mu M_m M_p}{4 \pi r^3} \left[\sin \left(\gamma _p\right) \sin \left(\gamma _s\right) R +\cos \left(\gamma _p\right) \cos \left(\gamma _s\right)\right]$
$R = \left[3 \cos \left(\varphi _p\right) \cos \left(-f+\phi _s+t \varphi '\right)-\cos \left(f+\varphi _p-\phi _s-t \varphi '\right)\right]$
Lagrangian
$\mathcal{L}=-\mathcal{P}-\mathcal{U}+\mathcal{T}$
Equation for r(t) -
$$ -\frac{9 \text{Gm}_s \left(T_p \cos (2 (f(t)-\psi (t)))+2 Q_p\right)}{r(t)^4}-\frac{3 \mu M_m M_p \left(\sin \left(\gamma _p\right) \sin \left(\gamma _s\right) \left(3 \cos \left(\varphi _p\right) \cos \left(f(t)-\phi _s-t \varphi '(t)\right)-\cos \left(f(t)+\varphi _p-\phi _s-t \varphi '(t)\right)\right)+\cos \left(\gamma _p\right) \cos \left(\gamma _s\right)\right)}{\pi r(t)^4}-\frac{4 \text{Gm}_s m_p}{r(t)^2} - \frac{4 m_p m_s r(t) f'(t)^2}{m_p+m_s}-\frac{9 \text{Gm}_p \left(T_s \cos (2 (f(t)-\varphi (t)))+2 Q_s\right)}{r(t)^4}-\frac{4 m_p m_s r''(t)}{m_p+m_s} =0 $$
Equation for f(t) -
$$ -\frac{4 m_p m_s r(t)^4 \left(r(t) f''(t)+2 f'(t) r'(t)\right)}{m_p+m_s}-6 \text{Gm}_p T_s \sin (2 (f(t)-\varphi (t)))-6 \text{Gm}_s T_p \sin (2 (f(t)-\psi (t)))+\frac{\mu M_m M_p}{\pi } [ \sin (\gamma _p) \sin (\gamma _s) [\sin (f(t)+\varphi _p-\phi _s-t \varphi '(t))-3 \cos (\varphi _p) \sin (f(t)-\phi _s-t \varphi '(t))] ] = 0 $$