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If I understand how to do the following problem, it will help with the real complicated problem I am actually doing. The problem I wish to understand is how to derive the equations of motion for a magnetic moment $\vec{m}$ that is in a changing magnetic field $\vec{B}$. I know I can use the Lagrangian to quite easily obtain the equations of motion i.e. the motion of $\ddot{\theta}$ and $\ddot{\phi}$ of the magnetic moment (the reason why I do not want to use the Lagrangian is because the actual problem that I am doing has a piecewise torque function so when the and when integrating the torque to find a potential energy the equation becomes super nasty). Instead I wish to obtain the equations of motion using $\vec{L}=\mathbf{I}\vec{\omega}$ where $\mathbf{I}$ is the moment of inertia tensor. In representing a the magnetic moment as a thin rod, the moment of inertia tensor becomes,

$$\mathbf{I}=\begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & 0 \end{bmatrix},$$

where $\lambda_1 = \lambda_2$.

The part that I am getting hung up on is the Euler's equation $\dot{\vec{L}}+\omega \times\vec{L}=\vec{\Gamma}=\vec{m}\times\vec{B}$:

$$\lambda_1\dot{\omega_1}-(\lambda_2 - \lambda_3)\omega_2\omega_3=\Gamma_1,$$ $$\lambda_2\dot{\omega_2}-(\lambda_3 - \lambda_1)\omega_3\omega_1=\Gamma_2,$$ $$\lambda_3\dot{\omega_3}=\Gamma_3,$$

where the $\omega$ terms are in the body frame.

So my question is how to implement the torque when the direction is constantly changing i.e. the magnetic field is a function of time and can be pointing in any direction. If I could get help on this one little spot (since the textbooks seem to have nice torques that are perpendicular to an axis) that would be appreciated.

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The relationship between the magnetic moment and the angular momentum is $\vec{m} = \gamma\vec{L}$. Here $\gamma$ is a constant of proportianalit that You can use this to rewrite your EoM as: $$ \dot{\vec{L}}+(\vec{\omega}+\gamma \vec{B})\times\vec{L}=0 $$ This equation doesn't care if $\vec{B}$ depends on time or not. Just plug in whatever functions you have for the magnetic field as a function of time, and solve the resulting set of coupled differential equations.

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