The equation of motion of a magnetic charge in the fixed external electromagnetic field $\mathbf{E},\mathbf{B}$ is $$ \gamma m \dot{\mathbf{v}}=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E}), $$ where $\gamma=\frac{1}{\sqrt{1-|\mathbf{v}|^2}}$ (using $c=\epsilon_0=\mu_0=1$), $q_m$ is the magnetic charge.

Of course, if one introduces the dual EM potential $(C^0,\mathbf{C})$, with $\mathbf{B}=-\nabla C^0-\dot{\mathbf{C}}$ and $\mathbf{E}=-\nabla\times\mathbf{C}$, then this equation of motion can be derived from the Lagrangian: $$ L=-\frac{m}{\gamma}-q_m(C^0-\mathbf{C}\cdot\mathbf{v}). $$ But is it possible to use instead a Lagrangian written in the original EM potential $(A^0,\mathbf{A})$, with $\mathbf{B}=\nabla\times\mathbf{A}$ and $\mathbf{E}=-\nabla A^0-\dot{\mathbf{A}}$ ? In other words, can we describe the dynamics of $\gamma m \dot{\mathbf{v}}=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E})+q_e(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ with a Lagrangian?

The equation of motion of a magnetic charge in the fixed external electromagnetic field $\mathbf{E},\mathbf{B}$ is $$ \frac{d}{dt}(\gamma m \mathbf{v})=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E}), $$ where $\gamma=\frac{1}{\sqrt{1-|\mathbf{v}|^2}}$ (using $c=\epsilon_0=\mu_0=1$), $q_m$ is the magnetic charge.

Of course, if one introduces the dual EM potential $(C^0,\mathbf{C})$, with $\mathbf{B}=-\nabla C^0-\dot{\mathbf{C}}$ and $\mathbf{E}=-\nabla\times\mathbf{C}$, then this equation of motion can be derived from the Lagrangian: $$ L=-\frac{m}{\gamma}-q_m(C^0-\mathbf{C}\cdot\mathbf{v}). $$ But is it possible to use instead a Lagrangian written in the original EM potential $(A^0,\mathbf{A})$, with $\mathbf{B}=\nabla\times\mathbf{A}$ and $\mathbf{E}=-\nabla A^0-\dot{\mathbf{A}}$ ? In other words, can we describe the dynamics of $\frac{d}{dt}(\gamma m \mathbf{v})=q_m(\mathbf{B}-\mathbf{v}\times\mathbf{E})+q_e(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ with a Lagrangian?