[QUE/CM-02023] Generalized potential for a charged particle in EM fields

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)

A point charge moves with velocity \(\vec{v}\) in presence of electric field \(\vec{E}\) and magentic field \(\vec{B}\). The Lorentz force on the charged particle is
\[\vec{F}= q\Big\{ \vec{E} + \frac{1}{c}\vec{v}\times\vec{B} \Big\}\]
In terms of the scalar potential \(\phi\) and vector  potential \(\vec{A}\) related to the electric and magnetic fields by
\[\vec{E}= -\nabla \phi- \frac{1}{c} \pp[\vec{A}]{t}, \quad
\vec{B}=\nabla\times \vec{A}.\]
the Lorentz force becomes \[\vec{F} = q\Big\{ - \nabla\phi -\frac{1}{c} \pp[\vec{A}]{t} +
\frac{1}{c} \vec{v}\times(\nabla \times \vec{A})\Big\} \]
Determine the generalzed potential for this system and write the Lagrangian.