The dynamics of a charged particle with velocity $\textbf{v}$ in electromagnetic field is dominated by the Lorentz force

$$\textbf{F} = q(\textbf{E}+{\textbf{v} \times \textbf{B}}), \tag{1}$$

and the corresponding Lagrangian is

$$L = \dfrac{1}{2}mv^2 - q\phi(\textbf{r},t)+q\textbf{v} \cdot \textbf{A}. \tag{2}$$

My question is: How to derive the equation $(1)$ just from $(2)$ ?

  • $\begingroup$ I just find the answer by this link:insti.physics.sunysb.edu/itp/lectures/01-Fall/PHY505/09c/… $\endgroup$ – Jack Jan 25 '17 at 11:37
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/80280/50583, physics.stackexchange.com/q/77325/50583 $\endgroup$ – ACuriousMind Jan 25 '17 at 17:14
  • $\begingroup$ I think it's not a duplicate. $\endgroup$ – Jack Jan 25 '17 at 23:42
  • $\begingroup$ Okay, could you then please edit your question to include what specifically you don't understand about the derivation? The other question gave basically the same information as you do and the hints there should be sufficient to work out the correct derivation. $\endgroup$ – ACuriousMind Jan 26 '17 at 0:14
  • $\begingroup$ I am sorry I just don't follow that question carefully. Almost we are asking the same question. You are right and it should be an duplicate question. $\endgroup$ – Jack Jan 27 '17 at 7:40