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, 2017 at 11:37
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/80280/50583, physics.stackexchange.com/q/77325/50583 $\endgroup$
    – ACuriousMind
    Jan 25, 2017 at 17:14
  • $\begingroup$ I think it's not a duplicate. $\endgroup$
    – Jack
    Jan 25, 2017 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, 2017 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, 2017 at 7:40