How to derive the equation of motion of Lorentz force by using Lagrange equation
14SEP20
Lagrangian of a particle in electric and magnetic field is
\[L=\frac12m \dot{\vec{r}}\cdot\dot{\vec{r}}-qV(\vec{r},t)+q\dot{\vec{r}}\cdot\vec{A}({\vec{r}},t)\]Lagrange equation is
\[\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{x}}}=\frac{\partial{L}}{\partial{x}}\]In this case,
\[\dot{x}\rightarrow\dot{\vec{r}} \\ x\rightarrow\vec{r}\]So, the equation of motion is
\[\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{\vec{r}}}}=\frac{\partial{L}}{\partial{\vec{r}}}\]On the left-hand side,
\[\frac{\partial{L}}{\partial{\dot{\vec{r}}}} = m\dot{\vec{r}}+q\cdot\vec{A}(\vec{r},t) \\ \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{\vec{r}}}} = \frac{dm\dot{\vec{r}}}{dt}+q\cdot\frac{d{\vec{A}}}{d{t}} \\\frac{d\vec{A}}{dt}=\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\frac{d{\vec{r}}}{d{t}}+\frac{\partial{\vec{A}}}{\partial{t}} \\ \therefore \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{\vec{r}}}} = \frac{dm\dot{\vec{r}}}{dt}+q\cdot(\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\frac{d{\vec{r}}}{d{t}}+\frac{\partial{\vec{A}}}{\partial{t}})\]On the right-hand side,
\[\frac{\partial{L}}{\partial{\vec{r}}}=-q\cdot\frac{\partial{V}}{\partial{\vec{r}}}+q\cdot\frac{\partial({\dot{\vec{r}}\cdot\vec{A})}}{\partial{\vec{r}}} \\ \frac{\partial{V}}{\partial{\vec{r}}} = \vec{\nabla}V\\\frac{\partial({\dot{\vec{r}}\cdot\vec{A})}}{\partial{\vec{r}}}=\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})\\ \therefore \frac{\partial{L}}{\partial{\vec{r}}}=-q\cdot\vec{\nabla}V+q\cdot\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})\]Then, the Equation of motion will be
\[\frac{dm\dot{\vec{r}}}{dt}+q\cdot(\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\cdot\dot{\vec{r}}+\frac{\partial{\vec{A}}}{\partial{t}}) =-q\cdot\vec{\nabla}V+q\cdot\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A}) \\ \rightarrow\frac{dm\dot{\vec{r}}}{dt} =-q\cdot(\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\cdot\dot{\vec{r}}+\frac{\partial{\vec{A}}}{\partial{t}})-q\cdot\vec{\nabla}V+q\cdot\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})\\ =q\cdot(-\vec{\nabla}V-\frac{\partial{\vec{A}}}{\partial{t}})+q\cdot(\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})-\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\cdot\dot{\vec{r}})\]On the right-hand side, the second term divided by q became
\[if \ \ \vec{r}=(x,y,z),\\\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})-\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\cdot\dot{\vec{r}}=\begin {bmatrix}\dot{x}\frac{\partial{A_x}}{\partial{x}}+\dot{y}\frac{\partial{A_y}}{\partial{x}}+\dot{z}\frac{\partial{A_z}}{\partial{x}}\\ \dot{x}\frac{\partial{A_x}}{\partial{y}}+\dot{y}\frac{\partial{A_y}}{\partial{y}}+\dot{z}\frac{\partial{A_z}}{\partial{y}}\\ \dot{x}\frac{\partial{A_x}}{\partial}+\dot{y}\frac{\partial{A_y}}{\partial{z}}+\dot{z}\frac{\partial{A_z}}{\partial{z}}\end{bmatrix}-\begin{bmatrix}\dot{x}\frac{\partial{A_x}}{\partial{x}}+\dot{y}\frac{\partial{A_x}}{\partial{y}}+\dot{z}\frac{\partial{A_x}}{\partial{z}}\\ \dot{x}\frac{\partial{A_y}}{\partial{x}}+\dot{y}\frac{\partial{A_y}}{\partial{y}}+\dot{z}\frac{\partial{A_y}}{\partial{z}}\\ \dot{x}\frac{\partial{A_z}}{\partial}+\dot{y}\frac{\partial{A_z}}{\partial{y}}+\dot{z}\frac{\partial{A_z}}{\partial{z}}\end{bmatrix}\\ =\begin{bmatrix}\dot{y}(\frac{\partial{A_y}}{\partial{x}}-\frac{\partial{A_x}}{\partial{y}})+\dot{z}(\frac{\partial{A_z}}{\partial{x}}-\frac{\partial{A_x}}{\partial{z}})\\ \dot{x}(\frac{\partial{A_x}}{\partial{y}}-\frac{\partial{A_y}}{\partial{x}})+\dot{z}(\frac{\partial{A_z}}{\partial{y}}-\frac{\partial{A_y}}{\partial{z}})\\\dot{x}(\frac{\partial{A_x}}{\partial{z}}-\frac{\partial{A_z}}{\partial{x}})+\dot{y}(\frac{\partial{A_y}}{\partial{z}}-\frac{\partial{A_z}}{\partial{y}})\end{bmatrix} \\=\dot{\vec{r}}\times\vec{B}\\\because\vec{B}=\vec{\nabla}\times\vec{A}=\begin {bmatrix}\frac{\partial{A_z}}{\partial{y}}-\frac{\partial{A_y}}{\partial{z}}\\\frac{\partial{A_x}}{\partial{z}}-\frac{\partial{A_z}}{\partial{x}}\\\frac{\partial{A_y}}{\partial{x}}-\frac{\partial{A_x}}{\partial{y}}\end{bmatrix}\]Also,
\[\vec{F} = \frac{dm{\dot{\vec{r}}}}{dt} = \frac{dm{\vec{v}}}{dt}\\ \vec{E}=-\vec{\nabla}V-\frac{\partial{\vec{A}}}{\partial{t}}\]Finally,
\[\frac{dm{\dot{\vec{r}}}}{dt}=q\cdot(-\vec{\nabla}V-\frac{\partial{\vec{A}}}{\partial{t}})+q\cdot(\vec{\nabla}(\dot{\vec{r}}\cdot\vec{A})-\frac{\partial{\vec{A}}}{\partial{\vec{r}}}\cdot\dot{\vec{r}}) \\\rightarrow\vec{F}=q\vec{E}+q\vec{v}\times\vec{B}\]