The Feynman Lectures on Physics III The Principle of Least Action 2 (action for relativistic motion in an erectromagnetic field)

http://file.buturi.blog.shinobi.jp/Feynmanhonbun3_slengl2.pdf
The Principle of Least Action 2 (action for relativistic motion in an erectromagnetic field)

For relativistic motion in an erectromagnetic field
\[
\mathcal{L}=-m_0c^2\sqrt{1-v^2/c^2}-q\phi (x, y, z, t)-\mathbf{v}\cdot \mathbf{A}(x, y, z, t).
\]

Let
\[
\mathbf{p}=\frac{m_0\mathbf{v}}{\sqrt{1-v^2/c^2}}.
\]



Let
$\mathbf{x}=\underline{\mathbf{x}}+\mathbf{\eta }$.

By using
\[
\mathbf{E}=-\nabla \phi -\frac{\partial \mathbf{A}}{\partial t},
\]
\[
\mathbf{B}=\nabla \times \mathbf{A},
\]

\begin{eqnarray*}
&&S(\mathbf{x})\\
&=&
-m_0c^2\int _{t_1}^{t2}\sqrt{1-\frac{(\dot{\underline{\mathbf{x}}}+\dot{\mathbf{\eta }})^2}{c^2}}dt
-q\int _{t_1}^{t_2}(\phi (\mathbf{x}(t)+\mathbf{\eta }(t), t)-(\dot{\underline{\mathbf{x}}}+\dot{\mathbf{\eta }})\cdot \mathbf{A}(\mathbf{x}(t)+\mathbf{\eta }(t), t))dt\\
& \fallingdotseq &
-m_0c^2\int _{t_1}^{t2}\sqrt{1-\frac{\dot{\underline{\mathbf{x}}}^2+2\dot{\underline{\mathbf{x}}}\cdot \dot{\mathbf{\eta }}}{c^2}}dt
-q\int _{t_1}^{t_2}(\phi +\nabla \phi \cdot \eta -(\dot{\underline{\mathbf{x}}}+\dot{\mathbf{\eta }})\cdot
\left( \mathbf{A}+
\left(
\begin{array}{c}
\nabla A_x\cdot \mathbf{\eta }\\
\nabla A_y\cdot \mathbf{\eta }\\
\nabla A_z\cdot \mathbf{\eta }\\
\end{array}
\right)
\right)dt\\
& \fallingdotseq &
-m_0c^2\int _{t_1}^{t2}\sqrt{1-\frac{\dot{\underline{\mathbf{x}}}^2}{c^2}}\left( 1-\frac{1}{1-\frac{\dot{\underline{\mathbf{x}}}^2}{c^2}}\frac{\dot{\underline{\mathbf{x}}}\cdot \dot{\mathbf{\eta}}}{c^2}\right) dt\\
&-&
q\int _{t_1}^{t_2}\left( \phi +\nabla \phi \cdot \eta -(\dot{\underline{\mathbf{x}}}+\dot{\mathbf{\eta }})\cdot
\left( \mathbf{A}+
\left(
\begin{array}{c}
\nabla A_x\cdot \mathbf{\eta }\\
\nabla A_y\cdot \mathbf{\eta }\\
\nabla A_z\cdot \mathbf{\eta }\\
\end{array}
\right)
\right) \right) dt\\
&\fallingdotseq &
S(\underline{\mathbf{x}})
+\int _{t_1}^{t2} \mathbf{p}\cdot \dot{\mathbf{\eta }}dt\\
&-&
q\int _{t_1}^{t_2}\left( \nabla \phi \cdot \eta -\dot{\mathbf{\eta }}\cdot \mathbf{A}-\dot{\underline{\mathbf{x}}}
\cdot
\left(
\begin{array}{c}
\nabla A_x\cdot \mathbf{\eta }\\
\nabla A_y\cdot \mathbf{\eta }\\
\nabla A_z\cdot \mathbf{\eta }\\
\end{array}
\right) \right) dt\\
&=&
S(\underline{\mathbf{x}})
-\int _{t_1}^{t2} \dot{\mathbf{p}}\cdot \mathbf{\eta }dt\\
&-&
q\int _{t_1}^{t_2}\left( \nabla \phi \cdot \mathbf{\eta} +\mathbf{\eta }\cdot \frac{\partial }{\partial t}\mathbf{A}
+
\mathbf{\eta }\cdot
\left(
\begin{array}{c}
\nabla A_x\cdot \dot{\mathbf{x}}\\
\nabla A_y\cdot \dot{\mathbf{x}}\\
\nabla A_z\cdot \dot{\mathbf{x}}\\
\end{array}
\right)
-
\dot{\underline{\mathbf{x}}}
\cdot
\left(
\begin{array}{c}
\nabla A_x\cdot \mathbf{\eta }\\
\nabla A_y\cdot \mathbf{\eta }\\
\nabla A_z\cdot \mathbf{\eta }\\
\end{array}
\right) \right) dt\\
&=&
S(\underline{\mathbf{x}})
-\int _{t_1}^{t2} \dot{\mathbf{p}}\cdot \mathbf{\eta }dt\\
&-&
q\int _{t_1}^{t_2}\left( \nabla \phi \cdot \mathbf{\eta} +\mathbf{\eta }\cdot \frac{\partial }{\partial t}\mathbf{A}
-\mathbf{\eta}
\cdot (\mathbf{v}\times (\nabla \times \mathbf{A}))
\right) dt\\
&=&
S(\underline{\mathbf{x}})
-\int _{t_1}^{t2} \dot{\mathbf{p}}\cdot \mathbf{\eta }dt\\
&+&
\int _{t_1}^{t_2}q\left( \mathbf{E}+\mathbf{v}\times \mathbf{B}
\right) \cdot \mathbf{\eta} dt
\end{eqnarray*}

Hence the path that has the minimum action is the one satisfying
\[
\dot{\mathbf{p}}
=q\left( \mathbf{E}+\mathbf{v}\times \mathbf{B}\right) .
\]