[NOTES/CM-02006] Generalized Force and Lagrange Equations -- An Example

Recall generalized force and Lagrange equations

d' Alembert's principle has led to the equations of motion for each generalized  coordinate in the form \begin{equation} \dd{t}\Big(\pp[T]{\dot{q}_k}\Big) - \pp[T]{q_k} = Q_k \end{equation} where \(Q_k\) is the generalized force given by \begin{equation} Q_k=\sum_\alpha F^\text{(e)}_\alpha \pp[\vec{x}_\alpha]{q_k}. \end{equation} The generalized potential \(U\) is defined  by  equation \begin{equation} Q_k = \dd{t}\Big(\pp[U]{\dot{q}_k}\Big)-\pp[U]{q_k} . \end{equation} Whenever velocity dependent forces can be derived from generalized potential, the Lagrangian becomes \(L=T-U.\)

Generalized potential for Lorentz force

The Lorentz force on a charged particle in electric field \(\vec{E}\) and magnetic field \(\vec{B}\) is given by \begin{equation}\label{EQ01} \vec{F}= q (\vec{E}  + \vec{v}\times \vec{B}) \end{equation} where \(q\) is the electric charge of the particle. We need to express the fields in terms of {\it vector and scalar potentials}, \(\phi, \vec{A}\) related to the fields by means of the equations \begin{equation}\label{EQ02} \vec{E}=-\dd[\vec{A}(\vec{x},t)]{t} - \nabla \phi(\vec{x},t), \qquad \vec{B}=\nabla \times \vec{A}. \end{equation}

 
The components of the Lorentz force are given by

\begin{eqnarray}\nonumber
 F_1 &=& qE_1 + q (\vec{v}\times\vec{B})_1 \\
 &=& - q \Big(\pp[\phi]{x_1}+\pp[A_1]{t}\Big) + q (v_2B_3-v_3B_2)\\\nonumber
 &=& - q \Big(\pp[\phi]{x_1}+\pp[A_1]{t}\Big) +
    q v_2\Big(\pp[A_2]{x_1}- \pp[A_1]{x_2}\Big)
 -q v_3\Big(\pp[A_1]{x_3}- \pp[A_3]{x_1}\Big)\\
 &=& -q\Big(\pp[\phi]{x_1}+\pp[A_1]{t}\Big) + q\Big\{v_2 \pp[A_2]{x_1} + v_3\pp[A_3]{x_1}\Big\}-
       q\Big\{v_2\pp[A_1]{x_2}+  v_3\pp[A_1]{x_3}\Big\}
\end{eqnarray}

By adding and subtracting \(qv_1\pp[A_1]{x_1}\), we rewrite the above expression as

\begin{eqnarray}
 F_1&=& - q \pp[\phi]{x_1}-q\pp[A_1]{t} +
         q\Big\{v_1\pp[A_1]{x_1} + v_2 \pp[A_2]{x_1} + v_3\pp[A_3]{x_1}\Big\}
      - q\Big\{v_1 \pp[A_1]{x_1}+v_2\pp[A_1]{x_2}+  v_3\pp[A_1]{x_3}\Big\}\\\nonumber
     &=& -q   \pp[\phi]{x_1} -q\pp[A_1]{t}  
           +  q v_k\pp[A_k]{x_1} - q v_k\pp[A_1]{x_k} \\\nonumber
     &=& -q   \pp[\phi]{x_1} +  q \pp[(v_k A_k)]{x_1} -\Big(\pp[A_1]{t} +
      \pp[x_k]{t}\pp[A_1]{x_k} \Big)  \\\nonumber
      &&\qquad\qquad \text{Combine the last terms into a single expression} \\
     &=& -q   \pp[\phi]{x_1} +  q \pp[(v_k A_k)]{x_1} -
     \dd[A_1]{t} \label{EQ09}
\end{eqnarray}


Our manipulations are almost complete. Noting that the vector potential does not depend on velocity components\(v_k\) we will have  \(\pp[A_1]{v_1}=0\), therefore

\begin{eqnarray}
 A_1 = \pp[A_1v_1]{v_1} = \pp{v_1}(A_kv_k) \qquad \because k=2,3 \text{ terms are zero}
\end{eqnarray}

Substituting for \(A_1\) in the last term \eqref{EQ09}, we are led to \begin{equation} F_1 =  -q   \pp[\phi]{x_1} +  q \pp[(A_kv_k)]{x_1} - q \dd{t}\Big[\pp{v_1}\big(A_kv_k\big)\Big]       \end{equation} Since \(\phi(\vec{x})\) depends only on \(\vec{x}\) and is independent of velocities, \(\pp[\phi(\vec{x})]{v_k}=0\). Adding an extra vanishing term we arrive at

\begin{eqnarray}\nonumber
 F_1&=&  -\pp[\phi]{x_1} +  q \pp[(A_kv_k)]{x_1} - q \dd{t}\Big[\pp{v_1}\big(A_kv_k\big)\Big]
  + \mHighLight{$\dd{t} \Big[ \pp[\phi]{v_1}\Big]$}\\\nonumber
  && \qquad \mHighLight{Extra term added here is zero}\\
  &=& -\pp[U]{x_1} + \dd{t}\Big[\pp[U]{\dot{x}_1}\Big]
\end{eqnarray}


where \(U =q \phi - q\vec{v}\cdot\vec{A}\) is generalized potential. We get similar expressions for other components of Lorentz force and \begin{equation} F_k = - \pp[U]{x_k} + \dd{t} \Big[\pp[U]{\dot{x}_k}\Big]. \end{equation} This give the Lagrangian for the charged particle to be

\begin{eqnarray}
 L &=& \frac{1}{2}m \Big(\dd[\vec{x}]{t}\Big)^2- U\\
   &=& \frac{1}{2}m \Big(\dd[\vec{x}]{t}\Big)^2 - q\phi + q\pp[\vec{x}]{t}\cdot\vec{A}.
\end{eqnarray}

For a system of several charged particle with charges \(q_\alpha\),  the Lagrangian takes the form

\begin{eqnarray}
   L= \sum_\alpha\left\{\frac{1}{2}m_\alpha \Big(\dd[\vec{x}_\alpha]{t}\Big)^2 -
   q_\alpha\phi(\vec{x}_\alpha) +
   q_\alpha\big(\vec{v}_\alpha\cdot\vec{A_\alpha}\big)\right\} + \ldots .
\end{eqnarray}

where \(\ldots\) represent potential term, \(\frac{1}{2}\sum_{\alpha\ne\beta}\frac{q_\alpha q_\beta}{|\vec{x}_\alpha-\vec{x}_\beta|}\) corresponding to the Coulomb interactions of  the charged particles.