[NOTES/CM-02010] Lagrangian for Conservative Forces

Generalized Force

The Newton's equations of motion for a system of point particles acted upon by forces \(\vec F_\alpha\) can be cast into the form
\begin{equation}\label{EQ23A} \Big(\frac{d}{dt}\frac{\partial{T}}{\partial\dot{q_k}}-\frac{\partial{T}}{ \partial{q_k}}\Big)=Q_k \end{equation}
where \(T\) is the kinetic energy, \(q_k, k=1,2,..,N\) are generalized coordinates.  \(Q_k\) are generalized forces given by
\begin{equation}\label{EQ21A} \sum_\alpha{F_\alpha}^{(e)}\cdot\frac{ \partial\vec{r}_\alpha}{\partial{q_k}}\equiv Q_k \end{equation}

Systems with Conservative Forces

Forces are conservative then there exists a function "$V$" called potential energy such that
\begin{equation}\label{Eq23} \vec{F}_{\alpha}^{(e)}=-\vec{\nabla}_\alpha{V} \end{equation}
\begin{equation}\label{Eq24} =-\Big(\frac{\partial{V}}{\partial{x_\alpha}},\frac{\partial{V}}{\partial{ y_\alpha}},\frac{\partial{V}}{\partial{z_\alpha}}\Big) \end{equation} where $({x_\alpha},{y_\alpha}, {z_\alpha})\equiv \vec{r}_\alpha$ are the components of position for the particle $\alpha$. The generalized force, defined  by l.h.s of \eqref{EQ21A} becomes

\begin{eqnarray}
Q_j={\sum_\alpha}\vec{F_\alpha}\cdot\frac{\partial\vec{r}_\alpha}{\partial{q_j}}
&=&-{\sum_\alpha}\Big(\frac{\partial{v}}{\partial{x_\alpha}}\frac{\partial{x_\alpha
}}{\partial{q_j}}+\frac{\partial{v}}{\partial{y_\alpha}}\frac{\partial{y_\alpha}
}{\partial{q_j}}+\frac{\partial{v}}{\partial{z_\alpha}}\frac{\partial{z_\alpha}}
{\partial{q_j}}\Big)\label{Eq25A}
\end{eqnarray}
The right hand side of \eqref{Eq25A} takes a simple form and \begin{eqnarray}
Q_j= -\frac{\partial{V}}{\partial{q_j}}  \label{Eq25}
\end{eqnarray}

Using \eqref{Eq25} in
\begin{equation}\label{EQ23} \Big(\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{ \partial{q_k}}\Big)=Q_k \end{equation}
we get
\begin{equation}\label{Eq26} \Big(\frac{d}{dt}\frac{\partial{T}}{\partial\dot{q_k}}-\frac{\partial{T}}{ \partial{q_k}}\Big)=-\frac{\partial{V}}{\partial{q_k}} \end{equation}
Thus \eqref{Eq26} can be rewritten as
\begin{equation}\label{Eq27} \frac{d}{dt}\frac{\partial{T}}{\partial\dot{q_k}}-\frac{\partial{T}}{\partial{ q_k}}=\frac{d}{dt} \frac{\partial{V}}{\partial\dot{q_k}}-\frac{\partial{V}}{\partial{q_k}} \end{equation}
because $V$ is a function of \(q\)'s alone and $\frac{\partial{V}}{\partial\dot{q}_k}=0$.

Lagrangian $L=T-V$
Therefore, rearranging the above equation we get
\begin{equation}\label{Eq28} \frac{d}{dt}\frac{\partial{L}}{\partial\dot{q_k}}-\frac{\partial{L}}{\partial{ q_k}}=0 \end{equation}
where $L=T-V,$ is called the Lagrangian of the system. \Eqref{Eq28} are called Euler Lagrange equation of motion.