4.9: Non-uniqueness of the Lagrangian

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The Lagrangian is not uniquely defined: two Lagrangians differing by the total derivative with respect to time of some function will give the same identical equations on minimizing the action,

\begin{equation}
S^{\prime}=\int_{t_{1}}^{t_{2}} L^{\prime}(q, \dot{q}, t) d t=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t+\int_{t_{1}}^{t_{2}} \frac{d f(q, t)}{d t} d t=S+f\left(q\left(t_{2}\right), t_{2}\right)-f\left(q\left(t_{1}\right), t_{1}\right)
\end{equation}

and since \(\begin{equation}
q\left(t_{1}\right), t_{1}, q\left(t_{2}\right), t_{2}
\end{equation}\) are all fixed, the integral over \(\begin{equation}
d f / d t
\end{equation}\) is trivially independent of path variations, and varying the path to minimize \(\begin{equation}
S^{\prime}
\end{equation}\) gives the same result as minimizing S. This turns out to be important later—it gives us a useful new tool to change the variables in the Lagrangian.