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In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does not depend on $x(t')$ or its derivatives at other times $t'\ne t$). This is no longer true if you use $\exp(S)$ instead of $S$. There is no dispute that $S$ and $\...


In a sense, all the Lagrangians giving the same Euler-Lagrange equations are exhausted by transformations of your type (b), which adds a total derivative/total divergence/boundary term/... Transformations of your type (a) can alter the Euler-Lagrange equations, for instance if $a\ne 1$, then the EL equations get rescaled by the same constant $a$. Perhaps ...


This problem is discussed in Bryant, Griffiths, Hsu, Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, for Lagrangians for scalar fields.

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