Kostik
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We obtain equations of motion by solving the Euler-Lagrange equations; along this path the action ##S## is an extremum. We are usually interested in the path ##x(t)## (in one dimension) connecting ##x(0)=x_0## and ##x(T)=x_1##. But does the numerical (extremal) value of the resulting action have any significance? Its units are plainly energy-time.
In the extremely simple case of a particle with mass ##m## under a constant force ##F##, the Euler-Lagrange equation gives the obvious ##m \ddot x = F##, but the action itself has the complicated form:
$$S = \frac {(x_1+x_0)FT} {2} + \frac {(x_1-x_0)^2 m} {2T} - \frac {F^2 T^3} {24m}.$$
This strikes me as a peculiarly complicated quantity to have some significance to a particle moving under a constant force.
In the extremely simple case of a particle with mass ##m## under a constant force ##F##, the Euler-Lagrange equation gives the obvious ##m \ddot x = F##, but the action itself has the complicated form:
$$S = \frac {(x_1+x_0)FT} {2} + \frac {(x_1-x_0)^2 m} {2T} - \frac {F^2 T^3} {24m}.$$
This strikes me as a peculiarly complicated quantity to have some significance to a particle moving under a constant force.