  In this paper, we show that the difficulties of interpretation of the
principle of least action concerning "final causes" or "efficient causes" are
due to the existence of two different actions, the "Euler-Lagrange action" (or
classical action) and the "Hamilton-Jacobi action". These two actions, which
are not clearly differentiated in the texbooks, are solutions to the same
Hamilton-Jacobi equation, but with very different initial conditions: smooth
conditions for the Hamilton-Jacobi action, singular conditions for the
Euler-Lagrange action. There are related by the Minplus Path Integral which is
the analog in classical mechanics of the Feynmann Path Integral in quantum
mechanics. Finally, we propose a clear-cut interpretation of the principle of
least action: the Hamilton-Jacobi action does not use "final causes" and seems
to be the action used by Nature; the Euler-Lagrange action uses "final causes"
and is the action used by an observer to retrospectively determine the
trajectory of the particle.
