  First, we show that there exists in classical mechanics three actions
corresponding to different boundary conditions: two well-known actions, the
Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position
x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which
links a family of particles of initial action S_0(x) to their various positions
x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which
links a particle in initial position x_0 and initial velocity v_0 to its
position x at time t. We study, in the semi-classical approximation, the
convergence of the quantum density and the quantum action, solutions to the
Madelung equations, when the Planck constant h tends to 0. We find two
different solutions which depend on the initial density. In the first case,
where the initial quantum density is a classical density, the quantum density
and the quantum action converge to a classical action and a classical density
which satisfy the statistical Hamilton-Jacobi equations. These are the
equations of a set of classical particles whose initial positions are known
only by the initial density. In the second case where initial density converges
to a Dirac density, the density converges to the Dirac function and the quantum
action converges to a deterministic action. Therefore we introduce into
classical mechanics non-discerned particles, which satisfy the statistical
Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned
particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally,
we propose an interpretation of the Schrodinger wave function that depends on
the initial conditions (preparation). This double interpretation seems to be
the interpretation of Louis de Broglie's "double solution" idea.
