We construct an operational formulation of classical mechanics without
presupposing previous results from analytical mechanics. In doing so, several
concepts from analytical mechanics will be rediscovered from an entirely new
perspective. We start by expressing the basic concepts of the position and
velocity of point particles as the eigenvalues of self-adjoint operators acting
on a suitable Hilbert space. The concept of Holonomic constraint is shown to be
equivalent to a restriction to a linear subspace of the free Hilbert space. The
principal results we obtain are: (1) the Lagrange equations of motion are
derived without the use of D'Alembert or Hamilton principles, (2) the
constraining forces are obtained without the use of Lagrange multipliers, (3)
the passage from a position-velocity to a position-momentum description of the
movement is done without the use of a Legendre transformation, (4) the
Koopman-von Neumann theory is obtained as a result of our ab initio operational
approach, (5) previous work on the Schwinger action principle for classical
systems is generalized to include holonomic constraints