  We give a geometrical interpretation for the principle of stationary action
in classical Lagrangian particle mechanics. In a nutshell, the difference of
the action along a path and its variation effectively ``counts'' the possible
evolutions that ``go through'' the area enclosed. If the path corresponds to a
possible evolution, all neighbouring evolutions will be parallel, making them
tangent to the area enclosed by the path and its variation, thus yielding a
stationary action. This treatment gives a full physical account of the geometry
of both Hamiltonian and Lagrangian mechanics which is founded on three
assumptions: determinism and reversible evolution, independence of the degrees
of freedom and equivalence between kinematics and dynamics. The logical
equivalence between the three assumptions and the principle of stationary
action leads to a much cleaner conceptual understanding.
