We've already used the "1st Picture" of classical mechanics in the last lecture. It's reproduced here (without its F=dp/dt consequence) to demonstrate that Newton's famous equation is a 2nd order differential equation; not a particularly nice mathematical entity to solve. But what's worse is that the very form shown is only guaranteed to work for Cartesian (xyz) coordinate systems. The form becomes messier in other systems such as the circular symmetry of the solar system, given the Sun as the center of force.Even the momentum isn't p=mdx/dt unless you're working in a Cartesian System. But the 2nd Picture of Classical Mechanics, Lagrange's solved that.
Lagrange constructed a mathematical object (called the Lagrangian) from the kinetic energy expression, K, (in terms of coordinates and their velocities) minus the potential, V, in terms of the coordinates (at least in a "conservative," i.e., friction-free system).
While his solutions to mechanical problems still involved 2nd derivatives in time (pooh), it had two wonderful consequences: the definition of general momentum and a definition of mechanics which would've satified Occam himself! The path taken by a classical trajectory could be none other than would minimize the "Classical Action" (that integral of the Lagrangian over the time the trajectory traverses).
The 3rd Picture of classical mechanics by Hamilton built on Lagrange's by describing a "Hamiltonian" as kinetic K (in terms of coordinates and momenta) PLUS the potential V to yield the total Energy function. Since the 1st Law says that's conserved, nice consequences might be expected to flow from this choice! And they did. Not only did Hamilton's solutions to motion become much simpler (as twice as many 1st Order equations), but it served as the springboard for Schrödinger's Wave Mechanics!