Discussed equilibrium points, motion near equilibrium, and began developing quantitative techniques to solve for motion in a potential.
We noted that the system is in equilibrium at all local maxima and minima of the potential, since the force locally vanishes. Thus a particle at rest at an equilibrium point stays there forever, according to NII.
Taylor expanded for motion near equilibrium, and found motion corresponded to that of a simple harmonic oscillator with spring constant k = V'' (evaluated at the equilibrium point). As long as the Taylor expansion is good (that is, the third order terms are negligible compared to retained terms), motion is well approximated by simple harmonic oscillator motion. Discussed stable equilibrium, where k > 0 produces restoring force, opposing initial displacement; and unstable equilibrium, where k < 0 acts to increase any initial deviation from equilibrium.
Introduced quantitative potential methods. Solved algebraically for particle's velocity, which gave integral equation for the duration t of a particle's motion. Performing the integral gives t(x), which can be inverted to solve for the particle's motion x(t).
Note that you will see a generalization of this potential method when you learn about tunneling in quantum mechanics.
-- KB