The conservative force in one dimension is the negative slope of the potential energy curve: $F_{x} = -\frac{dU(x)}{dx}$

Sum the potential energies of all pairs of interacting objects: $U_{total} = U_{12} + U_{13} + U_{23} + ...$

General: $$U = -\frac{GMm}{r}$$ (variable distance) | Near Earth: $$Delta U = mgDelta y$$ (constant gravitational field).

Conservative: store energy, path-independent | Non-conservative: dissipate energy, path-dependent.

Conservative Forces: Work done is path-independent, net work in a closed loop is zero, potential energy can be defined. Non-Conservative Forces: Work done is path-dependent, net work in a closed loop is non-zero, potential energy cannot be defined.

Near Earth's Surface: $$Delta U_{g} = mg\Delta y$$, valid for small changes in height. Point Masses: $$U_{g} = -G \frac{m_{1}m_{2}}{r}$$, valid for any separation distance, uses a zero point at infinite separation.

Stable Equilibrium: Local minimum, system returns to equilibrium after a small disturbance, concave up. Unstable Equilibrium: Local maximum, system moves away from equilibrium after a small disturbance, concave down.