1. Find the slope of the potential energy curve at the point of interest. 2. Take the negative of that slope: $$F_{x} = -\frac{dU(x)}{dx}$$.

Sum the potential energies of all pairs of interacting objects: U_total = U_12 + U_13 + U_23 + ...

The change in potential energy (ΔU) is equal to the negative work done by conservative forces: $$\Delta U = - \int_{a}^{b} \vec{F}_{cf}(r) \cdot d\vec{r}$$

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.

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.