Definition: Work \& Potential Energy

The work done by a force $\mathbf F$ is defined as $$W = \mathbf F \cdot \mathbf d = \int \mathbf F \cdot d \mathbf x.$$ If the force is conservative, i.e. the works is path-independent, we can define a potential energy: $$\Delta U = -\int \mathbf F \cdot d \mathbf x = - \Delta W_{\text{ext}}.$$ We can also reverse this to get $$F = -\frac{dU}{dx}.$$

Some common potential energies for use:

Gravity, measured close to the surface of the earth with $h \ll R_E$, is $U(h) = mgh$.
Spring potential is $U(x) = \frac 12 kx^2$.

Theorem: Conservation of Energy and Work-Energy Theorem

Conservation of energy tells us that the total change in kinetic and potential energy is equal to external, non-conservative work: $$\Delta U + \Delta K = W_{nc},$$ and an equivalent statement is the Work-Energy theorem, which tells us $$\Delta K = W_{\text{net}}.$$

Tip: Graphs

Potential energy graphs can help you visualize:

Equilibrium points: local maxima (unstable) and local minima (stable)
Force: just the slope
Turning points: where the total energy is the same as the potential energy, resulting in zero velocity.