Key Equations

Difference of potential energy	$Δ U_{A B} = U_{B} - U_{A} = − W_{A B}$
Potential energy with respect to zero of potential energy at ${\vec{r}}_{0}$	$Δ U = U (\vec{r}) - U ({\vec{r}}_{0})$
Gravitational potential energy near Earth’s surface	$U (y) = m g y + const .$
Potential energy for an ideal spring	$U (x) = \frac{1}{2} k x^{2} + const .$
Work done by conservative force over a closed path	$W_{closed path} = \int {\vec{F}}_{cons} \cdot d \vec{r} = 0$
Condition for conservative force in two dimensions	$(\frac{d F_{x}}{d y}) = (\frac{d F_{y}}{d x})$
Conservative force is the negative derivative of potential energy	$F_{l} = - \frac{d U}{d l}$
Conservation of energy with no non-conservative forces	$0 = W_{n c, A B} = Δ {(K + U)}_{A B} = Δ E_{A B} .$