Key Equations

Difference of potential energy	$\text{Δ}{U}_{AB}=U_B-U_A=\text{−}{W}_{AB}$
Potential energy with respect to zero of potential energy at ${\overrightarrow{r}}_0$	$\text{Δ}U=U\left(\overrightarrow{r}\right)-U\left({\overrightarrow{r}}_0\right)$
Gravitational potential energy near Earth’s surface	$U\left(y\right)=mgy+\text{const}.$
Potential energy for an ideal spring	$U\left(x\right)=\frac{1}{2}k{x}^2+\text{const}.$
Work done by conservative force over a closed path	$W_{\text{closed path}}={\displaystyle \int {\overrightarrow{F}}_{\text{cons}}·d\overrightarrow{r}=0}$
Condition for conservative force in two dimensions	$\left(\frac{d{F}_x}{dy}\right)=\left(\frac{d{F}_y}{dx}\right)$
Conservative force is the negative derivative of potential energy	$F_l=-\frac{dU}{dl}$
Conservation of energy with no non-conservative forces	$0=W_{nc,AB}=\text{Δ}{\left(K+U\right)}_{AB}=\text{Δ}{E}_{AB}.$