### Key Equations

Difference of potential energy | $\text{\Delta}{U}_{AB}={U}_{B}-{U}_{A}=\text{\u2212}{W}_{AB}$ |

Potential energy with respect to zero of potential energy at ${\overrightarrow{r}}_{0}$ |
$\text{\Delta}U=U\left(\overrightarrow{r}\right)-U\left({\overrightarrow{r}}_{0}\right)$ |

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 | ${W}_{\text{closed path}}={\displaystyle \int {\overrightarrow{F}}_{\text{cons}}\xb7d\overrightarrow{r}=0}$ |

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 |
$0={W}_{nc,AB}=\text{\Delta}{\left(K+U\right)}_{AB}=\text{\Delta}{E}_{AB}.$ |