Potential Energy

Energy stored due to its position. For example energy due to being in a gravity field, or energy in a spring.

$$\begin{array}{rlr}& \text{Spring:}& V(q)=\frac{1}{2}k(\Delta x(q){)}^2\\ & \text{Gravity:}& V(q)=mgh(q)\end{array}$$

Or more generally

$$V_i(q)=m{g}^T{r}_{cg}(q)+f_{ki}(q)$$

Where $r_{cg}$ is the center of gravity position, and $f_{ki}(q)$ potential energy due to other things.

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