Energy

Conservation §

A powerful invariance.

$$E=K+U$$

$K$ is kinetic energy, $U$ is potential energy.

$$\begin{array}{c}K=\frac{1}{2}m{\vec{v}}^2\\ v=v_x+v_y+v_z\end{array}$$

Force §

$$F=-\frac{dU}{dx}$$

Examples §

1. Spring force

$$F_{\text{Hooke}}=-kx,U=\frac{1}{2}k{x}^2$$

2. Earth’s Gravity

$$F_g=-mg,U=mgy$$

3. Gravity

$$F_g=-\frac{GMm}{r^2},U=-\frac{GMm}{r}$$