Cross product

calculator

Law of Cosines

\(c=\sqrt{a^{2}+b^{2}-2 a b \cdot \cos \gamma}\)

Small angle approximations:

\(\sin\theta \approx \theta, \tan\theta \approx \theta, \cos\theta \approx 1-\frac{\theta^2}{2}\) \(\sin(u)\sin(v) = \frac{1}{2}[\cos(u-v) - \cos(u+v)]\) \(\cos(u)\cos(v) = \frac{1}{2}[\cos(u-v) + \cos(u+v)]\)

Physics prequisite

force to torque:

\(\tau = r \times F = rF\sin\theta\)

friction torque(viscous) >Wiki:

\(\tau = -k\dot{\theta}\)

弹性力（k 为弹簧的劲度系数，x 为振子偏离平衡位置的位移）： \(F = -kx\) 阻尼力（c 为阻尼系数，v 为振子速度）： \(F = -cv\)

Potential energy

Springs: $\frac{1}{2}k(x-x_0)^2$

Gravity: $mgh$

Kinetic energy

Linear motion: $\frac{1}{2}mv^Tv$

Angular motion: $\frac{1}{2}\omega^TI\omega$

Inertia

$I$ is always symmetric. Diagonal if CoM aligns with primary axes.

For cylinder: $I_z = \frac{1}{2}MR^2$, $I_x=I_y=\frac{1}{4}MR^2+\frac{1}{12}ML^2$