User:Corkgkagj/disp

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and transverse velocity, the component of velocity along a circle centered at the origin, and equal to the distance to the origin times the angular velocity.

$\mathbf {v} _{trans}=|\mathbf {x} |\omega$

where

$\mathbf {v} _{trans}$ is the transverse velocity

$|\mathbf {x} |$ is the distance from the origin

$\omega \,$ is the angular velocity

Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with positive quantities representing counter-clockwise direction and negative quantities representing clockwise direction (in a right-handed coordinate system).

If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion