角动量守恒

The angular momentum $L$ of a system about a point $P$ is expressed as follows: $$L=\sum_{i}m_i(\vec{r_i}\times\vec{v_i})$$ where $m_i$ are the masses of the small elements, $\vec{r_i}$ are the position vectors from $P$, and $\vec{v_i}$ are the velocities.

A physical law known as the Law of Conservation of Angular Momentum states that in any closed system, the angular momentum with respect to a given point must stay constant. A closed system in this context means that there is no net external torque on the system, and that no mass is entering or leaving.

The mathematics of why this is true is quite complicated, so I will not try to prove it here.

This can have some fairly nonintuitive consequences. Here is an example:

Suppose that you sit on a chair holding a spinning bicycle wheel. Suppose that the plane of rotation is parallel to the floor, and that the wheel is spinning counterclockwise when looked at from above. Then the angular momentum of the wheel is a vector pointing directly upward.

Now suppose that you flip the wheel over, so that the wheel is spinning clockwise as seen from above. The angular momentum vector flips over, but you and the wheel are an isolated system! Thus, since angular momentum is conserved, you will actually start to spin counterclockwise in your chair to make up for the change.

It seems rather nonintuitive, but it makes sense when one considers angular momentum. This is true in many cases: the conservation of angular momentum is certainly not very intuitive, but it has many strange and beautiful consequences.