# External movement commutes with internal movement

There is a very interesting general principle of mechanics (which is not exactly new, but which I have not seen formally written anywhere this way):

External movement commutes with internal movement.

For example, consider a person in a plane which flies. When the person moves in the plane, it’s an internal movement with respect to the plane. And when the plane moves in the sky, it’s an external movement. The total movement of the person (with respect to a fixed reference system, say on the earth) is composed of these 2 movements. The principle is that, the order of the composition doesn’t matter: the plane can move first, and then the person moves inside the plane later, or the person can move inside the plane first, and the plane moves later, the end result (position of the person with respect to the earth) will be the same.

From the point of view of group theory, this principle can be understood as: left action commutes with right action (say multiplication on the left commutes with multiplication on the right).

But “Internal” and “external” are relative words. A movement can be internal with respect to something, and external with respect to another thing at the same time. The movement of the plane (in the sky) is external with respect to the plane, but internal to the earth, the movement of the Earth around the Sun is external to the Earth, and the movement of blood inside a person in the plane is internal to the person, and so on. So actually one can have a “chain” of movements, such that each two consecutive mevements in this chain form an internal-external pair. All the movements in such a chain commute with each other.

From the Hamiltonian systems point of view, this fact may be of great interest, because such a chain of movements may give rise to commuting Hamiltonian flows, and when the chain is long enough, these flows form an integrable Hamiltonian system (à la Liouville). This is the case with the so called Gelfand-Cetlin system.

Probably, this idea can also be applied to a “vortex inside a vortex” system. Consider a small vortex A living inside a large vortex B  (imagine a tornado if you like). Then A has 2 movements: the external one created by B, and the internal one given by A itself. These two movements commute with each other ? Now onto a “vortex inside a vortex inside a vortex …” ?!