What does it mean to say that two things are coupled or not decoupled ?

[AxiomOfChoice]

What does it mean to say that two things are "coupled" or "not decoupled"?

In a paper I'm reading, it says that "the vibrations and rotations are no longer decoupled for large angular momentum." (This is discussing a diatomic molecule.) What, exactly, does this mean?

 

[Civilized]

This means that the equations of motion (or the Lagrangian itself) has a term that mixes vibrational and rotational degrees of freedom, and that this terms is negligible at small angular momentum.

Imagine that x is a degree of freedom for rotations, and y is a degree of freedom for vibrations. Then the equations of motion:

x' = x + x^2
y' = y + Sin(y)

are decoupled in x & y, x & y are essentially independent (although there may still be a constraint involving both of them e.g. a boundary condition). Compare this to the situation:

x' = x + x^2 + y
y' = y + Sin(y) + Cos(x)

Now x & y are coupled! Although the example I'm presenting is artificial and non-quantum, coupled/decoupled equations are general terms in systems of differential equations, I'm just giving an example. In all cases decoupled subsystems behave independently of one another after the initial conditions have been set.