Chaos and controllability

Consider a system governed by systems of differential equations that evolves chaotically. Thus small variations in initial conditions are amplified to produce large changes in later states. Suppose humans can make small purposeful changes. Then perhaps the later states can be controlled. Suppose some initial conditions for a system are specified, and we can compute the future history. Small deviations from the initial conditions grow exponentially with time. Thus making a large change with a very small expenditure of energy requires a long time, but the time grows only logarithmically with the inverse of the energy expended. We can ask what states are reachable by small changes in initial conditions.

Take the solar system as an example. There are certain integrals of the motion of the solar system as a whole, e.g. energy and angular momentum. It is clear that we cannot change the total energy of the system by more than the energy we have available to expend. However, we can in principle change the energy of one planet at the expense of energy of some others.

Angular momentum is also conserved, but it isn't obvious to me what the trade-off is between expending energy to change angular momentum. Taking a mass a distance from the sun allows making a change of angular momentum of . If we want the changed angular momentum to be effective in the inner solar system, then must not be so large that the mass escapes. The time for the mass to return to the inner solar system is proportional to .

As a concrete problem, consider moving Mars to an orbit at the same distance from the sun as the earth, keeping it on the other side of the sun from the earth to eliminate gravitational interaction with the earth. This would make Mars warmer, which would facilitate human settlement.

2007-10-06