Suppose that a star is encompassed by a Dyson Sphere. Do we need a position control system for the Dyson Sphere to keep its origin always aligned with the center of the star? Will it stay aligned itself and automatically neutralize small perturbational forces? Or will a small perturbation remove it out of the orbit of the star, so that it will be eventually destroyed and swallowed by the star?
Do we need a position control system for the Dyson Sphere to keep its origin always aligned with the center of the star?
The important principle for this question is spherical symmetry with the application of Gauss's law. For a electrically charged sphere the charges are accumulated on the surface, leading to the situation where the field is effectively zero everywhere within that sphere. Mathematically a Dyson sphere is equivalent to that. Of course, the field within the Dyson sphere wouldn't be zero because of the sun, but the contribution from the Dyson sphere sphere structure itself could be zero. I say "could" because this assumes the matter is perfectly distributed.
Such a system is neither stable or unstable. It's just balanced.
If the perfect symmetry is satisfied (and the neighborhood is clear, more on that later), the location of the star within the sphere is irrelevant to its stability actually. The system would be just as stable with the star close to the edge as it would be exactly at the Dyson sphere's center.
If the matter was anything less than perfectly spherical it would be categorically unstable, not just balanced. I believe this is the same sort of consequence as Earnshaw's theorem. Any arrangement of matter won't create an inherently stable point in empty space where the field's divergence is zero.
For purposes of science fiction, I would say the answer is "yes", active stabilization of some form would be needed. Even if you assume perfect symmetry, there are ways other objects in the solar system would cause the sphere and the star to crash into each other in the absence of active controls. I will have to break this down into two cases:
Factors apply even if the sun is at the origin:
- the radiation in our part of the galaxy isn't isotropic, so it would push the sphere to hit the star eventually
- gravitational waves can act non-uniformly, and this can lead to other modes of instability (next list) to drive them to hit eventually
Factors that apply if the sun it slightly displaced from the origin:
- if the sun wasn't dead center, tidal forces from other astronomical bodies would accelerate it toward the side
- if the Dyson Sphere was non-uniform and the sun was not at the CM, it would be accelerating, and that would be unstable acceleration
However, if we are assuming an advanced civilization built the Dyson sphere, it shouldn't be a difficult task to use controlled reflected radiation from the sun itself to keep it in the center. There are other, much more major, problems with the physicality of such a structure.
You have ignored outside influences. Something as simple as asteroids striking the outside of the sphere would push it out of concentric balance and cause a drift toward eccentricity, eventually leading to a contact between the star and the shere (or more likely a vaporization of part of the sphere well before a contact). There would need to be active re-correction/stabilization.