Häiriöteoria

Häiriöteoria, or perturbation theory, is a mathematical method used to find approximate solutions to problems that cannot be solved exactly. It is particularly useful in physics and engineering when dealing with systems that are close to a solvable, simpler system. The core idea is to start with a known solution to a simplified problem and then introduce small corrections, or "perturbations," to account for the more complex aspects of the real system.

The method involves expanding the solution in terms of a small parameter, often denoted by epsilon (ε), which represents the magnitude of the perturbation. The exact solution is then approximated by a series, where each term represents a higher order correction due to the perturbation. For instance, if the unperturbed system has a solution $S_0$, the perturbed solution might be expressed as $S = S_0 + \epsilon S_1 + \epsilon^2 S_2 + ...$, where $S_1, S_2$, etc., are corrections that are progressively smaller.

Häiriöteoria is widely applied in various fields. In quantum mechanics, it's used to calculate energy levels and wave functions of atoms and molecules when external fields or interactions are present. In celestial mechanics, it helps predict the orbits of planets and satellites by accounting for the gravitational influence of other celestial bodies. It also finds application in fluid dynamics and elasticity for analyzing complex flow patterns or stress distributions. The validity of the approximation depends crucially on the smallness of the perturbation parameter; if the perturbation is too large, the series may not converge or provide an accurate representation of the true solution.