Perturbation

Perturbation refers to a modification or disturbance added to a system, often small in magnitude, used to study how the system responds, to analyze stability, or to approximate complex problems by building on a known baseline solution.

In mathematics and physics, perturbation theory develops solutions as a series in a small parameter ε that measures the size of the disturbance. When ε is small, the leading term approximates the original problem, and higher-order terms provide successive corrections. This approach exploits the idea that a difficult problem can be understood by starting from a simpler, exactly solvable one.

Two main types of perturbation techniques are regular and singular. Regular perturbations assume a smooth dependence on ε and yield well-behaved expansions. Singular perturbations occur when taking ε to zero fundamentally changes the problem’s character, requiring specialized methods such as matched asymptotics or multiple scales to capture different regimes.

In quantum mechanics, time-independent perturbation theory computes shifts in energy and changes to states due to a weak perturbing Hamiltonian, while time-dependent perturbation theory yields transition probabilities between states under a perturbation that varies in time. In classical mechanics and dynamical systems, perturbations analyze how trajectories or equilibria change when the system is slightly altered; orbital perturbations in celestial mechanics describe how gravity from other bodies modifies an orbit.

Perturbation methods are widely used in engineering and applied sciences to solve differential equations, perform stability analyses, and approximate control problems. Limitations include the requirement of a small parameter and potential non-convergence of series, in which case nonperturbative or numerical approaches may be necessary.