In this example, <math>A_0</math> would be the known solution to the exactly solvable initial problem and <math>A_1,A_2,\ldots</math> represent the "higher orders" which are found iteratively by some systematic procedure. For small <math>\epsilon</math> these higher orders become successively more unimportant.
Examples for the "mathematical description" are: an algebraic equation[?], a differential equation (e.g. the equations of motion in celestial mechanics or a wave equation), a free energy (in statistical mechanics), a Hamiltonian operator (in quantum mechanics).
Examples for the kind of solution to be found perturbatively: the solution of the equation (e.g. the trajectory of a particle), the statistical average[?] of some physical quantity (e.g. average magnetization), the ground state energy of a quantum mechanical problem.
Examples for the exactly solvable problems to start with: Linear equations, including linear equations of motion ([[harmonic oscillator]], linear wave equation), statistical or quantummechanical systems of noninteracting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).
Examples of "perturbations" to deal with: Nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/Free Energy.
For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.
Consider the following equation for the unknown variable <math>x</math>:
For the initial problem with <math>\epsilon=0</math>, the solution is <math>x_0=1</math>. For small <math>\epsilon</math> the lowest order approximation may be found by inserting the ansatz
into the equation and demanding the equation to be fulfilled up to terms that involve powers of <math>\epsilon</math> higher than the first. This yields <math>x_1=1</math>. In the same way, the higher orders may be found. However, even in this simple example it may be observed that for (arbitrarily) small <math>\epsilon>0</math> there are two other solutions to the equation (with very large magnitude) which cannot be found using perturbation theory.
The same problem occurs in many real applications in physics and elsewhere: Perturbation theory may only be used to find those solutions of a problem that evolve smoothly out of the initial solution when changing the parameter (that are "adiabatically connected" to the initial solution). In physics, this fails whenever the system may go to a different "phase" of matter, with a qualitatively different behaviour that cannot be understood by perturbation theory (e.g. a solid crystal melting into a liquid).
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