## Introduction to Perturbation Theory

Real problems rarely have Hamiltonians with equations of motion having simple solutions depending upon elementary functions, so some method of approximating the solutions is needed. Typically we start with a Hamiltonian of the form

(1)

where the unperturbed Hamiltonianis soluble but the perturbed Hamiltonian (1) is not. The methods of finding approximate solutions constitute perturbation theory. The perturbationmay be independent of the time, in which case the perturbed system is conservative, or may depend on the time – the methods used to find approximate solutions are different in each case.

The most famous perturbation problem is the motion of the planets. The greatest influence on the motion of the planets is the Sun, with each planet modifying the orbit of each other planet by only a small amount. For example, the motion of Venus around the Sun is perturbed most by Jupiter, whose mean force on Venus is less thantimes the force exerted by the Sun, and the Earth, which exerts an average force less thantimes that of the Sun. As a first approximation, only the influence of the Sun is considered. The Hamiltonian is one of central force type and simple to solve. The effects of Jupiter and the other planets are small perturbations to the motion.

Often the set of solutions to the unperturbed system H-0 form a complete set, so that any function, within limits can be expressed as a sum of these solutions. This extends to the perturbed Hamiltonian, so that an solution to the perturbed Hamiltonian can be expressed as a sum of solutions to the unperturbed solutions. The problem then becomes finding the coefficients $epsilon in the expansion. Finding the solutions of equations may mean treatingas the variable and finding the coefficientsof

Perturbation theory is particularly useful when a system contains parameters and we need to know the effect of these on the solutions, and also when we want to analyse the qualitative behaviour of solutions. For example, is is relatively easy to solve the equation

for various values ofto reasonable accuracy, but there are often circumstances when it is useful to know that the root behaves as

where