PERTURBATION

in astronomy, deviation in the motion of a celestial object caused either by the gravitational force of a passing object or by a collision with it. For example, predicting the Earth's orbit around the Sun would be rather straightforward were it not for the slight perturbations in its orbital motion caused by the gravitational influence of the other planets. The search for a 10th planet continues because some astronomers believe that the known outer planets are being gravitationally perturbed by something beyond Pluto. in mathematics, method for solving a problem by comparing it with a similar one for which the solution is known. Usually the solution found in this way is only approximate. Perturbation is used to find the roots of an algebraic equation that differs slightly from one for which the roots are known. Other examples occur in differential equations. In a physical situation, an unknown quantity is required to satisfy a given differential equation and certain auxiliary conditions that define the values of the unknown quantity at specified times or positions. If the equation or auxiliary conditions are varied slightly, the solution to the problem will also vary slightly. The process of iteration is one way in which a solution of a perturbed equation can be obtained. Let D represent an operation, such as differentiation, performed on a function, and let D + eP represent a new operation differing slightly from the first, in which e represents a small constant. Then, if f is a solution of the common type of problem Df = cf, in which c is a constant, the perturbed problem is that of determining a function g such that (D + eP)g = cg. This last equation can also be written as (D - c)g = -ePg. Then the function g1 that satisfies the equation (D - c)g1 = -ePf is called a first approximation to g. The function g2 that satisfies the equation (D - c)g2 = -ePg1 is called a second approximation to g, and so on, with the nth approximation gn satisfying (D - c)gn = -ePgn-1. If the sequence g1, g2, g3, . . . , gn, . . . converges to a specific function, that function will be the required solution of the problem. The largest value of e for which the sequence converges is called the radius of convergence of the solution. Another perturbation method is to assume that there is a solution to the perturbed equation of the form f + eg1 + e2g2 + . . . etc., in which the g1, g2, . . . etc., are unknown, and then to substitute this series into the equation, resulting in a collection of equations to solve corresponding to each power of e.