Mathematical Methods for Physics and Engineering - Matematica.NET

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES21.3 Separation of variables in polar coordinatesSo far we have considered the solution of PDEs only in Cartesian coordinates,but many systems in two and three dimensions are more naturally expressedin some form of polar coordinates, in which full advantage can be taken ofany inherent symmetries. For example, the potential associated with an isolatedpoint charge has a very simple expression, q/(4πɛ 0 r), when polar coordinates areused, but involves all three coordinates and square roots when Cartesians areemployed. For these reasons we now turn to the separation of variables in planepolar, cylindrical polar and spherical polar coordinates.Most of the PDEs we have considered so far have involved the operator ∇ 2 ,e.g.the wave equation, the diffusion equation, Schrödinger’s equation and Poisson’sequation (and of course Laplace’s equation). It is therefore appropriate that werecall the expressions for ∇ 2 when expressed in polar coordinate systems. Fromchapter 10, in plane polars, cylindrical polars and spherical polars, respectively,we have∇ 2 = 1 (∂ρ ∂ )+ 1 ∂ 2ρ ∂ρ ∂ρ ρ 2 ∂φ 2 , (21.23)∇ 2 = 1 (∂ρ ∂ )+ 1 ∂ 2ρ ∂ρ ∂ρ ρ 2 ∂φ 2 + ∂2∂z 2 , (21.24)∇ 2 = 1 (∂r 2 r 2 ∂ )+ 1 (∂∂r ∂r r 2 sin θ ∂ )1 ∂ 2+sin θ ∂θ ∂θ r 2 sin 2 θ ∂φ 2 . (21.25)Of course the first of these may be obtained from the second by taking z to beidentically zero.21.3.1 Laplace’s equation in polar coordinatesThe simplest of the equations containing ∇ 2 is Laplace’s equation,∇ 2 u(r) =0. (21.26)Since it contains most of the essential features of the other more complicatedequations, we will consider its solution first.Laplace’s equation in plane polarsSuppose that we need to find a solution of (21.26) that has a prescribed behaviouron the circle ρ = a (e.g. if we are finding the shape taken up by a circular drumskinwhen its rim is slightly deformed from being planar). Then we may seek solutionsof (21.26) that are separable in ρ and φ (measured from some arbitrary radiusas φ = 0) and hope to accommodate the boundary condition by examining thesolution for ρ = a.725