Some variational principles for problems in hydrodynamic and hydromagnetic stability

Abstract

A number of problems in hydrodynamic and hydromagnettc stability give rise to characteristic value problems with differential equations of high order but with constant coefficients. It is shown that these characteristic value problems can be formulated as variational problems which can be solved exactly without the necessity of integrating any differential equation. The method of solution depends primarily on the expansion of the unknown functions in the variational principles in complete sets of appropriate orthogonal functions. It is also shown how these methods can be extended to a non-self adjoint boundary value problem by using a variational principle dependent on both the original problem and the adjoint boundary value problem. A method of solving the characteristic value problem by a direct series substitution in the differential equation is illustrated. (M.C.G.)