Advanced Calculus fi..

Chapter 10 Partial Differential Equations 705physics, both models are an oversimplification of what is observed in nature. If eitherone serves to describe the phenomena concerned with sufficient accuracy, then it canbe regarded as a useful one.The equilibrium solution for systems (10.139) and the analogous ones in twoand three dimensions can be considered as problems of minimizing a function$(ul, . . . , uN). For, as was remarked in Section 10.4, Eqs. (10.139) can be writtenin the formd2~, dun avma-+ha-+-= F,(t) (a= 1 ,..., N). (10.141)dt2 dt au,The equilibrium problem is then the problem2rlwhere the F, are constants. The end-values uo and UN+I are also given as constants;we can consider them to be 0 by modifying the definition of Fl and FN. If we nowlet~(uI,. .., UN) = V(UI,. .., UN) - (Flu1 + --. + FNUN), (10.143)then (10.142) is simply the condition that - ~ l c j ,For (10.1 39) we haveand we can verify that (10.144) has precisely one solution u;, . . . , u*, and thatthis critical point is a minimum point is also easily verified (Problem 6 followingSection 10.18). A similar statement applies to the analogous problems in two or threedimensions. Indeed, the physical picture that leads to equations of form (10.141) isalmost always that of a system of particles capable of an equilibrium state, at whichthe potential energy V has its smallest value; Eqs. (10.141) then describe the forcedoscillations about this equilibrium state. When the applied forces are constant, thepotential energy V is replaced by a modified one 4; the Eqs. (10.141) then becomethe new equilibrium state is now the minimum of $.By appropriate passage to the limit it can be shown that the equilibrium problemfor (10.138) is equivalent to minimizing the expression