This address discusses variational methods for solving equilibrium and vibration problems, emphasizing the connection between boundary value problems of partial differential equations and variational problems. It highlights the work of Rayleigh and Ritz, who used variational principles to approximate solutions numerically. The key idea is to replace complex problems with simpler ones that involve a finite number of parameters. The text explains how variational problems can be transformed into quadratic functionals and how boundary conditions are determined through natural conditions derived from variational principles. It also discusses the Rayleigh-Ritz method, which involves approximating solutions using a set of coordinate functions and solving linear equations to find the minimum of a functional. The method is compared with finite difference methods, which approximate derivatives by differences and solve discrete equations. The text also covers the method of gradients, which uses the concept of steepest descent to find solutions to variational problems. The address concludes with an appendix that demonstrates the numerical treatment of a plane torsion problem for multiply-connected domains, showing the effectiveness of various methods in solving such problems.This address discusses variational methods for solving equilibrium and vibration problems, emphasizing the connection between boundary value problems of partial differential equations and variational problems. It highlights the work of Rayleigh and Ritz, who used variational principles to approximate solutions numerically. The key idea is to replace complex problems with simpler ones that involve a finite number of parameters. The text explains how variational problems can be transformed into quadratic functionals and how boundary conditions are determined through natural conditions derived from variational principles. It also discusses the Rayleigh-Ritz method, which involves approximating solutions using a set of coordinate functions and solving linear equations to find the minimum of a functional. The method is compared with finite difference methods, which approximate derivatives by differences and solve discrete equations. The text also covers the method of gradients, which uses the concept of steepest descent to find solutions to variational problems. The address concludes with an appendix that demonstrates the numerical treatment of a plane torsion problem for multiply-connected domains, showing the effectiveness of various methods in solving such problems.