This chapter discusses the application of variational methods to solve problems of equilibrium and vibrations, emphasizing the synthesis of theoretical and applied mathematics. The author, R. Courant, highlights the historical development of these methods, starting from the work of Gauss and W. Thompson, and the contributions of Lord Rayleigh and Walther Ritz. The chapter covers the formulation of variational problems, including quadratic functionals and boundary conditions, and discusses the advantages and limitations of the Rayleigh-Ritz method. It also explores alternative numerical methods such as the method of finite differences and the method of gradients, providing a comprehensive overview of the techniques available for solving boundary value problems in elasticity and vibrations. The appendix includes a numerical example of the plane torsion problem for multiply-connected domains, demonstrating the practical application of the methods discussed.This chapter discusses the application of variational methods to solve problems of equilibrium and vibrations, emphasizing the synthesis of theoretical and applied mathematics. The author, R. Courant, highlights the historical development of these methods, starting from the work of Gauss and W. Thompson, and the contributions of Lord Rayleigh and Walther Ritz. The chapter covers the formulation of variational problems, including quadratic functionals and boundary conditions, and discusses the advantages and limitations of the Rayleigh-Ritz method. It also explores alternative numerical methods such as the method of finite differences and the method of gradients, providing a comprehensive overview of the techniques available for solving boundary value problems in elasticity and vibrations. The appendix includes a numerical example of the plane torsion problem for multiply-connected domains, demonstrating the practical application of the methods discussed.