Finite Element Method with Applications in Engineering

Appendix B

Elements of Calculus of Variations

B.1 GENERAL

Governing equations in engineering are generally second (or fourth) order ordinary or partial differential equations. This requires the function to be continuous having derivatives up to order two (or four). Description of a problem is not complete until correct associated boundary conditions are specified. If there is a function of the functions (functional) for the governing equation, the governing equations as well as the appropriate boundary conditions can be derived through the technique of Calculus of Variations (CV). The functional requires the functions to be continuous having lower order derivatives. Rich history of the CV of the last 300 years can be found in the book by Goldstine ...