19. Calculus of Variations Revisited Plus the Gamma and Bessel Functions

A variety of topics that use the Calculus of Variations are explored. After developing general conditions for constrained optimization in a pastoral interlude they are applied to Queen Dido’s isoperimetric problem. Then the Euler-Lagrange equations of motion for conservative dynamical systems are derived in a pastoral interlude and applied to a linear spring, a double plane pendulum, and a vibrating string, respectively. Finally Bessel functions needed in the problems are introduced and examined which requires a similar preliminary examination of the gamma function as well since the former are defined in terms of the latter. In this context the Method of Stationary Phase and Laplace’s Method are developed and employed to derive asymptotic representations for the Bessel functions and Stirling’s formula for n!, respectively. There are eight problems: The first seven deal with a variety of Calculus of Variations applications and asymptotic representations for the special functions as well as a complete examination of the Bessel equation of order one-half. The eighth problem requires a Calculus of Variations approach to derive the wave equation governing the motion of a vibrating circular membrane which when solved by separation of variables gives rise to the eigenvalue example involving Bessel’s equation of order zero handled in the final section of the chapter.