(1) Based on calculus of variation, the approximate formula of lower rank nature frequencies is deduced for the free RCB with local dilapidation of concrete.

(2) The differential equation for the vibration of the riser system is derived by means of functional calculus of variation.

(3) According to the variation of magnetic energy, differential equation of dual resonance was gotten by means of calculus of variation in this paper.

(4) Ordinary differential equations, Partial differential equations, Linear algebra, Special functions, Calculus of variation.

(5) Calculation of resonant frequencies of sloshing in a container have been achieved by FEM with calculus of variation, which is able to solve sloshing in variety complex containers.