My question will be less informative because right now I myself don't know where I am heading. Basically I want to know that whether it is possible to think a chi-square minimization problem as a calculus of variation problem.
Not really. A chi-square minimization problem seeks the parameterized function that minimizes some quantity. A calculus of variation problem seeks the arbitrary function that minimizes some quantity. The degrees of freedom of the former are a finite number of parameters. The degrees of freedom of the latter are infinite.