APPLICATION DIRECT METHOD CALCULUS OF VARIATION FOR KLEIN-GORDON FIELD

Klein-Gordon field is often used to study the dynamics of elementary particles. The Klein-Gordon equation was first considered as a quantum wave equation by Schrodinger in his search for an equation describing de Broglie waves. The equation was found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation failed to predict the fine structure of the hydrogen atom, and overestimated the overall magnitude of the splitting pattern energy. This paper will describe in detail using the Direct Method of Calculus Variation as an alternative to solve the Klien-Gordon field equations. The Direct Method simplified the calculation because the variables are calculated and expressed in functional form of energy. The result of the calculation of Klien-Gordon Feld provided the existence of the minimizer, i.e. (phi) over bar = phi(0) + u with u epsilon W-0(1,p) and phi(0) epsilon W-1p Explicit form of the minimizer was calculated by the Ritz method through rows of convergent density.