SIGNIFICANCE AND LIMITATIONS OF SANTILLI' METHOD AND FIRST ENGELS' METHOD IN ANALYTICAL MECHANICS

Two methods for the construction of a Lagrangian in the theory of inverse problem, the Santilli' method and the first Engels'method, are discussed. (1) It is pointed out that the theoretical significance of the Santilli' method is in the fact that by a direct construction of Lagrangian one may prove that the self-adjoint differential equations can be derived by the variational principle, that is, represented in terms of the Lagrange's equations. (2) It is manifested that what constructed by the Santilli' method is not a unique Lagrangian, but a family of gauge equivalent Lagrangians, and the Santilli' method is modified. (3) The defects of the Santilli' method in practical applications are pointed out, especially for some mechanical systems, because the definite integral depending on a parameter is divergent, so the Lagrangian can not be constructed. (4) The significance and advantages of the first Engels' method are discussed, and the defects of this method are also pointed out. (5) Two examples are given to illustrate the application of the result.