A simplified method of solving Birkhoffian function and Lagrangian

Abstract

By using the calculus of variations, the conservative mechanical systems can be formulated by Lagrange's equations or Hamilton's equations, which are the basis of establishing, simplifying and integrating the equations of motion. Thus it is important to find the solutions of inverse problems for different dynamical systems so as to construct the most of the Lagrange's equations and Hamilton's equations. However, the Lagrangian or Hamiltonian formulation for a dynamical system, limited by the conditions of self-adjointness, is not directly universal if the physical variables remain without using Darboux transformations. Fortunately, Refs. [7, 11] show that based on the Cauchy-Kovalevsky theorem of the integrability conditions for partial differential equations and the converse of the Poincar lemma, it can be proved that there exists a direct universality of Birkhoff's equation for local Newtonian system by reducing the Newton's equations to a first-order form, which means that all local, analytic, regular, finite-dimensional, unconstrained or holonomic, conservative or non-conservative forms always admit, in a star-shaped neighborhood of a regular point of their variables, a representation in terms of first-order Birkhoff's equations in the coordinate and time variables of the experiment. The systems whose equations of motion are represented by the first-order Birkhoff's equations on a symplectic or a contact manifold spanned by the physical variables are called Birkhoffian systems. At present, one of the most important tasks of Birkhoffian mechanics is to study the method of constructing the Birkhoffian and Birkhoffian functions. However, due to the complexity of Birkhoffian system, there exist only a few of results in the literature. Among them, the most famous main methods in this problem are achieved by Santilli[Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp12-15]. But the redundant term in Santilli's second method which is used as the classical construction method, is always neglected. As a result, the calculation procedure is tedious and complicated, and the efficiency is not high. Therefore, it is necessary to simplify the Santilli's second method. In Section 2, we will review the first-order standard form of holonomic system in the frame of Cartesian coordinates, which is the starting point of our studying the Birkhoffian systems. In Section 3, the Birkhoff's equations and the key role of Birkhoffian dynamics functions for deriving Birkhoff's equations are introduced. In Section 4, the redundant items are eliminated by using some mathematical operation skills, and then a more simplified constructing method is put forward. In Section 5, the findings in this study are summarized. Through simplifying the Santilli's second method, we realize that the determining of the Birkhoff's equations by constructing the Birkhoffian functions is equivalent to the determining of its symplectic matrix. This view provides a new perspective for solving the problem of constructing the Birkhoffian functions. Finally, the simplified method is applied to Lagrangian inverse problem, and a simplified method of solving Lagrangian function is obtained.