On Santilli's methods in Birkhoffian inverse problem

In this paper, we mainly study the simplification and improvement of Santilli's methods in Birkhoffian system, which is a more general type of basic dynamic system. The theories and methods of Birkhoffian dynamics have been used in hadron physics, quantum physics, rotational relativity theory, and fractional dynamics. As is well known, Lagrangian inverse problem, Hamiltonian inverse problem, and Birkhoffian inverse problem are the main objects of the dynamic inverse problems. The results given by Douglas (Douglas J 1941 Trans. Amer. Math. Soc. 50 71) and Havas [Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363] show that only the self-adjoint Newtonian systems can be represented by Lagrange's equations, so the Lagrangian inverse problem is not universal for a holonomic constrained mechanical system. Furthermore, from the equivalence between Lagrange's equation and Hamilton's equation, Hamiltonian inverse problem is not universal. A natural question is then raised: whether there exists a self-adjoint dynamical model whose inverse problem is universal for holonomic constrained mechanical systems, in the field of analytical mechanics. An in-depth study of this issue in the 1980s by R. M. Santilli shows that a universal self-adjoint model exists for a holonomic constrained mechanic system that satisfies the basic conditions of locality, analyticity, and formality. The Birkhoff's equation is a natural extension of the Hamilton's equation, which shows the geometric properties of a nonconservative system as a general symplectic structure. This more general symplectic structure provides the geometry for the study of the non-conservative system preserving structure algorithms. Therefore, it is particularly important to study the problem of the Birkhoffian representation for the holonomic constrained system. For the inverse problem of Birkhoff's dynamics, studied mainly are the condition under which the mechanical systems can be represented by Birkhoff's equations and the construction method of Birkhoff's functions. However, due to the extensiveness and complexity of the holonomic nonconservative system, Birkhoff's dynamical functions do not have so simple construction method as Lagrange function and Hamilton function. The research results of this issue are very few. The existing construction methods are mainly for three constructions proposed by Santilli [Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp25-28], and there are still many technical problems to be solved in the applications of these methods. In order to solve these problems, this article mainly focuses on the following content. First, according to the existence theorem of Cauchy-Kovalevskaya type equations, we prove that the autonomous system always has an autonomous Birkhoffian representation. Second, a more concise method is given to prove that Santilli's second method can be simplified. An equivalent relationship implied in Santilli's third method is found, an improved Santilli's third method is proposed, and the MATLAB programmatic calculation of the method is studied. Finally, the full text is summarized and the results are discussed.