A method of judging a Birkhoffian to be a first integral of constrained mechanical system
Authors: Cui Jin-Chao  , Liao Cui-Cui  , Liu Shi-Xing  , Mei Feng-Xiang  ,   School of Science, Jiangnan University, Wuxi 214122, China;,   College of Physics, Liaoning University, Shenyang 110036, China;,   School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100018, China
Published: December 2020

As is well known, the development of analysis mechanics from Lagrangian systems to Birkhoffian systems, achieved the self-adjointness representations of the constrained mechanical systems. 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 equations for local Newtonian system by reducing Newton's equations into 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. The theory and method of Birkhoffian dynamics are used in hadron physics, quantum physics, relativity, rotational relativity, and fractional-order dynamics.
At present, for a given dynamical system, it is important and essential to determine whether a Birkhoffian function is the first integral of the system. Although the numerical approximation is an important method of solving the differential equations, the direct theoretical analysis is more helpful for refining the general integral method, and more consistent with the usual way of solving problems of analysis mechanics. In this paper, we study how to judge whether a given Birkhoffian dynamical function to be a first integral of Birkhoff's equations, based on the point of Birkhoffian dynamical functions carrying all the informationabout motion of the system, and use the thought of deriving the first integrals of Hamiltonian systems. In Section 2, the normal first-order form and the Birkhoff's equations of the equations of motion of holonomic systems are introduced. In Section 3, we prove that the Birkhoffian function of an autonomous Birkhoffian system must be a first integral, and the Birkhoffian function of a semi-autonomous system must not be a first integral. Moreover, the energy integral, cyclic integral and Hojman integral of the non-autonomous Birkhoffian systems are given. In Section 4, two examples are given to illustrate the applications of the results. In Section 5, the full text is summarized and the results are discussed. It is necessary to point out that the judging method is effective to determine whether a given Birkhoffian functions can be identified to be a first integral of Birkhoff's equations, but other new first integral cannot be found with this method. One possible method of covering the shortage is to obtain other equivalent Birkhoffian functions in terms of isotopic transformations of Birkhoff's equations, and then use our results to seek the new first integral. In addition, we also hope to develop a more direct method of obtaining the first integrals of Birkhoff's equations in the next study.