The Birkhoff’s Equation and Its Integral Theory for Constrained Systems

This paper investigates the integral theory of Birkhoff's equations. The existence of energy-like integrals and cyclic-like integrals for Birkhoffian systems is examined, along with the relationship between Santilli's integrals and cyclic-like integrals. The first integral theorem shows the existence of energy-like integrals in autonomous Birkhoff's equations, while semi-autonomous Birkhoff's equations lack such integrals. The second integral theorem provides conditions for the presence of cyclic-like integrals in Birkhoff's equations. Cyclic integrals and Santilli's integrals are discussed, with Corollary 2 demonstrating the relationship between cyclic coordinates and cyclic integrals. The paper concludes with a discussion on finding first integrals for Birkhoff's equations. The presented theorems and construction methods offer valuable insights into Birkhoffian mechanics and its dynamic properties.