Integrating Factors and Conservation Laws for Relativistic Mechanical System

In this paper, we present a new method to construct the conservation laws for relativistic mechanical systems by finding corresponding integrating factors. First, the Lagrange equations of relativistic mechanical systems are established, and the definition of integrating factors of the systems is given; second, the necessary conditions for the existence of conserved quantities of the relativistic mechanical systems are studied in detail, and the relation between the conservation laws and the integrating factors of the systems is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the systems are established, and an example is given to illustrate the application of the results.

A General Approach to the Construction of Conservation Laws for Birkhoffian Systems in Event Space

For a Birkhoffing system in the event space, a general approach to the construction of conservation laws is presented. The conservation laws are constructed by finding corresponding integrating factors for the parametric equations of the system. First, the parametric equations of the Birkhoffian system in the event space are established, and the definition of integrating factors for the system is given; second the necessary conditions for the existence of conservation laws are studied in detail, and the relation between the conservation laws and the integrating factors of the system is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the results.

Unification of Gravitational and Electromagnetic Fields in Curved Space-Time Using Gauge Symmetry of Bianchi Identities

This paper deals with the generalization of the linear theory of the unification of gravitational and electromagnetic fields using 4-dimensional gauge symmetry in order to solve the contradictions from the Kaluza-Klein theory’s unification of the gravitational and electromagnetic fields. The unification of gravitational and electromagnetic fields in curved space-time starts from the Bianchi identity, which is well known as a mathematical generalization of the gravitational equation, and by using the existing gauge symmetry condition, equations for the gravitational and electromagnetic fields can be obtained. In particular, the homogeneous Maxwell’s equation can be obtained from the first Bianchi identity, and the inhomogeneous Maxwell’s equation can be obtained from the second Bianchi identity by using Killing’s equation condition of the curved space-time. This paper demonstrates that gravitational and electromagnetic fields can be derived from one equation without contradiction even in curved space-time, thus proving that the 4-dimensional metric tensor using the gauge used for this unification is more complete. In addition, geodesic equations can also be derived in the form of coordinate transformation, showing that they are consistent with the existing equations, and as a result, they are consistent with the existing physical equations.