Rotational Relativistic Birkhoffian Research Articles

A Geometrical Approach to Hojman Theorem of a Rotational Relativistic Birkhoffian System**The project supported by National Natural Science Foundation of China under Grant No. 10272021 and the ‘Qing Lan’ Project Foundation of Jiangsu Province of China

A geometrical approach to the Hojman theorem of a rotational relativistic Birkhoffian system is presented. The differential equations of motion of the system are established. According to the invariance of differential equations under infinitesimal transformation, the determining equations of Lie symmetry are constructed. A new conservation law of the system, called Hojman theorem, is obtained, which is the generalization of previous results given sequentially by Hojman, Zhang, and Luo et al. In terms of the theory of modern differential geometry a proof of the theorem is given.

Perturbation of symmetries of rotational relativistic Birkhoffian systems and its inverse problem

This Letter investigates the perturbation of symmetries and adiabatic invariants for the rotational relativistic Birkhoffian systems. The Pfaff–Birkhoff principle, equation of motion and equation of small disturbance are established. Lie symmetries, laws of conservations, symmetrical perturbations and adiabatic invariants are presented, based on the invariance of differential equations under infinitesimal transformations. The inverse problems are also discussed for the perturbation of the symmetries of these systems. The connection is demonstrated between the symmetrical perturbations of rotational relativistic Birkhoffian systems and those of classical rotational Birkhoffian systems. Finally, an example is discussed to further illustrate the applications.

A set of the Lie symmetrical conservation laws for the rotational relativistic Birkhoffian system

For a rotational relativistic Birkhoffian system, a set of the Lie symmetries and conservation laws is given under infinitesimal transformations. On the basis of the invariance of rotational relativistic Birkhoffian equations under infinitesimal transformations, Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The determining equations of Lie symmetries are given and a new type of non-noether conserved quantities are directly obtained from Lie symmetries of the system. An example is given to illustrate the application of the results.

Cyclic integrals and reduction of rotational relativistic Birkhoffian system

The order reduction method of the rotational relativistic Birkhoffian equations is studied. For a rotational relativistic Birkhoffian system, the cyclic integrals can be found by using the perfect differential method. Through these cyclic integrals, the order of the system can be reduced. If the rotational relativistic Birkhoffian system has a cyclic integral, then the Birkhoffian equations can be reduced at least two degrees and the Birkhoffian form can be kept. An example is given to illustrate the application of the results.

Stability for the equilibrium state of rotational relativistic Birkhoffian systems

The stability for equilibrium state of rotational relativistic Birkhoffian systems is studied.The equilibrium state equations, the disturbance equation and the first approximation equation are given.The stability criteria for equilibrium state,and the stability criteria of the direct method for the rotational relativistic Birkhoffian autonomous systems are obtained.The relationship of the stability of equilibrium state between rotational relativistic Birkhoffian systems and rotational classical Birkhoffian systems are discussed. Two examples are presented to illustrate the results.

Generalized energy integral and reduction of rotational relativistic Birkhoffian system

We study the order reduction method of the rotational relativistic Birkhoffian equations. For a rotational relativistic autonomous Birkhoffian system, if the conservative law of the Birkhoffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the rotational relativistic Birkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least two degrees and the Birkhoffian form can be kept. An example is given to illustrate the application of the result.

Form Invariance and Noether Symmetries of Rotational Relativistic Birkhoff Systems**The project supported by National Natural Science Foundation of China under Grant No. 19972010, and Natural Science Foundation of Henan Province under Grant Nos. 984053100 and 998040080

Under the infinitesimal transformations of groups, a form invariance of rotational relativistic Birkhoff systems is studied and the definition and criteria are given. In view of the invariance of rotational relativistic Pfaff-Birkhoff-D'Alembert principle under the infinitesimal transformations of groups, the theory of Noether symmetries of rotational relativistic Birkhoff systems are constructed. The relation between the form invariance and the Noether symmetries is studied, and the conserved quantities of rotational relativistic Birkhoff systems are obtained.

Algebraic structure and Poisson integrals of a rotational relativistic Birkhoff system

We have studied the algebraic structure of the dynamical equations of a rotational relativistic Birkhoff system. It is proven that autonomous and semi-autonomous rotational relativistic Birkhoff equations possess consistent algebraic structure and Lie algebraic structure. In general, non-autonomous rotational relativistic Birkhoff equations possess no algebraic structure, but a type of special non-autonomous rotational relativistic Birkhoff equation possesses consistent algebraic structure and consistent Lie algebraic structure. Then, we obtain the Poisson integrals of the dynamical equations of the rotational relativistic Birkhoff system. Finally, we give an example to illustrate the application of the results.

Theory of symmetry for a rotational relativistic Birkhoff system

The theory of symmetry for a rotational relativistic Birkhoff system is studied. In terms of the invariance of the rotational relativistic Pfaff-Birkhoff-D'Alembert principle under infinitesimal transformations, the Noether symmetries and conserved quantities of a rotational relativistic Birkhoff system are given. In terms of the invariance of rotational relativistic Birkhoff equations under infinitesimal transformations, the Lie symmetries and conserved quantities of the rotational relativistic Birkhoff system are given.

Form Invariance and Lie Symmetries of the Rotational RelativisticBirkhoff System

For a rotational relativistic Birkhoff system, the relation between theform invariance and the Lie symmetries are given under infinitesimaltransformations of groups. If the infinitesimal transformationgenerators ξ0 and ξµ satisfy the conditions of the forminvariance, and the determining equation of Lie symmetries holds, theform invariance leads to a Lie symmetry of the system. Furthermore, ifthe infinitesimal transformations generators ξ0 and ξµsatisfy the conditions of the form invariance and the determiningequation of Lie symmetry holds, and if there is a gauge function Gsatisfying the structure equation of Lie symmetry, then the forminvariance will lead to the Lie symmetrical conserved quantity of thesystem. An example is given to illustrate the application of the results.

Birkhoff's equations and geometrical theory of rotational relativistic system

The Birkhoffian and Birkhoff's functions of a rotational relativistic system are constructed, the Pfaff action of rotational relativistic system is defined, the Pfaff-Birkhoff principle of a rotational relativistic system is given and the Pfaff-Birkhoff-D'Alembert principles and Birkhoff's equations of rotational relativistic system are constructed. The geometrical description of a rotational relativistic system is studied and the exact properties of Birkhoff's equations and their forms on ×T*M for a rotational relativistic system are obtained. The global analysis of Birkhoff's equations for a rotational relativistic system is studied, the global properties of autonomous, semi-autonomous and non-autonomous rotational relativistic Birkhoff's equations and the geometrical properties of energy change for rotational relativistic Birkhoff's equations are given.

A FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF A ROTATIONAL REALTIVISTIC BIRKHOFFIAN SYSTEM

A field method for solving the equations of motion of a rotational relativistic Birkhoffian system is obtained.An example to illustrate the application of the method is given.

BASIC THEORY OF RELATIVISTIC BIRKHOFFIAN DYNAMICS OF ROTATIONAL SYSTEM

The basic theory of relativistic Birkhoffian dynamics of rotational system is constructed,and the Birkhoffian,Birkhoff's functions,Pfaff action,Pfaff-Birkhoff principle,Pfaff-Birkhoff-D'Alembert principle and Birkhoffian equations are given.The relations among relativistic Lagrangian mechanics,Hamiltonian mechanics and relativistic Birkhoffian dynamics of rotational system are studied.It is proved that the holonomic conserved and holonomic non-conserved rotational relativistic systems can all belong to the rotational relativistic Birkhoffian system.