Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented. The concepts of Lie symmetry and higher order adiabatic invariants of Poincaré equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.

Exact invariants and adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system

The exact invariants and the adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.

A new type of adiabatic invariants for nonconservative systems of generalized classical mechanics

The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invaxiant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invaxiant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invaxiant. An example is also given to illustrate the application of the results.

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.

PERTURBATION OF SYMMETRIES OF BIRKHOFF SYSTEM AND ADIABATIC INVARIANTS

The perturbation of symmetries of the free Birkhoff system under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of adiabatic invariants and the conditions for their existence are given. Then these results are generalized to the constrained Birkhoff system. One example is presented to illustrate these results.

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering d/dt in the structure equation and process of proof including infinitesimal parameter ε obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.

Perturbation to Noether Symmetries and Adiabatic Invariants for Generalized Birkhoff Systems Based on El-Nabulsi Dynamical Model

With the action of small perturbation on generalized El-Nabulsi-Birkhoff fractional equations,the perturbation to Noether symmetries and adiabatic invariants are studied under the framework of El-Nabulsi′s fractional model.Firstly,based on the invariance of El-Nabulsi-Pfaff action under the infinitesimal transformations of group,the exact invariants are given.Secondly,on the basis of the definition of higher order adiabatic invariants of a dynamical system,the adiabatic invariants of the Noether symmetric perturbation for disturbed generalized El-Nabulsi′s fractional Birkhoff system are presented under some conditions,and some special cases are discussed.Finally,an example known as Hojman-Urrutia problem is given to illustrate the application of the results.

Hojman Exact Invariants and Adiabatic Invariants of Hamilton System

The perturbation to Lie symmetry and adiabatic invariants are studied.Based on the concept of higher-order adiabatic invariants of mechanical systems with action of a small perturbation,the perturbation to Lie symmetryis studied,and Hojman adiabatic invariants of Hamilton system are obtained.An example is given to illustrate theapplication of the results.

Perturbation to Mei symmetry and adiabatic invariants for Hamilton systems

Based on the concept of adiabatic invariant, this paper studies the perturbation to Mei symmetry and adiabatic invariants for Hamilton systems. The exact invariants of Mei symmetry for the system without perturbation are given. The perturbation to Mei symmetry is discussed and the adiabatic invariants induced from the perturbation to Mei symmetry of the system are obtained.

Adiabatic invariants of generalized Lutzky type for disturbed holonomic nonconservative systems

Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariants, i.e. generalized Lutzky adiabatic invariants, of a disturbed holonomic nonconservative mechanical system are obtained by investigating the perturbation of Lie symmetries for a holonomic nonconservative mechanical system with the action of small disturbance. The adiabatic invariants and the exact invariants of the Lutzky type of some special cases, for example, the Lie point symmetrical transformations, the special Lie symmetrical transformations, and the Lagrange system, are given. And an example is given to illustrate the application of the method and results.

Perturbation to Noether Symmetry and Adiabatic Invariants for Dynamical Systems with Nonstandard Lagrangians

The perturbation to Noether symmetry and adiabatic invariants for dynamical systems with nonstandard Lagrangians are studied. Based on two kinds of nonstandard Lagrangians( i. e.exponential Lagrangians and power-law Lagrangians),the exact invariants of Noether type are given. Based on the definition of highorder adiabatic invariants,the relationship between the perturbation of Noether symmetry and the adiabatic invariants of the system under a small disturbance is studied,and then the corresponding theorems of adiabatic invariants are established. Finally, two examples are given to illustrate the methods and results appear in this paper.

Perturbation to symmetries and Hojman adiabatic invariant for nonholonomic controllable mechanical systems with non-Chetaev type constraints

This paper studies the perturbation to symmetries and adiabatic invariant for nonholonomic controllable mechanical systems with non-Chetaev type constraints. It gives the exact invariants introduced by the Lie symmetries of the nonholonomic controllable mechanical system with non-Chetaev type constraints without perturbation. Based on the definition of high-order adiabatic invarlants of mechanical system, the perturbation of Lie symmetries for nonholonomic controllable mechanical system with non-Chetaev type constraints with the action of small disturbances is investigated, and a new type of adiabatic invariant of system are obtained. In the end of this paper, an example is given to illustrate the application of the results.

Perturbation to Lie symmetry and another type of Hojman adiabatic invariants for Birkhoffian systems

The perturbation to Lie symmetry and another type of Hojman adiabatic invariants induced from the perturbation to Lie symmetry for Birkhoffian systems are studied. The exact invariants of Lie symmetry for the system without perturbation are given. Based on the concept of adiabatic invariant, the perturbation to Lie symmetry is discussed and another new type of Hojman adiabatic invariants that have the different form from that in [Acta Phys. Sin. 55 3833] for the perturbed system are obtained.

Perturbation to Lie Symmetry and Adiabatic Invariants for General Holonomic Mechanical Systems

Based on the concept of adiabatic invariant,the perturbation to the Lie symmetry and adiabatic invariantsfor general holonomic mechanical systems are studied.The exact invariants induced directly from the Lie symmetryof the system without perturbation are given.The perturbation to the Lie symmetry is discussed and the adiabaticinvariants that have the different form from that in[Act.Phys.Sin.55(2006)3236(in Chinese)]of the perturbedsystem,are obtained.

Area and entropy spectra of black holes via an adiabatic invariant

By considering and using an adiabatic invariant for black holes, the area and entropy spectra of static spherically- symmetric black holes are investigated. Without using quasi-normal modes of black holes, equally-spaced area and entropy spectra are derived by only utilizing the adiabatic invariant. The spectra for non-charged and charged black holes are calculated, respectively. All these results are consistent with the original Bekenstein spectra.

Adiabatic Invariant for Dynamic Systems on Time Scale

Perturbation to Noether symmetry and adiabatic invariants for Birkhoffian system,Hamiltonian system and Lagrangian system with delta derivative are investigated,respectively.Firstly,the definition and some related properties of time scale calculus are listed simply as preliminaries.Secondly,the Birkhoffian system with delta derivative is studied.Based on the differential equation of motion as well as Noether symmetry and conserved quantity,perturbation to Noether symmetry and adiabatic invariant are investigated.Thirdly,adiabatic invariants for the Hamiltonian system and the Lagrangian system are presented through some transformations.And finally,an example is given to illustrate the methods and results.

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system

According to the Herglotz variational principle and differential variational principle of Herglotz type, we study the adiabatic invariants for a non-conservative nonholonomic system. Firstly, the differential equations of motion of the non-conservative nonholonomic system based upon the generalized variational principle of Herglotz type are given, and the exact invariant for the non-conservative nonholonomic system is introduced. Secondly, a new type of adiabatic invariant for the system under the action of a small perturbation is obtained. Thirdly, the inverse theorem of the adiabatic invariant is given. Finally, an example is given.

Nonshifted Adiabatic Invariant on Time Scale

Three aspects which include differential equation of motion, Noether symmetry and conserved quantity, perturbation to Noether symmetry and adiabatic invariant are investigated for the nonshifted Birkhoffian system, nonshifted Hamiltonian system and nonshifted Lagrangian system, respectively. Then, some relationships among the three mechanic systems and some special cases of the main results are presented. Finally, an example is given to illustrate the application of the methods and results.

Area spectrum of the three-dimensional Godel black hole

The aim of this paper is to investigate the area spectrum of the three-dimensional Godel black hole by using two different methods. The result shows that the area spectrum of the black hole is △A = 8πl2p, which confirms the initial proposal of Bekenstein that the area spectrum is independent of black hole parameters and the spacing is 8πl2p.

A Study of Three Galaxy Types, Galaxy Formation, and Warm Dark Matter

We try to bridge the gap between the theory of linear density-velocity-gravitational perturbations in the early universe, and the relaxed galaxies we observe today. We succeed quantitatively for dark matter if dark matter is warm. The density runs of baryons and of dark matter of relaxed galaxies are well described by hydro-static equations. The evolution from initial linear perturbations to final relaxed galaxies is well described by hydro-dynamical equations. These equations necessarily include dark matter velocity dispersion. If the initial perturbation is large enough, the halo becomes self-gravitating. The adiabatic compression of the dark matter core determines the final core density, and provides a negative stabilizing feedback. The relaxed galaxy halo may form adiabatically if dark matter is warm. The galaxy halo radius continues to increase indefinitely, so has an ill-defined mass.