Hojman Conservation Research Articles

On the Hojman conservation quantities in Cosmology

We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.

Hojman symmetry in f ( T ) $f(T)$ theory

Today, $f(T)$ theory has been one of the popular modified gravity theories to explain the accelerated expansion of the universe without invoking dark energy. In this work, we consider the so-called Hojman symmetry in $f(T)$ theory. Unlike Noether conservation theorem, the symmetry vectors and the corresponding conserved quantities in Hojman conservation theorem can be obtained by using directly the equations of motion, rather than Lagrangian or Hamiltonian. We find that Hojman symmetry can exist in $f(T)$ theory, and the corresponding exact cosmological solutions are obtained. We find that the functional form of $f(T)$ is restricted to be the power-law or hypergeometric type, while the universe experiences a power-law or hyperbolic expansion. These results are different from the ones obtained by using Noether symmetry in $f(T)$ theory. Therefore, it is reasonable to find exact cosmological solutions via Hojman symmetry.

Hojman symmetry approach for scalar–tensor cosmology

Scalar–tensor cosmologies can be dealt under the standard of the Hojman conservation theorem that allows to fix the form of the coupling F(ϕ), of the potential V(ϕ) and to find out exact solutions for related cosmological models. Specifically, the existence of a symmetry transformation vector for the equations of motion gives rise to a Hojman conserved quantity on the corresponding minisuperspace and exact solutions for the cosmic scale factor a and the scalar field ϕ can be achieved. In particular, we take advantage of the fact that minimally coupled solutions, previously obtained in the Einstein frame, can be conformally transformed in non-minimally coupled solutions in the Jordan frame. Some physically relevant examples are worked out.

Exact cosmological solutions from Hojman conservation quantities

We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmological model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar–tensor models, we show that the Hojman conserved quantities exist for a wide range of V(ϕ)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.

Symmetries and Two Types of Non-Noether Conservation Lawsof Birkhoffian System with UnilateralConstraints

The symmetries and non-Noether conservation laws ofBirkhoffian system with unilateral constraints are studied. Thedifferential equations of motion of the system are established,and the criterions of Noether symmetry, Lie symmetry and Meisymmetry of the system are given. Two types of new conservationlaws, called the Hojman conservation law and the Mei conservationlaw respectively, are obtained, and the intrinsic relations amongthe symmetries and the new conservation laws are researched.At the end of the paper, an example is given to illustrate theapplication of the results.

Hojman Conservation Laws for Relativistic Birkhoffian System**The project supported by National Natural Science Foundation of Chinaunder Grant No. 10272021

For a relativistic Birkhoffian system, the Liesymmetry and Hojman conserved quantity are given under infinitesimaltransformations. On the basis of the invariance of relativistic Birkhoffianequations under infinitesimal transformations, Lie symmetricaltransformations of the system are constructed, which only depend on theBirkhoffian variables. The determining equations of Lie symmetry are given,and a Hojman conserved quantity is directly obtained from Lie symmetry ofthe system. An example is given to illustrate the application of the results.

Hojman's conservation theorems for generalized Raitzin canonical equations of motion

Using the Lie symmetry under infinitesimal transformations in which the time is not variable, Hojman's conservation theorems for Raitzin's canonical equations of motion in generalized classical mechanics are studied. The generalized Raitzin's canonical equations of motion are established. The determining equations of Lie symmetry under infinitesimal transformations are given. The Hojman conservation theorems of the system are established. Finally, an example is also presented to illustrate the application of the result.

The Unified Form of Hojman's Conservation Law and Lutzky's Conservation Law

In this paper, a unified form of Hojman's conservation law and Lutzky's one is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equation of motion ...

Hojman’s conservation theorems for Raitzin’s canonical equations of motion of nonlinear nonholonomic systems

Using the Lie symmetry under infinitesimal transformations in which the time is not variable, the Hojman's conservation theorems for Raitzin's canonical equations of motion of nonlinear nonholonomic systems are studied. The differential equations of the systems and the determining equations of Lie symmetry under infinitesimal transformations are given. The Hojman's conservation theorems of the systems are established. Finally, we give an example to illustrate the application of the result.

Non-Noether Conserved Quantity of Nonholonomic System Having Variable Mass and Unilateral Constraints**The project supported by Natural Science Foundation of Heilongjiang Province of China under Grant No. 9507

Using the Lie Symmetry under infinitesimal transformations in which the time is not variable, the non-Noether conserved quantity of nonholonomic system having variable mass and unilateral constraints is studied. The differential equations of motion of the system are given. The determining equations of Lie symmetrical transformations of the system under infinitesimal transformations are constructed. The Hojman's conservation theorem of the system is established. Finally, we give an example to illustrate the application of the result.

The conserved quantity of Hojman for mechanicalsystems with variable mass in phase space

In this paper, we study the Hojman conserved quantity of Lie symmetry for mechan ical systems with variable mass in phase space under a general infinitesimal tra ns formation. The determining equations and Hojman conservation theorem of Lie symm etry of the system in phase space are obtained. An example is given to illustrat e the application of the result.

Geometric foundations of Hojman theorem\=for Birkhoffian systems

This paper studies the geometric foundations of Hojman theorem for Brikhoffian s y stems. The differential equations of motion and the Hojman conservation law of t he systems are established. A proof of Hojman theorem of Birkhoffian systems is given by use of the modern differential geometry.

Comment on a theorem of Hojman and its generalizations

An invariant derived using Hojman's conservation law is shown to be trivial for all Noether symmetries.