The method of undetermined tensor for constructing Birkhoffian functions

Jin-Chao, Cui; Song, Duan; Guo, Yong‐Xin

doi:10.7498/aps.61.244501

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…(iii) The Birkhoffian functions R µ are easy to obtain via the method in Ref. [8]. Now, we specifically discuss the necessary and sufficient condition for transforming an autonomous system into linear autonomous Birkhoff equation (11) or (12).…”

Section: Linear Autonomous Birkhoff Equationsmentioning

confidence: 99%

“…The authors have expatiated on the undetermined tensor method in Ref. [8], in which the problem for constructing the Birkhoffian representation of a nonautonomous system, i.e., ȧµ = Ξ µ (t, a) , (µ = 1, . .…”

Section: A Condition For Transforming An Autonomous System Into Linea...mentioning

confidence: 99%

“…[1,2] Through comparing linear autonomous Birkhoff equations with nonautonomous Birkhoff equations, we can find that the former have a simpler structure and are easier to solve, and so they are the basis for investigating Birkhoff's equations, such as the constructive methods of Birkhoffian dynamical functions, integral theory, and structure preserving algorithms. [3][4][5][6][7][8][9] The well-known Whittaker and Hojman-Urrutia equations [10] represent a kind of system with a simple first-order form, but without the traditional Hamiltonian representations. Thus they are of the linear autonomous non-Hamiltonian type.…”

Section: Introductionmentioning

confidence: 99%

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A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

Jin-Chao¹,

Liu²,

Song³

2013

Chinese Phys. B

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The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.

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Section: Linear Autonomous Birkhoff Equationsmentioning

confidence: 99%

Section: A Condition For Transforming An Autonomous System Into Linea...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

Jin-Chao¹,

Liu²,

Song³

2013

Chinese Phys. B

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Discrete optimal control for Birkhoffian systems and its application to rendezvous and docking of spacecrafts

Kong¹,

Wu²

2017

Acta Phys. Sin.

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In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

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On Santilli's methods in Birkhoffian inverse problem

Jin-Chao¹,

Chen²,

Cui-Cui³

2018

Acta Phys. Sin.

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In this paper, we mainly study the simplification and improvement of Santilli's methods in Birkhoffian system, which is a more general type of basic dynamic system. The theories and methods of Birkhoffian dynamics have been used in hadron physics, quantum physics, rotational relativity theory, and fractional dynamics. As is well known, Lagrangian inverse problem, Hamiltonian inverse problem, and Birkhoffian inverse problem are the main objects of the dynamic inverse problems. The results given by Douglas (Douglas J 1941 Trans. Amer. Math. Soc. 50 71) and Havas[Havas P 1957 Nuovo Cimento Suppl. Ser. X5 363] show that only the self-adjoint Newtonian systems can be represented by Lagrange's equations, so the Lagrangian inverse problem is not universal for a holonomic constrained mechanical system. Furthermore, from the equivalence between Lagrange's equation and Hamilton's equation, Hamiltonian inverse problem is not universal. A natural question is then raised:whether there exists a self-adjoint dynamical model whose inverse problem is universal for holonomic constrained mechanical systems, in the field of analytical mechanics.An in-depth study of this issue in the 1980s by R. M. Santilli shows that a universal self-adjoint model exists for a holonomic constrained mechanic system that satisfies the basic conditions of locality, analyticity, and formality. The Birkhoff's equation is a natural extension of the Hamilton's equation, which shows the geometric properties of a nonconservative system as a general symplectic structure. This more general symplectic structure provides the geometry for the study of the non-conservative system preserving structure algorithms. Therefore, it is particularly important to study the problem of the Birkhoffian representation for the holonomic constrained system.For the inverse problem of Birkhoff's dynamics, studied mainly are the condition under which the mechanical systems can be represented by Birkhoff's equations and the construction method of Birkhoff's functions. However, due to the extensiveness and complexity of the holonomic nonconservative system, Birkhoff's dynamical functions do not have so simple construction method as Lagrange function and Hamilton function. The research results of this issue are very few. The existing construction methods are mainly for three constructions proposed by Santilli[Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York:Springer-Verlag) pp25-28], and there are still many technical problems to be solved in the applications of these methods.In order to solve these problems, this article mainly focuses on the following content. First, according to the existence theorem of Cauchy-Kovalevskaya type equations, we prove that the autonomous system always has an autonomous Birkhoffian representation. Second, a more concise method is given to prove that Santilli's second method can be simplified. An equivalent relationship implied in Santilli's third method is found, an improved Santilli's third method is proposed, and the MATLAB programmatic calculation of the method is studied. Finally, the full text is summarized and the results are discussed.

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The method of undetermined tensor for constructing Birkhoffian functions

Cited by 3 publications

References 19 publications

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

Discrete optimal control for Birkhoffian systems and its application to rendezvous and docking of spacecrafts

On Santilli's methods in Birkhoffian inverse problem

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