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Hamiltonian structures of dynamics of a gyrostat in a gravitational field

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Abstract

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

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Authors and Affiliations

  1. Department of Guidance, Navigation and Control, School of Astronautics, Beijing University of Aeronautics and Astronautics, Room B1024, New Main Building, 100191, Beijing, China

    Yue Wang & Shijie Xu

Authors
  1. Yue Wang
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Correspondence to Yue Wang.

Appendices

Appendix A: Calculation of Poisson tensor B 1(z 1)

The Poisson tensor B 1(z 1) can be obtained in a differential geometric method. The natural canonical bracket is given by Eq. (19) on the manifold T ∗ Q(Ξ). Considering functions f,g∈C ∞(ℝ18(z 1)), we define as follows:

(95)

According to Eqs. (19) and (24), we have

(96)

In order to calculate the differential of an arbitrary smooth function \(\tilde{f}\), we consider W∈T Ξ (T ∗ Q) defined as follows:

(97)

where variables in Ξ are rearranged as for convenience. W generates the curve in T ∗ Q

(98)

Thus, the differential is given by

(99)

where variables in z 1 are rearranged as (r,p,α 1,β 1,γ 1,Γ). Equation (99) can be written further as follows:

(100)

where is the jth column of the matrix . Rearranging Eq. (100) yields

(101)

where

(102)

The element can be denoted by

(103)

And then we get

(104)

According to Eqs. (101) and (104), we have

(105)

Noticing Eq. (103), we can obtain the derivatives of \(\tilde{f}\) as follows:

(106)

where (⋅)∧ is the isomorphism defined by Eq. (5). Then, according to Eq. (96), the Poisson bracket can be derived as

(107)

The term AM g in Eq. (107) can be written as

(108)

And then, we rewrite Eq. (107) in the following form:

(109)

Now the Poisson tensor B 1(z 1) has been derived.

Appendix B: Calculation of Poisson tensor B 2(z 2)

To get the Poisson tensor B 2(z 2), we consider f,g∈C ∞(ℝ18(z 2)) and define as follows:

(110)

According to Eqs. (19) and (38), we have

(111)

Then we consider W∈T Ξ (T ∗ Q) defined in Eq. (97) and the curve in T ∗ Q generated by W given by Eq. (98). Thus, the differential is given by

(112)

Equation (112) can be written as follows:

(113)

where is the jth row of the matrix . Rearranging Eq. (113) yields

(114)

where

(115)

We define by Eq. (103) and have the differential in Eq. (104). According to Eqs. (104) and (114), we have vectors a,b,c and d in Eq. (104) as follows:

(116)

According to Eq. (103), the derivatives of \(\tilde{f}\) can be obtained as follows:

(117)

Then Eq. (111) can be written as follows:

(118)

We rewrite Eq. (118) in the following form:

(119)

Now we have obtained the Poisson tensor B 2(z 2).

Appendix C: Calculation of Poisson tensor B 3(z 3)

The Poisson tensor B 3(z 3) can be derived in a similar method to Appendixes A and B. Based on f,g∈C ∞(ℝ18(z 3)), we define as

(120)

Using Eqs. (19) and (51), we have

(121)

Then we define W∈T Ξ (T ∗ Q) by Eq. (97) and the curve in T ∗ Q by Eq. (98). Thus, the differential can be written as

(122)

Equation (122) can be written as follows:

(123)

Equation (123) can be further rearranged as follows:

(124)

where

(125)

We have in Eq. (103) and the differential in Eq. (104). Using Eq. (124), we can obtain the vectors in Eq. (104) as follows:

(126)

Using Eq. (103), we can obtain the derivatives of \(\tilde{f}\) as follows:

(127)

Then according to Eq. (121), Poisson bracket can be derived as

(128)

We rewrite Eq. (128) in the following form:

(129)

Now the Poisson tensor B 3(z 3) has been obtained.

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Wang, Y., Xu, S. Hamiltonian structures of dynamics of a gyrostat in a gravitational field. Nonlinear Dyn 70, 231–247 (2012). https://doi.org/10.1007/s11071-012-0447-6

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