DYNAMICS OF A RIGID BODY IN A CENTRAL GRAVITATIONAL FIELD

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1 ROMAI J., 3, (007), DYAMICS OF A RIGID BODY I A CETRAL GRAVITATIOAL FIELD Ioan Ioja, Petronela Angela Ioja West University Vasile Goldiş, Arad; Theoretical Highschool Vasile Goldiş, Arad iojaioanaurel@yahoo.com Abstract The first section builds upon the fact that a fixed body C of mass M, with spherical symmetry influences the motion of a B rigid body of mass m. Using the expression for kinetic energy of a rigid body T and the angular speed Ω, we presume the expression of the gravitational energy of body of mass m. Then, starting with the Lie group SO(3), which canonically acts upon SE(3, R), it follows the identification of the Hamiltonian H depending on the angular moment of the rigid body B and the spatial linear moment of the same body. The second section presents practical applications in modeling planets and artificial satellites of the Earth. For this, we consider different orders of approximations of the gravitational potential, of the Taylor series expansions and truncate these expansions at a finite number of terms. In the final section, starting with the Hamiltonian field X H, and corresponding flows/fluxes, the Lie-Trotter integrator is considered, as a first order integrator.. GEOMETRICAL ASPECTS I THE DYAMICS OF A RIGID BODY SITUATED I A CETRAL GRAVITATIOAL FIELD Let C be a fixed rigid body of mass M, with spherical symmetry, influencing the motion of a B rigid body of mass m. The inertial reference of the observer is attached to C and the reference of the rigid body is fixed to B at its mass center. A material particle Q in the rigid is represented by the vector q = BQ + r, where B is an element of SO(3) (independently of the particle) and r is the vector of C at the mass center of body B (fig.). 75

2 76 Ioan Ioja, Petronela Angela Ioja Fig.. The rigid body in a central gravitational field At any moment the configuration of the rigid B is univocally determined by the pair (B, r) SE(3, R), where SE(3, R) is the special Euclidian group. The kinetic energy of the rigid body relative to an observer situated in C is given by T = q dm(q), (.) B where dm( ) represents a measure of body mass and represents the Euclidian norm in R 3. The above formula may be represented in the equivalent form T = λω, IΩ + m ṙ, (.) where Ω is the angular speed of the rigid body, m is its total mass and I is the inertial tensor of B in the reference of the body. ote that the angular speed Ω of a body is defined by Ḃ = B Ω, where 0 Ω 3 Ω Ω = Ω 3 0 Ω Ω Ω 0 is the antisymmetrical matrix associated with Ω. The spatial angular speed ω is defined by Ḃ = ωb. Thus, the relationship ω = BΩ follows. Moreover, the relationship RM = T defines a Riemannian metrics in/on the space of the configurations SE(3, R).

3 Dynamics of a rigid body in a central gravitational field 77 The gravitational energy of body B is given by V = q dm(q) = dm(q), (.) r + BQ B where G is the universal gravitational constant. SO(3) Lie Group acts in a canonical way on SE(3, R), namely Φ : SO(3) SE(3, R) SE(3, R), B Φ(P, (B, r)) def = (PB, Pr) and let Φ T be the lift of this action to T SE(3, R), meaning Φ T : SO(3) T SE(3, R) T SE(3, R), Φ T (R, (π, B, r, p)) def = (π, RB, Rr, Rp). The Hamiltonian H or the energy of the considered system is given by H = λπ, I π + p m dm(q), (.3) r + BQ where π = IΩ is the angular momentum of the rigid body B and p = mṙ is the spatial linear momentum of the body. Thus, it may be concluded Proposition. [5]. The rigid body in a central gravitational field is a Hamiltonian mechanical system with the phase space (T SE(3, R), ω = dθ), where ω is the canonical simplectical form on T SE(3, R) and the Hamiltonian H of the given system is represented by (.3). It may be easily verified that the action Φ T, previously defined, has the momentum application J given by B J : T SE(3, R) (so(3)) R 3 (.4), J(π, B, r, p) = Bπ + r p. Moreover, the Hamiltonian H is SO(3)-invariant, so it induces a Hamiltonian H on the quotient T SE(3, R)/SO(3) SE(3, R) (SE(3, R)) /SO(3) SO(3) R 3 R 3 R 3 /SO(3) R 3 R 3 R 3

4 78 Ioan Ioja, Petronela Angela Ioja and thus the dynamics of X H may be expressed in terms of the reduced variables {π, λ, µ}, where π is the angular momentum of the body, λ = B t r and µ = B t p. In these new variables, the Hamiltonian H is expressed by H(π, λ, µ) = λπ, I π + µ m dm(q) (.5) B λ + Q and the reduced dynamics reads π = π I FM(λ Q) π + B λ + Q 3 dm(q), λ = λ I π + µ m, µ = µ I (λ + Q) π B λ + Q 3 dm(q). (.6) A simple computation leads to the following result Proposition.. [5]. The dynamics (.6) has the following Hamilton- Poisson output (R 9, {, }, H), where {, } is the Poisson structure generated by the matrix Π = π λ µ λ O 3 µ O 3 (.7) and H is given by (.5).

5 Proof. Truly, we have Dynamics of a rigid body in a central gravitational field 79 = H π π λ µ Π H = λ O 3 H π = µ O λ λ 3 µ H µ π I (λ Q) π + B λ + Q 3 dm(q) λ I π + µ m µ I (λ + Q) π B λ + Q 3 dm(q) = π λ µ meaning the aimed relationship. Following the technique of [] it may be demonstrated Proposition.3 [4] Every Casimir function of the configuration (R 9, {, }) has the form C ϕ = ϕ( π + λ µ ), (.8) where ϕ C (R, R). Proof. Define D = {(π, λ, µ) R 9 π + λ µ 3 λ 3 µ 0}, where Thus, we have: (i) rang(π) = 8 π = π π π 3 ; λ = λ λ λ 3 ; µ = and, consequently, there is a single Casimir of our configuration (R 9, Π); µ µ µ 3.

6 80 Ioan Ioja, Petronela Angela Ioja (ii) det(π 8 ) = π + λ µ 3 λ 3 µ 0 on D, where matrix Π 8 has the form 0 π λ 3 0 λ µ 3 0 µ π 0 λ λ 0 µ µ 0 λ 3 λ Π 8 = 0 λ ; λ µ 3 µ µ µ [ (iii) Λ = (Π Π 8 )t = a 3 = a 5 = a 4 = a 6 = a 7 = a 8 = a = a = a a a 3 a 4 a 5 a 6 a 7 a 8 ] t, where π + λ µ 3 µ 3 µ ( π µ λ 3 + λ µ 3 ); π + λ µ 3 λ 3 µ ( π 3 + λ µ λ µ ); π + λ µ 3 λ 3 µ ( π 3 µ + π µ 3 + µ 3 λ 3 µ + µ µ λ λ µ 3 λ µ ); π + λ µ 3 λ 3 µ [π 3 µ λ µ µ 3 (π µ λ 3 + λ µ 3 ) + λ µ µ ]; π + λ µ 3 λ 3 µ [ π µ λ 3 µ + µ µ 3λ + µ (π µ λ 3 + λ µ 3 )]; π + λ µ 3 λ 3 µ (λ µ 3 λ 3 π µ λ 3 µ λ + λ λ 3 µ 3 + λ λ µ ); π + λ µ 3 λ 3 µ [ π 3 λ + λ 3 (π µ λ 3 + λ µ 3 ) + λ λ µ µ λ ]; π + λ µ 3 λ 3 µ [π λ + λ λ 3 µ λ (π µ λ 3 + λ µ 3 ) µ 3 λ ]; (iv) thus, the associated differential equation is (π + λ µ 3 λ 3 µ )dπ = ( π µ λ 3 + λ µ 3 )dπ +( π 3 + λ µ λ µ )dπ 3 +( π 3 µ + µ π 3 + µ 3 λ 3 µ + µ µ λ λ µ 3 λ µ )dλ

7 Dynamics of a rigid body in a central gravitational field 8 +(π 3 µ λ µ µ 3 π + µ µ 3 λ 3 µ 3λ + λ µ µ )dλ +( π µ λ 3 µ + µ µ 3 λ + µ π µ λ 3 + µ µ 3 λ )dλ 3 +(λ π 3 π λ 3 µ λ 3 µ λ + λ λ 3 µ 3 + λ λ µ )dµ +( π 3 λ + λ 3 π λ 3µ + λ λ 3 µ 3 + λ λ µ µ λ )dµ +(π λ + λ λ 3 µ λ π + λ λ 3 µ λ µ 3 µ 3 λ )dµ 3 which integrated leads to the Casimir C(π, µ, λ) = π + λxµ, and therefore any Casimir of the configuration has the form ϕ( π + λxµ ), where ϕ C (R, R).. ZERO ORDER APPROXIMATIO For practical applications in modeling planets and artificial satellites of the Earth it is useful to consider different order estimations of the gravitational potential. This may be accomplished through Taylor series expansions in the neighborhood of λ = or, equivalent, r = and by truncating these expansions at a finite number of terms. The only unsolved and still debated issue is that, as a consequence of the above described approximation process, symmetries as well as existing conservation laws are valid. Let us consider the reduced Poisson model as a starting point for further approximations (.6). Expanding H in Taylor series, we have Ṽ (λ) = B λ + Q dm(q) = { λq, λ dm(q) λ B λ Q λ + 3 } λq, λ λ 4 + O( λ 4 ) [ = ] [ λ λ 3Trace(I) + 3 ] λ 5λλ, I λ + O( λ 5 ). The subsequent zero order approximation of H will be considered in the following (.5) H 0 (π, λ, µ) = λπ, I π + µ m m λ.

8 8 Ioan Ioja, Petronela Angela Ioja Thus, the corresponding zero order dynamics has the following form: π = π I π, λ = λ I π + µ m, µ = µ I π m λ 3 λ, or equivalently, π = π π 3 π π 3, π = π π 3 I π π 3, π 3 = π π π π I, λ = λ π 3 λ 3π + µ m, λ = λ 3π I λ π 3 + µ m, λ 3 = λ π λ π + µ 3 I m, µ = µ π 3 µ 3π m λ 3 λ, (.) µ = µ 3π µ π 3 m I λ 3 λ, µ 3 = µ π µ π m I λ 3 λ 3. Proposition.. Dynamics (.) has the Hamilton Poisson output/realization (R 9, Π, H 0 ), where Π = π λ µ λ O 3 µ O 3, H0 (π, λ, µ) = λπ, I π + µ m m λ. Proof. Indeed, a simple computation shows H 0 π π I π Π H 0 = Π H 0 = λ I π + µ π λ m = λ, H 0 µ I π m µ λ 3 λ µ meaning the aimed relationship. A comprehensive and direct computation leads to Proposition. [5]. The following quantities of the dynamics (..) are constant:

9 Dynamics of a rigid body in a central gravitational field 83 (i) C ϕ (π, λ, µ) = ϕ( π + λ µ ), (.) where ϕ C (R, R); (ii) C ψ (π, λ, µ) = ψ( π ), (.3) where ψ C (R, R); (iii) K(π, λ, µ) = λπ, I π (.4) A long and direct computation, or using Maple V, shows that the equilibrium( points of the dynamics (.) are: ) I (i) 0, 0,, 0, ± 3 3 I, 0, ± 3 3 m, 0, 0, R ( ) I (ii) 0,, 0, 0, 0, ± 3 I, ± 3, 0, 0, R ; ( ) I (iii), 0, 0, 0, 0, ± 3 3 I, 0, ± 3 m, 0, R ; I (iv) 0, 0, ± +, R ; (v) 0, 0, ± +, R ; (vi) 0, ± +, R ; (vii) 0, ± +, R ; ,,, 0, ± m 8 +,,, 0, ± m 8 +, 0,, 0,, ± m 8 +, 0,, 0,, ± m 8 +, ± m, 0, 8 +, ± m, 8 +, 0, ± m, 8 +, 0, ± m, 8 +

10 84 Ioan Ioja, Petronela Angela Ioja ( (viii) ± I +, ( R ; (ix) ± I +, R., 0, 0, 0,,, 0, ± m 8 +, 0, 0, 0,,, 0, ± m 8 +, ± m ), 8 +, ± m ), 8 + Proposition.3. [4]. The equilibrium points (i)-(ix) of the dynamics (.) are unstable. Proof. Using Maple V it may be proved that zero is not a simple root of the minimal polynomial associated with the matrix of the system (.) linearization at any of the (i)-(ix) equilibrium points. Thus, according to Lyapunov proposition equilibrium points (i)-(ix) are unstable. 3. UMERICAL ITEGRATIO VIA LIE-TROTTER ALGORITHM Let us discuss the numerical integration of dynamics (.) via Lie-Trotter integrator. Firstly, it remark that X H Hamiltonian field splits as follows X H = X H + X H + X H3 + X H4 + X H5, where H (π, λ, µ) = π ; H (π, λ, µ) = π ; H 3 (π, λ, µ) = π 3 ; H 4 (π, λ, µ) = I m (µ + µ + µ 3); H 5 (π, λ, µ) = λ. The corresponding flows/fluxes are given by Φ t, = exp(tx H ) = A,

11 Dynamics of a rigid body in a central gravitational field 85 where cos π (0)t I 0 sin π (0)t I sin π (0)t I cos π (0)t I A = cos π (0)t I sin π (0)t I sin π (0)t I cos π (0)t I cos π (0)t I sin π (0)t I sin π (0)t I cos π (0)t I Φ t, = exp(tx H ) = A, where cos π (0)t 0 sin π (0)t sin π (0)t 0 cos π (0)t cos π (0)t 0 sin π (0)t A = sin π (0)t 0 cos π (0)t cos π (0)t 0 sin π (0)t sin π (0)t 0 cos π (0)t Φ 3 t, = exp(tx H 3 ) = A 3,

12 86 Ioan Ioja, Petronela Angela Ioja where cos π 3 (0)t sin π 3 (0)t sin π 3 (0)t cos π 3 (0)t A 3 = cos π 3 (0)t sin π 3 (0)t sin π 3 (0)t cos π 3 (0)t cos π 3 (0)t sin π 3 (0)t sin π 3 (0)t 0 cos π 3 (0)t Φ 4 t, = exp(tx H 4 ) = A 4, where t m 0 0 A 4 = t m t m

13 where Φ 5 t, Dynamics of a rigid body in a central gravitational field 87 = exp(tx H 5 ) = A , A 5 = where a a a 0 0 mt a = [(λ (0)) + (λ (0)) + (λ 3 (0)) ] 3/. Therefore the Lie-Trotter integrator (see [], [3]) is given by π n+ λ n+ µ n+ = Φ t, Φ t, Φ 3 t, Φ 4 t, Φ 5 t, π n λ n µ n or, equivalently, by π n+ λ n+ µ n+ = A A A 3 A 4 A 5 π n λ n µ n (3.) Proposition 3.. [4]. The first order (3.) integrator has the following properties:

14 88 Ioan Ioja, Petronela Angela Ioja (i) it keeps Casimir (.8); (ii) it maintains the motion constant (.3); (iii) it does not maintain Hamiltonian H 0 ; (iv) it does not maintain motion constant (.4); (v) it does not maintain the Poisson structure generated by the matrix: π λ µ Π = λ O 3. µ O 3 Proof. The proof is obtained via Maple V. Remark 3.. The proof of the previous result may be eventually obtained through a direct computation. Considering its complexity, the research team could not accomplish it without a computer. Remark 3.. Assertion (ii) is a consequence of the fact that H, H, H 3, H 4, H 5 are not in involution related to the Poisson parenthesis generated by matrix Π. References [] B. H. Bermejo, V. Fairen, Simple evaluation of Casimir invariants in finite dimensional Poisson systems, Phys. Lett. A, 4(998), [] M. Puta, An overview of some Poisson integrators, in EUMATH 97, nd European Conference on umerical Mathematics and Advanced Applications, Heidelberg, eds. H. Bock, G. Kanschat, R. Rannacher, F. Brezzi, R. Glowinski, Y. Kuznetsov and J. Periaux, World Scientific, 998, [3] M. Puta, Lie-Trotter formula and Poisson dynamics, Int. Journ. of Bifurcation and Chaos, 9, 3(999), [4] M. Puta, I. Casu, R. Tudoran, I. Ioja [00]; Some remarks on the rigid body dynamics in a central gravitational field, An. Univ. Timisoara, S. Math-Comp. Sci., XXXIX(00). [5] L. S. Wang, P.S. Krishnaprasad, J. H. Maddocks, Hamiltonian dynamics of a rigid body in a central gravitational field, Celest. Mech. and Dynamical Astronomy, 50(99),