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The gyrostat with a fixed point in a Newtonian force field: Relative equilibria and stability
J. Math. Anal. Appl. 401 (2013) 836–849
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The gyrostat with a fixed point in a Newtonian force field: Relative equilibria and stability J.A. Vera Centro Universitario de la Defensa (MDE-UPCT), Base Aérea de San Javier, C/General López Peña s/n, 30720 Santiago de la Ribera (Murcia), Spain
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Article history: Received 24 January 2012 Available online 19 December 2012 Submitted by Roman O. Popovych Keywords: Gyrostat Relative equilibria Energy-Casimir method Spectral stability Lyapunov stability
abstract In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat with a fixed point in a Newtonian force field. By means of geometric-mechanics methods we study the relative equilibria of these systems for different approximations of the potential function. In particular, we obtain all the equilibria of a generalized Lagrange–Poisson problem under different potentials. Also, we use the Energy-Casimir method to obtain sufficient conditions of stability of equilibria in complex problems of gyrostat dynamics. By means of this method and spectral stability analysis we have obtained necessary and sufficient conditions of stability for equilibria for any potential with axial symmetry U (k3 ) and a Newtonian potential U (3) . The advantages of the Energy-Casimir method, in opposition with the classical Lyapunov–Chetaev method in stability problems of gyrostat dynamics is clear and we illustrate it with several interesting examples. © 2012 Elsevier Inc. All rights reserved.
1. Introduction The general study of the dynamics of rigid bodies and gyrostats has been presented extensively in the classic literature. Eulerian, Lagrangian and Hamiltonian formulations of such dynamics have been the main tools used in the formulation of these problems (see for instance [5], or [18]). It is known that a gyrostat is a mechanical system S made of a rigid body S1 to which other bodies S2 are connected; these other bodies may be deformable or rigid, but must not be rigidly connected to S1 , so that the movements of S2 with respect to S1 do not modify the distribution of mass within the compound system S. For instance, we can envision a rigid main body S1 , designated as the platform, supporting additional bodies S2 , which possess axial symmetry and are designated as rotors. These rotors may rotate with respect to the platform in such a way that the mass distribution within the system as a whole is not altered; this will produce an internal angular momentum, designated as gyrostatic momentum, which will be normally regarded as a constant. Note that when this constant vector is zero, the motion of the system is reduced to the motion of a rigid body. Vito Volterra was the first to introduce the concept of a gyrostat in [20], in order to study the motion of the Earth’s polar axis and explaining variations in the Earth’s latitude by means of internal movements that do not alter the planets’s distribution of mass. Among the various aspects related to these problems that are discussed in the literature, we can highlight the following: 1. Equilibria and stabilities in rigid bodies and gyrostats, either with fixed point or in orbit (see [1,8,14,15,17,19]). 2. Periodic solutions, bifurcations, or chaos, in various gyrostat motion problems [4,6,9,16]. 3. Integrability and first integrals for the problem (see [2,3,10]).
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.11.003
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
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Fig. 1. Gyrostat with a fixed point.
These problems are undoubtedly appealing, not only in the field of astronautics because of the need for placing satellites (whether rigid or gyrostatic) in stable orbits with stable orientations, but also in hydrodynamics, specifically in the study of equilibria and their stabilities for underwater vehicles with symmetric rotors (see Fig. 1). Moving to the study of the motion of rigid bodies and gyrostats, we can consider problems that are simpler, yet not less important, related to the motion of a gyrostat with a fixed point. The study of its equilibria and stabilities in such gyrostats, under differing Newtonian potentials, is interesting as a first approximation towards addressing more complex complex problems. The interest in the study of gyrostats is absolutely justified since certain equilibrium solutions, which are unstable in the case of a rigid body, become stable through a proper selection of the gyrostatic momentum. In the line of this work, many works related to this problem are dedicated to study the position of equilibria and its stability for a gyrostat with a fixed point (see [10,14,15] among others). Using the second method of Lyapunov, Rumiantsev studies the motion of a heavy gyrostat of revolution, where the potential is approximated by U (1) (see Appendix for deduction of the approximate expression of the potential), obtaining for the stability of the equilibrium solution
π1 = 0,
π2 = 0,
π3 = π0 ,
k1 = 0,
k2 = 0,
k3 = 1,
the following necessary and sufficient condition of stability
(π0 + l)2 > 4I1 m0 z0 , where I1 , I2 , I3 , with I1 = I2 are the principal moments of inertia of S , (0, 0, l) is the gyrostatic momentum and (x0 , y0 , z0 ) are the coordinates of the center of masses of S. On the other hand in [1], the same method is applied to study the equilibria of the same problem when the potential is approximated by U (2) , also obtaining the following necessary and sufficient conditions of stability
(π0 + l)2 > 4I1 (m0 z0 + m1 (I3 − I1 )) , where m0 = mg and m1 = 3g /r, being m the total mass of the gyrostat and g the acceleration of the gravity to the distance r of the center of attraction. This condition shows, not only the influence of the election of the gyrostatic momentum, but also the importance of the approach of the potential that we are using. In this work new approaches, both for classical and modern problems, can arise through the introduction of new mathematical tools. Using mechanics-geometry and is one way for such problems to be addressed, as well as new perspectives in their study to be developed. The books [7,11] are standard references in geometric mechanics methods. In this paper we shall use such methods to describe the relative equilibria and sufficient conditions of stability in a generalized Lagrange–Poisson problem. By generalized Lagrange–Poisson problem we are specifically referring to a mechanical system formed by a symmetric gyrostat that has a fixed point in a Newtonian force field and is under an axially symmetric potential that is to say a potential U (k3 ), U being a smooth function like in the case of Lagrange–Poisson for a rigid body with a fixed point. In this way, we obtain all the equilibria and, depending on the case, we obtain necessary and sufficient conditions of stability. We are going to use as tools the Energy-Casimir method, that provides sufficient conditions for the Lyapunov stability of equilibria for Lie–Poisson systems and spectral analysis to determine necessary conditions. Finally, using the methods previously mentioned, we obtain sufficient conditions for stability of two relevant equilibria of a triaxial gyrostat under the potential U (3) which generalizes the classical results
(I3 − I1 )π02 + lπ0 > I32 (m0 z0 + m1 (I3 − I1 )) (I3 − I2 )π02 + lπ0 > I32 (m0 z0 + m1 (I3 − I2 )) obtained in previous mentioned works.
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2. Lie–Poisson equations of a gyrostat in a Newtonian force field Let the rigid part of a gyrostat S be fixed in one of its points O, which will be taken as the origin of two systems of reference; one fixed and inertial OX1 X2 X3 and another mobile Ox1 x2 x3 , fixed in the body, and whose axes are directed according to the principal inertia directions of the gyrostat at O. We will suppose that the only external forces come from the Newtonian attraction that a fixed point P (or a rigid body with a spherical distribution of masses) of mass M exerts over the gyrostat S, of total mass m. Besides, if the relative motion of its mobile part with respect to its rigid part is supposedly known, the system of axes OX1 X2 X3 is chosen in such a way that the point P is over the axis OX3 , in its negative part, at a constant distance r = |OP|, so that r = −rk, being k = (k1 , k2 , k3 ) the unitary vector of the axis OX3 expressed in the mobile system and the mutual potential between the point P and the gyrostat S is given by the function U, then according to the angular momentum theorem and the kinematic equations of Poisson, the equations of motion of this problem in the mobile system (see [10]) takes the form: dπ1 dt dπ2 dt dπ3 dt dk1 dt dk2 dt dk3 dt
∂U ∂U − k3 ∂ k3 ∂ k2 I3 − I1 l1 π3 ∂U ∂U l3 π1 = − + k3 − k1 π1 π3 + I3 I1 I1 I3 ∂ k1 ∂ k3 I1 − I2 l1 π2 l2 π1 ∂U ∂U = π1 π2 + − + k1 − k2 I2 I1 I2 I1 ∂ k2 ∂ k1 k2 π3 k3 π2 = −
=
= =
I2 − I3
I2 I3
I3 k3 π1 I1 k1 π2 I2
− −
π2 π3 +
l2 π3 I3
−
l3 π2 I2
+ k2
(1)
I2 k1 π3 I3 k2 π1 I1
where
• I = diag(I1 , I2 , I3 ) is the diagonal inertia tensor; • π = (π1 , π2 , π3 ) is the angular momentum of the gyrostat considered as a rigid body; • 5 = π + l is the total angular momentum vector for the gyrostat, where l = (l1 , l2 , l3 ) is the gyrostatic momentum, • • • • •
assumed to be constant; k = (k1 , k2 , k3 ) is the Poisson vector; |v| is the Euclidean norm of v ∈ R3 ; × is the cross (vector) product in R3 ; z = (π, k) = (π1 , π2 , π3 , k1 , k2 , k3 ); ∇z f is the gradient of f with respect to the vector z ∈ R6 .
In order to apply the Energy-Casimir method, we must declare the mechanical system in question to be a Lie–Poisson system with a certain Hamiltonian function H , in this case it is given by the formula
H=
1 2
π12 I1
+
π22 I2
+
π32 I3
+ U (k1 , k2 , k3 ).
Then, it is easy to see that the Eqs. (1) can be written in vector form using the following relations dπ dt dk dt
= (π + l) × ∇π H + k × ∇k H (2)
= k × ∇π H .
Finally, we provide (using [12]) the following result that completely describes the equations as a Lie–Poisson system. Proposition 1 (Lie–Poisson Brackets of a Gyrostat with a Fixed Point). The geometric structure associated to the motion of a gyrostat with a fixed point O and constant gyrostatic momentum, is given by the following Lie–Poisson brackets:
{F , G}(π, k) = −(π + l) · (∇π F × ∇π G) − k · (∇π F × ∇k G + ∇k F × ∇π G) defined in R3 × R3 , where F , G ∈ C ∞ (R3 × R3 ), π = (π1 , π2 , π3 ) is the angular momentum of the gyrostat S, considered as a rigid body, k = (k1 , k2 , k3 ) is the Poisson vector and l = (l1 , l2 , l3 ) is the gyrostatic momentum.
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The Poisson tensor associated to the brackets, is given by the matrix: 0
π2 + l2 −π1 − l1
0 k1
0 k2 −k 1 0
−π3 − l3
0
π3 + l3 −π2 − l2 B(π, k) = 0
π1 + l1 −k 3
k3
−k 2
−k3
0 k3 −k2 0 0 0
0 k1 0 0 0
k2 −k 1 0 . 0 0 0
And the Lie–Poisson equations associated to the Hamiltonian
H=
1
π12
2
I1
+
π22
π32
+
I2
I3
+ U (k1 , k2 , k3 )
are expressed by the formulas (2). Remark 2. Note that, if the gyrostatic momentum is zero then these equations are reduced to the equations of the rigid body with a fixed point. The problem has two Casimir functions given by (π + l) · k and |k|2 . This system has, in general, three integrals of motion in involution. In the particular case of a axisymmetric gyrostat (I1 = I2 ), with gyrostatic momentum l = (0, 0, l) and an axially symmetrical potential U (k3 ), we have: dπ1
∂U , ∂ k3 dπ2 I3 − I1 lπ1 ∂U = π1 π3 + − k1 , dt I3 I1 I1 ∂ k3 dk3 k1 π2 k2 π1 dπ3 = 0, = − dt
=
I1 − I3
I1 I3
dt
π2 π3 −
dt
lπ2 I2
+ k2
I2
dk1 dt dk2 dt
= =
k2 π3 I3 k3 π1 I1
− −
k3 π2 I2 k1 π3 I3
, ,
(3)
I1
and a fourth integral is given by π3 = cte. 3. The equilibrium solutions of a generalized Lagrange–Poisson problem According to the equations of motion, we can deduce that all equilibrium solutions must verify ω = λk, being π1 π2 π3 ω = I , I , I the angular velocity of S and λ ∈ R. Mor over, the coordinates of all possible equilibria are given by 1
1
3
the following two algebraic equations
π1 k1 + π2 k2 + (π3 + l)k3 = v with v ∈ R and k21 + k22 + k23 = 1. Of the Eqs. (3) we obtain in an immediate way the following result. Proposition 3. The following points of R3 × S 2 are equilibrium solutions of (3). E1 = 0, 0, π30 , 0, 0, 1 ,
E2 = 0, 0, π30 , 0, 0, −1 .
Equilibrium E1 corresponds to the motion of the gyrostat around the vertical in the upward direction, E2 with the motion of the gyrostat around the vertical in the downward direction. On the other hand, depending on the function U, we have other possible relative equilibria. To looking for different equilibria to E1 and E2 , we can assume that these have the expression E3 = (I1 ωu, I1 ωv, I3 ωw, u, v, w) with u2 + v 2 + w 2 = 1,
w > 0.
The equilibrium point E3 is the critical point of the function
Hλ,µ =
1 2
π12 I1
+
π22 I2
+
π32 I3
+ U (k3 ) + λ (π1 k1 + π2 k2 + (π3 + l)k3 ) + µ(k21 + k22 + k23 )
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with λ and µ to determine. After some computations, we obtain the following conditions
λ = −ω,
µ=
I1 ω 2 2
,
U ′ (ω) + ω2 w(I1 − I3 ) − ωl = 0.
The last equation is used to obtain ω in some particular cases related with the form of the potential U (k3 ). We now see certain particular cases, of great physical significance, in which ω can be calculated.
• We suppose that U (k3 ) = m0 k3 (heavy gyrostat), then we have m0 + ω2 w(I1 − I3 ) − ωl = 0, and
ω l − m0 , ω2 (I1 − I3 )
w=
verifying this equality if, and only if
ω l − m0 ω 2 (I − I ) < 1 1
3
since |w| < 1. • We suppose that U (k3 ) = m0 k3 + m21 (I3 − I1 )k23 (gyrostat with influence of Keplerian attraction) then we have the relation m0 + m1 (I3 − I1 )w + ω2 w(I1 − I3 ) − ωl = 0, and
w=
ωl − m0 , (ω2 − m1 )(I1 − I3 )
verifying this equality if, and only if
ω l − m0 (ω2 − m )(I − I 1
1
3
<1 )
since |w| < 1. 4. Necessary and sufficient conditions of stability to the equilibria Ei Remember that this solution E1 correspond to the motion of the gyrostat around the vertical in the upward direction and its expression is E1 = (0, 0, π0 , 0, 0, 1) . We are going to use the spectral analysis to determine the necessary conditions of stability and the Energy-Casimir method to obtain the sufficient conditions for the Lyapunov stability of this equilibrium solution. To obtain necessary conditions for stability, we compute the tangent flow of Eqs. (3) in the equilibrium E1 denoted by dz dt
= U(E1 )z
with U(E1 ) the Jacobian matrix
0 −a − b U(E1 ) = 0 −d
a+b 0 d 0
0 −c 0 −a
c 0 a 0
of (3) in E1 and a=−
π30 I3
,
b=
π30 + l I1
,
c = U ′ (1),
d=
1 I1
.
The characteristic polynomial associated to these equations is
λ4 + pλ2 + q with p = (2ab + b2 + 2a2 + 2dc ) q = (b2 a2 + 2a3 b − 2dca2 + a4 − 2adbc + c 2 d2 ).
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If the polynomial discriminant is less than zero, at least two characteristic roots have positive real part, then the equilibrium is unstable. The polynomial discriminant is (2a + b)2 (b2 + 4dc ) and therefore, when
(b2 + 4dc ) < 0 we have instability in the system. Substituting the values b, c and d in this expression, we have
π0 + l
2 −
I1
4U ′ (1)
< 0,
I1
then, the necessary conditions to have stability in the solution E1 is
(π0 + l)2 ≥ 4I1 U ′ (1). In order to study sufficient conditions for the stability of E1 , we use the Energy-Casimir method, see [13] for details. Theorem 4 (Energy-Casimir). Let (M, B, H ) be a Lie–Poisson system. We consider ze an equilibrium of the equations B(z)∇z H (z). Let C1 , C2 , . . . , Cr ∈ F(M ) be integrals of the system and
Hφ1 ,φ2 ,...,φr = H +
r
dz dt
=
φi (Ci )
i =1
where φi ∈ C ∞ (R), i = 1, . . . , r is such that dHφ1 ,...,φr (ze ) = 0. Then, if d2 Hφ1 ,...,φr (ze ) |W ×W being W = ker dC1 ∩ ker dC2 ∩ · · · ∩ ker dCr is defined positive or negative, then ze is stable in the sense of Lyapunov. In particular, if W = 0, then ze is stable in the sense of Lyapunov. Now, we consider the function: f =
1 2
π12
+
I1
π22 I1
+
π32
I3
+ U (k3 ) + φ1 ((π + l) · k) + φ2 |k|2 + φ3 (π3 )
where φ1 , φ2 , φ3 are smooth real functions and U (k3 ) is the previous potential, verifying that U has not a critical point in E1 . Imposing the condition d(f )(E1 ) = 0, realizing the corresponding calculates and using the notation x = φ1′ (π0 + l), y = 2φ2′ (1), z = φ3′ (π0 + l), we obtain z = φ3′ (π0 + l) = −
π0 + xI3 I3
,
y = 2φ2′ (1) = − U ′ (1) + x(π0 + l) .
We calculate a base of the subspace W = ker dφ1 ∩ ker dφ2 ∩ ker dφ3 and we obtain that one basis comes given by BW = {e1 , e2 , e4 , e5 } where B = {ei / i = 1, . . . , 6} the canonical basis of R6 , then, the matrix d2 (f )(E1 ) |W ×W is
1 d (f )(E1 ) |W ×W 2
0
I1 = 0 x
I1 0
0
x
1
x
0
0
x .
y
0
0
y
In order to deduce the conditions so this matrix is define positive, it is necessary to study the sign of its determinant which is always positive since
(y − x2 I1 )2 I12
,
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and the sign of the determinant of its maximum minor
1
0
x
I1 0
I1
, 0
x
0
y
1
in other words, the sign of
( y − x 2 I1 ) I12
.
Substituting and calculating in the previous expression we obtain that the sufficient condition to have a define positive matrix is that the following function d(x) =
−U ′ (1) − x[π0 + l + xI1 ] I12
is positive for all the values of x. If we calculate the minimum of this function, we obtain
−
(π0 + l) 2I1
,
and substituting this in the function d(x) we have
−
4U ′ (1)I1 − π02 − 2π0 l − l2 4I12
>0
therefore,
(π0 + l)2 > 4I1 U ′ (1). Then, we can conclude: Theorem 5. A necessary and sufficient condition so that the solution of equilibrium E1 is stable in the sense of Lyapunov is
(π0 + l)2 > 4I1 U ′ (1). 4.1. Stability of E2 Remember that this solution E2 corresponds to the motion of the gyrostat around the vertical in the downward direction and its expression is E2 = (0, 0, π0 , 0, 0, −1) . Analogously, linearizing the equations of motion for the equilibrium E2 , we obtain the necessary condition of stability
(π0 + l)2 ≥ −4I1 U ′ (−1), and using the Energy-Casimir method for E2 , the sufficient condition obtained is
(π0 + l)2 > −4I1 U ′ (−1), therefore, we can conclude: Theorem 6. A necessary and sufficient condition so that the solution of equilibrium E2 is stable in the sense of Lyapunov is
(π0 + l)2 > −4I1 U ′ (−1). 4.2. Stability of the family of equilibria E3 Similarly, the tangent flow of Eq. (3) in the equilibrium E3 is given by dz dt
= U(E3 )z
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with 0
a 0 0
−a 0 0 U(E3 ) = w − I1 v
b c 0
w
−
I1
I1
−U ′ (w)
0
−ωw
ωw
0
−ωw
ωw
I3 u
0
−
v
0 U ′ (w) 0
I3
u I1
0
0 0
−v U ′′ (w) uU ′′ (w) 0 ωu −ωu 0
and a=
(I3 − I1 )ωw + l I1
,
b=
(I3 − I1 )ωv I1
,
c=
(I3 − I1 )ωu I1
.
The characteristic polynomial is
λ2 (I12 λ4 + pλ2 + q) being p = I1 (1 − w 2 )U ′′ (w) − 2I1 w U ′ (w) + I12 (ω2 + a2 ), q = I1 aωw(1 − w 2 )U ′′ (w) − I1 aω(1 + w 2 )U ′ (w) + w 2 U ′ (w)2 − w(1 − w 2 )U ′′ (w)U ′ (w) + ω2 I12 a2 . Then, the system has spectral stability when p ≥ 0, q ≥ 0, p2 − 4q ≥ 0. If w = 0 and U (k3 ) = m0 k3 + U1 (k3 ) with U1′ (0) = 0, then the system has only one equilibrium point given by
E3 =
I1 m 0 l
u,
I1 m0 l
v, 0, u, v, 0 .
The characteristic polynomial associated to this equilibrium solution is
λ4 (λ2 I12 l2 + (l4 + m20 I12 − U ′′ (0)l2 I1 )) and the necessary condition of stability for this solution is l4 + m20 I12 − U ′′ (0)l2 I1 ≥ 0. The Energy-Casimir method does not give us information about sufficient conditions of stability because the quadratic form is semidefinite. 5. Stability of two equilibrium solutions of a triaxial gyrostat under the potential U (3) In this section we consider a triaxial gyrostat with center of mass coordinates (0, 0, z0 ) in the body frame under the Newtonian potential U (3) = mgz0 k3 +
3g 2r
(I1 k21 + I2 k22 + I3 k23 ) −
3g 2r 2
((Jxyy + Jxzz + Jxxx )k1 + (Jyyy + Jyzz + Jyxx )k2
5g (Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 + 3Jxxz k21 k3 + 3Jyyx k22 k1 2r 2 + 3Jyyz k22 k3 + 3Jzzx k23 k1 + 3Jzzy k23 k2 + 3Jxyz k1 k2 k3 ).
+ (Jzzz + Jzxx + Jzyy )k3 ) +
The following proposition is a clear consequence of the Eq. (1). Proposition 7. E1 = (0, 0, π0 , 0, 0, 1) is a equilibrium solution if and only if the triple inertia products verify the following relations 5Jxzz − (Jxxx + Jxyy + Jxzz ) = 0 5Jyzz − (Jyyy + Jyxx + Jyzz ) = 0.
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In this case the potential U (3) is reduced to 3g
U3 = mgz0 k3 +
r
(I1 k21 + I2 k22 + I3 k23 ) −
3g r2
(5Jxzz k1 + 5Jyzz k2 + (Jzzz + Jzxx + Jzyy )k3 )
5g
(Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 + 3Jxxz k21 k3 + 3Jyyx k22 k1 2r 2 + 3Jyyz k22 k3 + 3Jzzx k23 k1 + 3Jzzy k23 k2 + 3Jxyz k1 k2 k3 ). +
Using the Energy-Casimir method we consider Hφ1 ,φ2 = H + φ1 ((π + l) · k) + φ2 (|k|2 ) with H =
1
5t I5 + U (3)
2
and φ1 , φ2 ∈ C ∞ (R). Calculating dHφ1 ,φ2 (E1 ) = 0 we obtain the following relations
φ1p (π0 + l) = − 2φ2p (1)
=
π0 I3
π0 (π0 + l) I3
− mgz0 +
3gI3 r
−
3g Jxxz + Jyyz + Jzzz 2r 2
+
15gJzzz 2r 2
.
A basis BW of the subspace W given by W = ker dφ1 ((π + l) · k)(E1 ) ∩ ker dφ2 (|k|2 )(E1 ) are
BW = {e1 , e2 , e4 , e5 } being B = {ei / i = 1, . . . , 6} the canonical basis of R6 . On the other hand, the matrix d2 Hφ1 ,φ2 (ze ) |W× W is given by the following expression
1
0
I1 0 x
I2
0
x
x
1
0 3gI1
0
r
+
0 x
15gJxxz r2
15gJxyz
+y
15gJxyz
3gI2
r2
r
+
r2 15gJyyz r2
+y
being x=−
y=
π0 I3
π0 (π0 + l) I3
− mgz0 +
3gI3 r
−
3g Jxxz + Jyyz + Jzzz 2r 2
+
15gJzzz 2r 2
.
By means of Sylvester’s Criterion we obtain the following relations for the positive definiteness of d2 Hφ1 ,φ2 (ze ) |W× W 3gI1 r + 15gJxxz + yr 2 − x2 I1 r 2 > 0
(3gI2 r + 15gJyyz + yr 2 − x2 I2 r 2 )(3gI1 r + 15gJxxz + yr 2 − x2 I1 r 2 ) > (15gJxyz )2 .
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We summarize all these results in the following proposition. Proposition 8. The equilibrium E1 = (0, 0, π0 , 0, 0, 1) is stable in the sense of Lyapunov if
Φ1 > 0 Φ1 Ψ1 > (15gJxyz )2 where
Φ1 = (I3 − I1 )π02 + lπ0 − Ψ1 = (I3 − I2 )π02 + lπ0 −
gI32 (2mz0 r 2 + 6(I3 − I1 )r − 3Jyyz − 33Jxxz + 12Jzzz ) gI32
2r 2 (2mz0 r + 6(I3 − I2 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2
2r 2
.
If Jxyz = 0 we have
Φ1 > 0 Ψ1 > 0. Employing the same methodology we obtain for the equilibrium E2 = (0, 0, π0 , 0, 0, −1) the following result. Proposition 9. The equilibrium E2 = (0, 0, π0 , 0, 0, −1) is stable in the sense of Lyapunov if
Φ2 > 0 Φ2 Ψ2 > (15gJxyz )2 where
Φ2 = (I3 − I1 )π02 + lπ0 + Ψ2 = (I3 − I2 )π02 + lπ0 +
gI32 (2mz0 r 2 + 6(I3 − I1 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2r 2 gI32 (2mz0 r 2 + 6(I3 − I2 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2r 2
.
If Jxyz = 0 we have
Φ2 > 0 Ψ2 > 0. Remark 10. If Jijk are all null we obtain the results
(I3 − I1 )π02 + lπ0 − (I3 − I2 )π02 + lπ0 −
gI32 (2mz0 r 2 + 6(I3 − I1 )r ) 2r 2 2 2 gI3 (2mz0 r + 6(I3 − I2 )r ) 2r 2
>0 >0
for E1 and
(I3 − I1 )π02 + lπ0 + (I3 − I2 )π02 + lπ0 +
gI32 (2mz0 r 2 + 6(I1 − I3 )r ) gI32
2r 2 (2mz0 r + 6(I2 − I3 )r ) 2
2r 2
>0 >0
for E2 . 6. Conclusions In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat with a fixed point in a Newtonian force field. By means of geometric-mechanics methods we study the relative equilibria for different approximations of the potential function, in particular we obtained all the equilibria of a generalized Lagrange–Poisson problem under different potentials.
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J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
Also, we use the Energy-Casimir method to obtain sufficient conditions of stability of equilibria in complex problems of gyrostat dynamics. By means of this method and spectral stability analysis we have obtained necessary and sufficient conditions of stability for equilibria of a family of problems for any potential with axial symmetry U (k3 ); particularizing adequately the potential U (k3 ), the results agree with [14,15] for a heavy symmetric gyrostat and [1]. Plus in fact Potential U (k3 ) = mgz0 k3 U (k3 ) = mgz0 k3 +
3g (I3 −I1 ) 2 k3 2r
Sufficient condition (π0 + l)2 > 4I1 mgz 0
(π0 + l)2 > 4I1 mgz0 +
3g (I3 −I1 ) r
In [10] the following potential is considered m1 2 U (k3 ) = (m0 − m3 )k3 + k3 + m4 k33 2 3g α
with m0 = mgz0 , m1 = r , m3 = 2r 2 , m4 = Immediately the sufficient condition is 3g
5g α 2r 2
and α = Jzzz − 3Jzxx , Jzxx = Jzyy , Jijk = 0 for other triple inertia products.
(π0 + l)2 > 4I1 (m0 − m3 + m1 + 3m4 ) for the equilibria E1 . This formula is new in the literature. We emphasize that our methodology leads to much easier calculations when compared to those employed in classical approaches. We have also obtained sufficient conditions of stability for a potential U (3) . To the best of our knowledge, these conditions are new. The derivation of these conditions within the classical Lyapunov–Chetaev framework is tedious and less systematic when compared to using the Energy-Casimir method. The advantages of the last method in stability problems of gyrostat dynamics is clear. Acknowledgments The author is grateful to the reviewers for their comments, which have measurably improved the paper. Appendix Potential energy of a Newtonian force field. We consider the Newtonian attraction of a point P(x, y, z ), of mass M, on a body S, of total mass m. The elementary force df with the point P attracts the point P′ of mass dm, of the body S, is given by
3 |3|3
df = −GM
being 3 = r′ − r, the vector of P′ related to P, and G the universal gravitational constant. The total force f of the point P on the body S, is
3 dm. |3|3
f = −GM S
If we keep in mind that, with regard to the coordinates (x′ , y′ , z ′ ) of P′ and (x, y, z ) of P the following relationship is verified
∇r′
1
= −∇r
|3|
1
|3|
=−
3 |3|3
we can introduce the potential function V given by
1
V = −GM S
|3|
dm
obtaining f = ∇r (V ). Let us see that the function V can be expressed by means of a series. We consider r ′ = |r′ |,
r = |r|,
h=
r′ r
with θ is the angle formed for P, O and P′ and
|3|2 = r 2 + r ′2 − 2rr ′ cos θ r · r′ cos θ = . ′ rr
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
847
Fig. 2. The multipolar series of the potential.
For r ′ < r we obtain the following power series 1
|3|
1
= √
1 − 2h cos θ + h2
r
=
∞ 1
r n=0
hn Pn (cos θ )
with Pn (x) the Legendre polynomials given by Pn (x) =
1 dn [(x2 − 1)n ] n!2n
dxn
.
The first three Legendre polynomials are (see Fig. 2) P 0 ( x) = 1 P 1 ( x) = x 1 P2 (x) = (3x2 − 1) 2 1 P3 (x) = x(5x2 − 3). 2 The potential function V is given by the following power series V =
∞
U (n) = −
∞ GM
r
n =0
hn Pn (cos θ )dm S
n =0
where the series converges absolutely for r > R, being R the maximum of r ′ corresponding to points of S. The first terms of the series are U
(0)
=−
GM
r
U (1) = −
= −
GM r GM r3
dm = −
S
r′
GM r
cos θ dm = −
GM
r3 S GMm r· r′ dm = − 3 r · r0 r S S
r
rr ′ cos θ dm
with r0 = (x0 , y0 , z0 ) the position of the mass center of S. If the origin of the body frame is taken in the mass center of S, then U (1) = 0. For U (2) , we have U (2) = −
=− =−
GM
′ 2
r GM
S
2r 3 S GM 2r 3
S
r
1
r
2
(3 cos2 θ − 1)dm
(r ′ )2 (3 cos2 θ − 1)dm (r ′ )2 (2 − 3 sin2 θ )dm.
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J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
In a similar form for the U (3) U
(3)
GM
=−
′ 2
r
S
GM
=−
r
1
r
2
5
2r 4
′ 3 r
cos θ (5 cos2 θ − 3)dm
cos3 θ dm − 3
′ 3 r
cos θ dm
S
S
′2 r ′ ( rr cos θ ) dm − 3 rr cos θ dm 2r 4 r3 r S S 5GM 3GM 3 2 =− 7 r · r′ dm + r ′ (r · r′ )dm 2r 2r 5 S S 5GM 3GM =− 7 (xx′ + yy′ + zz ′ )3 dm + (x′2 + y′2 + z ′2 )(xx′ + yy′ + zz ′ )dm 2r 2r 5 S S GM
=−
5
5GM
=−
2r 7
1
′
3
x′3 dm + y3
x3
x′2 z ′ dm + 3xy2
× S
′ ′ ′
×
x y z dm + S
+y
U U
(0) (1)
x′ y′2 dm + 3y2 z
′2 ′
3GM
x′2 y′ dm + 3x2 z
S
y′2 z ′ dm + 3z 2 x
S
2r 5 ′3
x y dm + y S
GM r2
z ′3 dm + 3x2 y
S
S
If g =
y′3 dm + z 3
S
S
S
S
′ ′2
y z dm + z S
y′ z ′2 dm + 6xyz
S
x′ z ′2 dm
x y dm + x S
y dm + y S
′ ′2
x dm + x
x
S
′3
x′ z ′2 dm + 3z 2 y
′2 ′
′2 ′
x z dm + z S
z dm . ′3
y z dm + z S
S
and P is located on the negative part of the axis OZ , we will have r = −rk, being k = (k1 , k2 , k3 ). Therefore
= −mgr = mg (x0 k1 + y0 k2 + z0 k3 )
U (2) = −
3g 2r
(I1 k21 + I2 k22 + I3 k23 ).
Notice U (0) is constant if r is constant and if Jxyz = the form U (3) =
−3g
S
x′ y′ z ′ dm, Jxxz =
S
x′2 z ′ dm, . . . , we will be able to express U (3) in
[(Jxxx + Jxyy + Jxzz )k1 + (Jxxy + Jyyy + Jyzz )k2 + (Jxxz + Jyyz + Jzzz )k3 ]
2r 2 5g
[Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 2r 2 + 3Jxxz k21 k3 + 3Jyyx k1 k22 + 3Jyyz k22 k3 + 3Jxzz k21 k23 + 3Jyzz k2 k23 + 6Jxyz k1 k2 k3 ]. +
In a similar form we obtain the next U (i) in the series. References [1] J.A. Cavas, A. Vigueras, Stability of certain rotations of a gyrostat analogous to that of Lagrange and Poisson in Newtonian force field, in: Proceedings of the Fourth International Workshop on Positional Astronomy and Celestial Mechanics, 1996, pp. 129–136. [2] J.A. Cavas, A. Vigueras, An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field, Celestial Mech. Dynam. Astronom. 60 (1994) 317–330. [3] J.E. Cochran, P.H. Shu, S.D. Rews, Attitude motion of asymmetric dual-spin spacecraft, J. Guid. Control Dyn. 5 (1982) 37–42. [4] A. Elipe, V. Lanchares, Two equivalent problems: gyrostats in free motion and parametric Hamiltonians, Mech. Res. Commun. 24 (1997) 583–590. [5] J.L.G. Guirao, J.A. Vera, Lagrangian relative equilibria for a gyrostat in the three body problem, J. Phys. A 43 (19) (2010) 1–16. [6] J.L.G. Guirao, J.A. Vera, Sufficient conditions for a nondegenerate Hopf bifurcation in a generalized Lagrange–Poisson problem, J. Math. Phys. 52 (2011) 032701. http://dx.doi.org/10.1063/1.3559064 (p. 11). [7] D.D. Holm, T. Schmah, C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press, 2009. [8] T.J. Kalvouridis, Stationary solutions of a small gyrostat in the Newtonian field of two bodies with equal masses, Nonlinear Dynam. 61 (3) (2010) 373–381. [9] T.J. Kalvouridis, V. Tsogas, Rigid body dynamics in the restricted ring problem of n + 1 bodies, Astrophys. Space Sci. 282 (4) (2002) 751–765. [10] E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer Verlag, Berlin, 1965. [11] J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, in: Texts in Applied Mathematics, vol. 17, Springer Verlag, 1999. [12] A. Marsden, T. Ratiu, A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984) 147–177. [13] J.P. Ortega, T.S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity 12 (3) (1999) 693–720. [14] V.V. Rumiantsev, On the stability of motion of gyrostats, J. Appl. Math. Mech. (1961) 9–19. [15] V.V. Rumiantsev, On the stability of motion of certain types of gyrostats, J. Appl. Math. Mech. (1961) 1158–1169.
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[16] V. Tsogas, T.J. Kalvouridis, A.G. Mavraganis, Equilibrium states of a gyrostat satellite moving in the gravitational field of an annular configuration of n big bodies, Acta Mech. 175 (1–4) (2005) 181–195. [17] J.A. Vera, Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria, Nonlinear Dynam. 55 (1–2) (2009) 191–201. [18] J.A. Vera, A. Vigueras, Hamiltonian dynamics of a gyrostat in the n-body problem: relative equilibria, Celestial Mech. Dynam. Astronom. 94 (3) (2006) 289–315. [19] J.A. Vera, A. Vigueras, Soluciones de equilibrio de un problema generalizado de Lagrange–Poisson: condiciones necesarias y suficientes de estabilidad, Monogr. Real Acad. Cienc. Zaragoza 22 (2003) 141–150. [20] V. Volterra, Sur la théorie des variations des latitudes, Acta Math. 22 (1899) 201–357.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
The gyrostat with a fixed point in a Newtonian force field: Relative equilibria and stability J.A. Vera Centro Universitario de la Defensa (MDE-UPCT), Base Aérea de San Javier, C/General López Peña s/n, 30720 Santiago de la Ribera (Murcia), Spain
article
info
Article history: Received 24 January 2012 Available online 19 December 2012 Submitted by Roman O. Popovych Keywords: Gyrostat Relative equilibria Energy-Casimir method Spectral stability Lyapunov stability
abstract In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat with a fixed point in a Newtonian force field. By means of geometric-mechanics methods we study the relative equilibria of these systems for different approximations of the potential function. In particular, we obtain all the equilibria of a generalized Lagrange–Poisson problem under different potentials. Also, we use the Energy-Casimir method to obtain sufficient conditions of stability of equilibria in complex problems of gyrostat dynamics. By means of this method and spectral stability analysis we have obtained necessary and sufficient conditions of stability for equilibria for any potential with axial symmetry U (k3 ) and a Newtonian potential U (3) . The advantages of the Energy-Casimir method, in opposition with the classical Lyapunov–Chetaev method in stability problems of gyrostat dynamics is clear and we illustrate it with several interesting examples. © 2012 Elsevier Inc. All rights reserved.
1. Introduction The general study of the dynamics of rigid bodies and gyrostats has been presented extensively in the classic literature. Eulerian, Lagrangian and Hamiltonian formulations of such dynamics have been the main tools used in the formulation of these problems (see for instance [5], or [18]). It is known that a gyrostat is a mechanical system S made of a rigid body S1 to which other bodies S2 are connected; these other bodies may be deformable or rigid, but must not be rigidly connected to S1 , so that the movements of S2 with respect to S1 do not modify the distribution of mass within the compound system S. For instance, we can envision a rigid main body S1 , designated as the platform, supporting additional bodies S2 , which possess axial symmetry and are designated as rotors. These rotors may rotate with respect to the platform in such a way that the mass distribution within the system as a whole is not altered; this will produce an internal angular momentum, designated as gyrostatic momentum, which will be normally regarded as a constant. Note that when this constant vector is zero, the motion of the system is reduced to the motion of a rigid body. Vito Volterra was the first to introduce the concept of a gyrostat in [20], in order to study the motion of the Earth’s polar axis and explaining variations in the Earth’s latitude by means of internal movements that do not alter the planets’s distribution of mass. Among the various aspects related to these problems that are discussed in the literature, we can highlight the following: 1. Equilibria and stabilities in rigid bodies and gyrostats, either with fixed point or in orbit (see [1,8,14,15,17,19]). 2. Periodic solutions, bifurcations, or chaos, in various gyrostat motion problems [4,6,9,16]. 3. Integrability and first integrals for the problem (see [2,3,10]).
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.11.003
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
837
Fig. 1. Gyrostat with a fixed point.
These problems are undoubtedly appealing, not only in the field of astronautics because of the need for placing satellites (whether rigid or gyrostatic) in stable orbits with stable orientations, but also in hydrodynamics, specifically in the study of equilibria and their stabilities for underwater vehicles with symmetric rotors (see Fig. 1). Moving to the study of the motion of rigid bodies and gyrostats, we can consider problems that are simpler, yet not less important, related to the motion of a gyrostat with a fixed point. The study of its equilibria and stabilities in such gyrostats, under differing Newtonian potentials, is interesting as a first approximation towards addressing more complex complex problems. The interest in the study of gyrostats is absolutely justified since certain equilibrium solutions, which are unstable in the case of a rigid body, become stable through a proper selection of the gyrostatic momentum. In the line of this work, many works related to this problem are dedicated to study the position of equilibria and its stability for a gyrostat with a fixed point (see [10,14,15] among others). Using the second method of Lyapunov, Rumiantsev studies the motion of a heavy gyrostat of revolution, where the potential is approximated by U (1) (see Appendix for deduction of the approximate expression of the potential), obtaining for the stability of the equilibrium solution
π1 = 0,
π2 = 0,
π3 = π0 ,
k1 = 0,
k2 = 0,
k3 = 1,
the following necessary and sufficient condition of stability
(π0 + l)2 > 4I1 m0 z0 , where I1 , I2 , I3 , with I1 = I2 are the principal moments of inertia of S , (0, 0, l) is the gyrostatic momentum and (x0 , y0 , z0 ) are the coordinates of the center of masses of S. On the other hand in [1], the same method is applied to study the equilibria of the same problem when the potential is approximated by U (2) , also obtaining the following necessary and sufficient conditions of stability
(π0 + l)2 > 4I1 (m0 z0 + m1 (I3 − I1 )) , where m0 = mg and m1 = 3g /r, being m the total mass of the gyrostat and g the acceleration of the gravity to the distance r of the center of attraction. This condition shows, not only the influence of the election of the gyrostatic momentum, but also the importance of the approach of the potential that we are using. In this work new approaches, both for classical and modern problems, can arise through the introduction of new mathematical tools. Using mechanics-geometry and is one way for such problems to be addressed, as well as new perspectives in their study to be developed. The books [7,11] are standard references in geometric mechanics methods. In this paper we shall use such methods to describe the relative equilibria and sufficient conditions of stability in a generalized Lagrange–Poisson problem. By generalized Lagrange–Poisson problem we are specifically referring to a mechanical system formed by a symmetric gyrostat that has a fixed point in a Newtonian force field and is under an axially symmetric potential that is to say a potential U (k3 ), U being a smooth function like in the case of Lagrange–Poisson for a rigid body with a fixed point. In this way, we obtain all the equilibria and, depending on the case, we obtain necessary and sufficient conditions of stability. We are going to use as tools the Energy-Casimir method, that provides sufficient conditions for the Lyapunov stability of equilibria for Lie–Poisson systems and spectral analysis to determine necessary conditions. Finally, using the methods previously mentioned, we obtain sufficient conditions for stability of two relevant equilibria of a triaxial gyrostat under the potential U (3) which generalizes the classical results
(I3 − I1 )π02 + lπ0 > I32 (m0 z0 + m1 (I3 − I1 )) (I3 − I2 )π02 + lπ0 > I32 (m0 z0 + m1 (I3 − I2 )) obtained in previous mentioned works.
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2. Lie–Poisson equations of a gyrostat in a Newtonian force field Let the rigid part of a gyrostat S be fixed in one of its points O, which will be taken as the origin of two systems of reference; one fixed and inertial OX1 X2 X3 and another mobile Ox1 x2 x3 , fixed in the body, and whose axes are directed according to the principal inertia directions of the gyrostat at O. We will suppose that the only external forces come from the Newtonian attraction that a fixed point P (or a rigid body with a spherical distribution of masses) of mass M exerts over the gyrostat S, of total mass m. Besides, if the relative motion of its mobile part with respect to its rigid part is supposedly known, the system of axes OX1 X2 X3 is chosen in such a way that the point P is over the axis OX3 , in its negative part, at a constant distance r = |OP|, so that r = −rk, being k = (k1 , k2 , k3 ) the unitary vector of the axis OX3 expressed in the mobile system and the mutual potential between the point P and the gyrostat S is given by the function U, then according to the angular momentum theorem and the kinematic equations of Poisson, the equations of motion of this problem in the mobile system (see [10]) takes the form: dπ1 dt dπ2 dt dπ3 dt dk1 dt dk2 dt dk3 dt
∂U ∂U − k3 ∂ k3 ∂ k2 I3 − I1 l1 π3 ∂U ∂U l3 π1 = − + k3 − k1 π1 π3 + I3 I1 I1 I3 ∂ k1 ∂ k3 I1 − I2 l1 π2 l2 π1 ∂U ∂U = π1 π2 + − + k1 − k2 I2 I1 I2 I1 ∂ k2 ∂ k1 k2 π3 k3 π2 = −
=
= =
I2 − I3
I2 I3
I3 k3 π1 I1 k1 π2 I2
− −
π2 π3 +
l2 π3 I3
−
l3 π2 I2
+ k2
(1)
I2 k1 π3 I3 k2 π1 I1
where
• I = diag(I1 , I2 , I3 ) is the diagonal inertia tensor; • π = (π1 , π2 , π3 ) is the angular momentum of the gyrostat considered as a rigid body; • 5 = π + l is the total angular momentum vector for the gyrostat, where l = (l1 , l2 , l3 ) is the gyrostatic momentum, • • • • •
assumed to be constant; k = (k1 , k2 , k3 ) is the Poisson vector; |v| is the Euclidean norm of v ∈ R3 ; × is the cross (vector) product in R3 ; z = (π, k) = (π1 , π2 , π3 , k1 , k2 , k3 ); ∇z f is the gradient of f with respect to the vector z ∈ R6 .
In order to apply the Energy-Casimir method, we must declare the mechanical system in question to be a Lie–Poisson system with a certain Hamiltonian function H , in this case it is given by the formula
H=
1 2
π12 I1
+
π22 I2
+
π32 I3
+ U (k1 , k2 , k3 ).
Then, it is easy to see that the Eqs. (1) can be written in vector form using the following relations dπ dt dk dt
= (π + l) × ∇π H + k × ∇k H (2)
= k × ∇π H .
Finally, we provide (using [12]) the following result that completely describes the equations as a Lie–Poisson system. Proposition 1 (Lie–Poisson Brackets of a Gyrostat with a Fixed Point). The geometric structure associated to the motion of a gyrostat with a fixed point O and constant gyrostatic momentum, is given by the following Lie–Poisson brackets:
{F , G}(π, k) = −(π + l) · (∇π F × ∇π G) − k · (∇π F × ∇k G + ∇k F × ∇π G) defined in R3 × R3 , where F , G ∈ C ∞ (R3 × R3 ), π = (π1 , π2 , π3 ) is the angular momentum of the gyrostat S, considered as a rigid body, k = (k1 , k2 , k3 ) is the Poisson vector and l = (l1 , l2 , l3 ) is the gyrostatic momentum.
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
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The Poisson tensor associated to the brackets, is given by the matrix: 0
π2 + l2 −π1 − l1
0 k1
0 k2 −k 1 0
−π3 − l3
0
π3 + l3 −π2 − l2 B(π, k) = 0
π1 + l1 −k 3
k3
−k 2
−k3
0 k3 −k2 0 0 0
0 k1 0 0 0
k2 −k 1 0 . 0 0 0
And the Lie–Poisson equations associated to the Hamiltonian
H=
1
π12
2
I1
+
π22
π32
+
I2
I3
+ U (k1 , k2 , k3 )
are expressed by the formulas (2). Remark 2. Note that, if the gyrostatic momentum is zero then these equations are reduced to the equations of the rigid body with a fixed point. The problem has two Casimir functions given by (π + l) · k and |k|2 . This system has, in general, three integrals of motion in involution. In the particular case of a axisymmetric gyrostat (I1 = I2 ), with gyrostatic momentum l = (0, 0, l) and an axially symmetrical potential U (k3 ), we have: dπ1
∂U , ∂ k3 dπ2 I3 − I1 lπ1 ∂U = π1 π3 + − k1 , dt I3 I1 I1 ∂ k3 dk3 k1 π2 k2 π1 dπ3 = 0, = − dt
=
I1 − I3
I1 I3
dt
π2 π3 −
dt
lπ2 I2
+ k2
I2
dk1 dt dk2 dt
= =
k2 π3 I3 k3 π1 I1
− −
k3 π2 I2 k1 π3 I3
, ,
(3)
I1
and a fourth integral is given by π3 = cte. 3. The equilibrium solutions of a generalized Lagrange–Poisson problem According to the equations of motion, we can deduce that all equilibrium solutions must verify ω = λk, being π1 π2 π3 ω = I , I , I the angular velocity of S and λ ∈ R. Mor over, the coordinates of all possible equilibria are given by 1
1
3
the following two algebraic equations
π1 k1 + π2 k2 + (π3 + l)k3 = v with v ∈ R and k21 + k22 + k23 = 1. Of the Eqs. (3) we obtain in an immediate way the following result. Proposition 3. The following points of R3 × S 2 are equilibrium solutions of (3). E1 = 0, 0, π30 , 0, 0, 1 ,
E2 = 0, 0, π30 , 0, 0, −1 .
Equilibrium E1 corresponds to the motion of the gyrostat around the vertical in the upward direction, E2 with the motion of the gyrostat around the vertical in the downward direction. On the other hand, depending on the function U, we have other possible relative equilibria. To looking for different equilibria to E1 and E2 , we can assume that these have the expression E3 = (I1 ωu, I1 ωv, I3 ωw, u, v, w) with u2 + v 2 + w 2 = 1,
w > 0.
The equilibrium point E3 is the critical point of the function
Hλ,µ =
1 2
π12 I1
+
π22 I2
+
π32 I3
+ U (k3 ) + λ (π1 k1 + π2 k2 + (π3 + l)k3 ) + µ(k21 + k22 + k23 )
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J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
with λ and µ to determine. After some computations, we obtain the following conditions
λ = −ω,
µ=
I1 ω 2 2
,
U ′ (ω) + ω2 w(I1 − I3 ) − ωl = 0.
The last equation is used to obtain ω in some particular cases related with the form of the potential U (k3 ). We now see certain particular cases, of great physical significance, in which ω can be calculated.
• We suppose that U (k3 ) = m0 k3 (heavy gyrostat), then we have m0 + ω2 w(I1 − I3 ) − ωl = 0, and
ω l − m0 , ω2 (I1 − I3 )
w=
verifying this equality if, and only if
ω l − m0 ω 2 (I − I ) < 1 1
3
since |w| < 1. • We suppose that U (k3 ) = m0 k3 + m21 (I3 − I1 )k23 (gyrostat with influence of Keplerian attraction) then we have the relation m0 + m1 (I3 − I1 )w + ω2 w(I1 − I3 ) − ωl = 0, and
w=
ωl − m0 , (ω2 − m1 )(I1 − I3 )
verifying this equality if, and only if
ω l − m0 (ω2 − m )(I − I 1
1
3
<1 )
since |w| < 1. 4. Necessary and sufficient conditions of stability to the equilibria Ei Remember that this solution E1 correspond to the motion of the gyrostat around the vertical in the upward direction and its expression is E1 = (0, 0, π0 , 0, 0, 1) . We are going to use the spectral analysis to determine the necessary conditions of stability and the Energy-Casimir method to obtain the sufficient conditions for the Lyapunov stability of this equilibrium solution. To obtain necessary conditions for stability, we compute the tangent flow of Eqs. (3) in the equilibrium E1 denoted by dz dt
= U(E1 )z
with U(E1 ) the Jacobian matrix
0 −a − b U(E1 ) = 0 −d
a+b 0 d 0
0 −c 0 −a
c 0 a 0
of (3) in E1 and a=−
π30 I3
,
b=
π30 + l I1
,
c = U ′ (1),
d=
1 I1
.
The characteristic polynomial associated to these equations is
λ4 + pλ2 + q with p = (2ab + b2 + 2a2 + 2dc ) q = (b2 a2 + 2a3 b − 2dca2 + a4 − 2adbc + c 2 d2 ).
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If the polynomial discriminant is less than zero, at least two characteristic roots have positive real part, then the equilibrium is unstable. The polynomial discriminant is (2a + b)2 (b2 + 4dc ) and therefore, when
(b2 + 4dc ) < 0 we have instability in the system. Substituting the values b, c and d in this expression, we have
π0 + l
2 −
I1
4U ′ (1)
< 0,
I1
then, the necessary conditions to have stability in the solution E1 is
(π0 + l)2 ≥ 4I1 U ′ (1). In order to study sufficient conditions for the stability of E1 , we use the Energy-Casimir method, see [13] for details. Theorem 4 (Energy-Casimir). Let (M, B, H ) be a Lie–Poisson system. We consider ze an equilibrium of the equations B(z)∇z H (z). Let C1 , C2 , . . . , Cr ∈ F(M ) be integrals of the system and
Hφ1 ,φ2 ,...,φr = H +
r
dz dt
=
φi (Ci )
i =1
where φi ∈ C ∞ (R), i = 1, . . . , r is such that dHφ1 ,...,φr (ze ) = 0. Then, if d2 Hφ1 ,...,φr (ze ) |W ×W being W = ker dC1 ∩ ker dC2 ∩ · · · ∩ ker dCr is defined positive or negative, then ze is stable in the sense of Lyapunov. In particular, if W = 0, then ze is stable in the sense of Lyapunov. Now, we consider the function: f =
1 2
π12
+
I1
π22 I1
+
π32
I3
+ U (k3 ) + φ1 ((π + l) · k) + φ2 |k|2 + φ3 (π3 )
where φ1 , φ2 , φ3 are smooth real functions and U (k3 ) is the previous potential, verifying that U has not a critical point in E1 . Imposing the condition d(f )(E1 ) = 0, realizing the corresponding calculates and using the notation x = φ1′ (π0 + l), y = 2φ2′ (1), z = φ3′ (π0 + l), we obtain z = φ3′ (π0 + l) = −
π0 + xI3 I3
,
y = 2φ2′ (1) = − U ′ (1) + x(π0 + l) .
We calculate a base of the subspace W = ker dφ1 ∩ ker dφ2 ∩ ker dφ3 and we obtain that one basis comes given by BW = {e1 , e2 , e4 , e5 } where B = {ei / i = 1, . . . , 6} the canonical basis of R6 , then, the matrix d2 (f )(E1 ) |W ×W is
1 d (f )(E1 ) |W ×W 2
0
I1 = 0 x
I1 0
0
x
1
x
0
0
x .
y
0
0
y
In order to deduce the conditions so this matrix is define positive, it is necessary to study the sign of its determinant which is always positive since
(y − x2 I1 )2 I12
,
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and the sign of the determinant of its maximum minor
1
0
x
I1 0
I1
, 0
x
0
y
1
in other words, the sign of
( y − x 2 I1 ) I12
.
Substituting and calculating in the previous expression we obtain that the sufficient condition to have a define positive matrix is that the following function d(x) =
−U ′ (1) − x[π0 + l + xI1 ] I12
is positive for all the values of x. If we calculate the minimum of this function, we obtain
−
(π0 + l) 2I1
,
and substituting this in the function d(x) we have
−
4U ′ (1)I1 − π02 − 2π0 l − l2 4I12
>0
therefore,
(π0 + l)2 > 4I1 U ′ (1). Then, we can conclude: Theorem 5. A necessary and sufficient condition so that the solution of equilibrium E1 is stable in the sense of Lyapunov is
(π0 + l)2 > 4I1 U ′ (1). 4.1. Stability of E2 Remember that this solution E2 corresponds to the motion of the gyrostat around the vertical in the downward direction and its expression is E2 = (0, 0, π0 , 0, 0, −1) . Analogously, linearizing the equations of motion for the equilibrium E2 , we obtain the necessary condition of stability
(π0 + l)2 ≥ −4I1 U ′ (−1), and using the Energy-Casimir method for E2 , the sufficient condition obtained is
(π0 + l)2 > −4I1 U ′ (−1), therefore, we can conclude: Theorem 6. A necessary and sufficient condition so that the solution of equilibrium E2 is stable in the sense of Lyapunov is
(π0 + l)2 > −4I1 U ′ (−1). 4.2. Stability of the family of equilibria E3 Similarly, the tangent flow of Eq. (3) in the equilibrium E3 is given by dz dt
= U(E3 )z
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with 0
a 0 0
−a 0 0 U(E3 ) = w − I1 v
b c 0
w
−
I1
I1
−U ′ (w)
0
−ωw
ωw
0
−ωw
ωw
I3 u
0
−
v
0 U ′ (w) 0
I3
u I1
0
0 0
−v U ′′ (w) uU ′′ (w) 0 ωu −ωu 0
and a=
(I3 − I1 )ωw + l I1
,
b=
(I3 − I1 )ωv I1
,
c=
(I3 − I1 )ωu I1
.
The characteristic polynomial is
λ2 (I12 λ4 + pλ2 + q) being p = I1 (1 − w 2 )U ′′ (w) − 2I1 w U ′ (w) + I12 (ω2 + a2 ), q = I1 aωw(1 − w 2 )U ′′ (w) − I1 aω(1 + w 2 )U ′ (w) + w 2 U ′ (w)2 − w(1 − w 2 )U ′′ (w)U ′ (w) + ω2 I12 a2 . Then, the system has spectral stability when p ≥ 0, q ≥ 0, p2 − 4q ≥ 0. If w = 0 and U (k3 ) = m0 k3 + U1 (k3 ) with U1′ (0) = 0, then the system has only one equilibrium point given by
E3 =
I1 m 0 l
u,
I1 m0 l
v, 0, u, v, 0 .
The characteristic polynomial associated to this equilibrium solution is
λ4 (λ2 I12 l2 + (l4 + m20 I12 − U ′′ (0)l2 I1 )) and the necessary condition of stability for this solution is l4 + m20 I12 − U ′′ (0)l2 I1 ≥ 0. The Energy-Casimir method does not give us information about sufficient conditions of stability because the quadratic form is semidefinite. 5. Stability of two equilibrium solutions of a triaxial gyrostat under the potential U (3) In this section we consider a triaxial gyrostat with center of mass coordinates (0, 0, z0 ) in the body frame under the Newtonian potential U (3) = mgz0 k3 +
3g 2r
(I1 k21 + I2 k22 + I3 k23 ) −
3g 2r 2
((Jxyy + Jxzz + Jxxx )k1 + (Jyyy + Jyzz + Jyxx )k2
5g (Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 + 3Jxxz k21 k3 + 3Jyyx k22 k1 2r 2 + 3Jyyz k22 k3 + 3Jzzx k23 k1 + 3Jzzy k23 k2 + 3Jxyz k1 k2 k3 ).
+ (Jzzz + Jzxx + Jzyy )k3 ) +
The following proposition is a clear consequence of the Eq. (1). Proposition 7. E1 = (0, 0, π0 , 0, 0, 1) is a equilibrium solution if and only if the triple inertia products verify the following relations 5Jxzz − (Jxxx + Jxyy + Jxzz ) = 0 5Jyzz − (Jyyy + Jyxx + Jyzz ) = 0.
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In this case the potential U (3) is reduced to 3g
U3 = mgz0 k3 +
r
(I1 k21 + I2 k22 + I3 k23 ) −
3g r2
(5Jxzz k1 + 5Jyzz k2 + (Jzzz + Jzxx + Jzyy )k3 )
5g
(Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 + 3Jxxz k21 k3 + 3Jyyx k22 k1 2r 2 + 3Jyyz k22 k3 + 3Jzzx k23 k1 + 3Jzzy k23 k2 + 3Jxyz k1 k2 k3 ). +
Using the Energy-Casimir method we consider Hφ1 ,φ2 = H + φ1 ((π + l) · k) + φ2 (|k|2 ) with H =
1
5t I5 + U (3)
2
and φ1 , φ2 ∈ C ∞ (R). Calculating dHφ1 ,φ2 (E1 ) = 0 we obtain the following relations
φ1p (π0 + l) = − 2φ2p (1)
=
π0 I3
π0 (π0 + l) I3
− mgz0 +
3gI3 r
−
3g Jxxz + Jyyz + Jzzz 2r 2
+
15gJzzz 2r 2
.
A basis BW of the subspace W given by W = ker dφ1 ((π + l) · k)(E1 ) ∩ ker dφ2 (|k|2 )(E1 ) are
BW = {e1 , e2 , e4 , e5 } being B = {ei / i = 1, . . . , 6} the canonical basis of R6 . On the other hand, the matrix d2 Hφ1 ,φ2 (ze ) |W× W is given by the following expression
1
0
I1 0 x
I2
0
x
x
1
0 3gI1
0
r
+
0 x
15gJxxz r2
15gJxyz
+y
15gJxyz
3gI2
r2
r
+
r2 15gJyyz r2
+y
being x=−
y=
π0 I3
π0 (π0 + l) I3
− mgz0 +
3gI3 r
−
3g Jxxz + Jyyz + Jzzz 2r 2
+
15gJzzz 2r 2
.
By means of Sylvester’s Criterion we obtain the following relations for the positive definiteness of d2 Hφ1 ,φ2 (ze ) |W× W 3gI1 r + 15gJxxz + yr 2 − x2 I1 r 2 > 0
(3gI2 r + 15gJyyz + yr 2 − x2 I2 r 2 )(3gI1 r + 15gJxxz + yr 2 − x2 I1 r 2 ) > (15gJxyz )2 .
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
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We summarize all these results in the following proposition. Proposition 8. The equilibrium E1 = (0, 0, π0 , 0, 0, 1) is stable in the sense of Lyapunov if
Φ1 > 0 Φ1 Ψ1 > (15gJxyz )2 where
Φ1 = (I3 − I1 )π02 + lπ0 − Ψ1 = (I3 − I2 )π02 + lπ0 −
gI32 (2mz0 r 2 + 6(I3 − I1 )r − 3Jyyz − 33Jxxz + 12Jzzz ) gI32
2r 2 (2mz0 r + 6(I3 − I2 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2
2r 2
.
If Jxyz = 0 we have
Φ1 > 0 Ψ1 > 0. Employing the same methodology we obtain for the equilibrium E2 = (0, 0, π0 , 0, 0, −1) the following result. Proposition 9. The equilibrium E2 = (0, 0, π0 , 0, 0, −1) is stable in the sense of Lyapunov if
Φ2 > 0 Φ2 Ψ2 > (15gJxyz )2 where
Φ2 = (I3 − I1 )π02 + lπ0 + Ψ2 = (I3 − I2 )π02 + lπ0 +
gI32 (2mz0 r 2 + 6(I3 − I1 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2r 2 gI32 (2mz0 r 2 + 6(I3 − I2 )r − 3Jyyz − 33Jxxz + 12Jzzz ) 2r 2
.
If Jxyz = 0 we have
Φ2 > 0 Ψ2 > 0. Remark 10. If Jijk are all null we obtain the results
(I3 − I1 )π02 + lπ0 − (I3 − I2 )π02 + lπ0 −
gI32 (2mz0 r 2 + 6(I3 − I1 )r ) 2r 2 2 2 gI3 (2mz0 r + 6(I3 − I2 )r ) 2r 2
>0 >0
for E1 and
(I3 − I1 )π02 + lπ0 + (I3 − I2 )π02 + lπ0 +
gI32 (2mz0 r 2 + 6(I1 − I3 )r ) gI32
2r 2 (2mz0 r + 6(I2 − I3 )r ) 2
2r 2
>0 >0
for E2 . 6. Conclusions In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat with a fixed point in a Newtonian force field. By means of geometric-mechanics methods we study the relative equilibria for different approximations of the potential function, in particular we obtained all the equilibria of a generalized Lagrange–Poisson problem under different potentials.
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Also, we use the Energy-Casimir method to obtain sufficient conditions of stability of equilibria in complex problems of gyrostat dynamics. By means of this method and spectral stability analysis we have obtained necessary and sufficient conditions of stability for equilibria of a family of problems for any potential with axial symmetry U (k3 ); particularizing adequately the potential U (k3 ), the results agree with [14,15] for a heavy symmetric gyrostat and [1]. Plus in fact Potential U (k3 ) = mgz0 k3 U (k3 ) = mgz0 k3 +
3g (I3 −I1 ) 2 k3 2r
Sufficient condition (π0 + l)2 > 4I1 mgz 0
(π0 + l)2 > 4I1 mgz0 +
3g (I3 −I1 ) r
In [10] the following potential is considered m1 2 U (k3 ) = (m0 − m3 )k3 + k3 + m4 k33 2 3g α
with m0 = mgz0 , m1 = r , m3 = 2r 2 , m4 = Immediately the sufficient condition is 3g
5g α 2r 2
and α = Jzzz − 3Jzxx , Jzxx = Jzyy , Jijk = 0 for other triple inertia products.
(π0 + l)2 > 4I1 (m0 − m3 + m1 + 3m4 ) for the equilibria E1 . This formula is new in the literature. We emphasize that our methodology leads to much easier calculations when compared to those employed in classical approaches. We have also obtained sufficient conditions of stability for a potential U (3) . To the best of our knowledge, these conditions are new. The derivation of these conditions within the classical Lyapunov–Chetaev framework is tedious and less systematic when compared to using the Energy-Casimir method. The advantages of the last method in stability problems of gyrostat dynamics is clear. Acknowledgments The author is grateful to the reviewers for their comments, which have measurably improved the paper. Appendix Potential energy of a Newtonian force field. We consider the Newtonian attraction of a point P(x, y, z ), of mass M, on a body S, of total mass m. The elementary force df with the point P attracts the point P′ of mass dm, of the body S, is given by
3 |3|3
df = −GM
being 3 = r′ − r, the vector of P′ related to P, and G the universal gravitational constant. The total force f of the point P on the body S, is
3 dm. |3|3
f = −GM S
If we keep in mind that, with regard to the coordinates (x′ , y′ , z ′ ) of P′ and (x, y, z ) of P the following relationship is verified
∇r′
1
= −∇r
|3|
1
|3|
=−
3 |3|3
we can introduce the potential function V given by
1
V = −GM S
|3|
dm
obtaining f = ∇r (V ). Let us see that the function V can be expressed by means of a series. We consider r ′ = |r′ |,
r = |r|,
h=
r′ r
with θ is the angle formed for P, O and P′ and
|3|2 = r 2 + r ′2 − 2rr ′ cos θ r · r′ cos θ = . ′ rr
J.A. Vera / J. Math. Anal. Appl. 401 (2013) 836–849
847
Fig. 2. The multipolar series of the potential.
For r ′ < r we obtain the following power series 1
|3|
1
= √
1 − 2h cos θ + h2
r
=
∞ 1
r n=0
hn Pn (cos θ )
with Pn (x) the Legendre polynomials given by Pn (x) =
1 dn [(x2 − 1)n ] n!2n
dxn
.
The first three Legendre polynomials are (see Fig. 2) P 0 ( x) = 1 P 1 ( x) = x 1 P2 (x) = (3x2 − 1) 2 1 P3 (x) = x(5x2 − 3). 2 The potential function V is given by the following power series V =
∞
U (n) = −
∞ GM
r
n =0
hn Pn (cos θ )dm S
n =0
where the series converges absolutely for r > R, being R the maximum of r ′ corresponding to points of S. The first terms of the series are U
(0)
=−
GM
r
U (1) = −
= −
GM r GM r3
dm = −
S
r′
GM r
cos θ dm = −
GM
r3 S GMm r· r′ dm = − 3 r · r0 r S S
r
rr ′ cos θ dm
with r0 = (x0 , y0 , z0 ) the position of the mass center of S. If the origin of the body frame is taken in the mass center of S, then U (1) = 0. For U (2) , we have U (2) = −
=− =−
GM
′ 2
r GM
S
2r 3 S GM 2r 3
S
r
1
r
2
(3 cos2 θ − 1)dm
(r ′ )2 (3 cos2 θ − 1)dm (r ′ )2 (2 − 3 sin2 θ )dm.
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In a similar form for the U (3) U
(3)
GM
=−
′ 2
r
S
GM
=−
r
1
r
2
5
2r 4
′ 3 r
cos θ (5 cos2 θ − 3)dm
cos3 θ dm − 3
′ 3 r
cos θ dm
S
S
′2 r ′ ( rr cos θ ) dm − 3 rr cos θ dm 2r 4 r3 r S S 5GM 3GM 3 2 =− 7 r · r′ dm + r ′ (r · r′ )dm 2r 2r 5 S S 5GM 3GM =− 7 (xx′ + yy′ + zz ′ )3 dm + (x′2 + y′2 + z ′2 )(xx′ + yy′ + zz ′ )dm 2r 2r 5 S S GM
=−
5
5GM
=−
2r 7
1
′
3
x′3 dm + y3
x3
x′2 z ′ dm + 3xy2
× S
′ ′ ′
×
x y z dm + S
+y
U U
(0) (1)
x′ y′2 dm + 3y2 z
′2 ′
3GM
x′2 y′ dm + 3x2 z
S
y′2 z ′ dm + 3z 2 x
S
2r 5 ′3
x y dm + y S
GM r2
z ′3 dm + 3x2 y
S
S
If g =
y′3 dm + z 3
S
S
S
S
′ ′2
y z dm + z S
y′ z ′2 dm + 6xyz
S
x′ z ′2 dm
x y dm + x S
y dm + y S
′ ′2
x dm + x
x
S
′3
x′ z ′2 dm + 3z 2 y
′2 ′
′2 ′
x z dm + z S
z dm . ′3
y z dm + z S
S
and P is located on the negative part of the axis OZ , we will have r = −rk, being k = (k1 , k2 , k3 ). Therefore
= −mgr = mg (x0 k1 + y0 k2 + z0 k3 )
U (2) = −
3g 2r
(I1 k21 + I2 k22 + I3 k23 ).
Notice U (0) is constant if r is constant and if Jxyz = the form U (3) =
−3g
S
x′ y′ z ′ dm, Jxxz =
S
x′2 z ′ dm, . . . , we will be able to express U (3) in
[(Jxxx + Jxyy + Jxzz )k1 + (Jxxy + Jyyy + Jyzz )k2 + (Jxxz + Jyyz + Jzzz )k3 ]
2r 2 5g
[Jxxx k31 + Jyyy k32 + Jzzz k33 + 3Jxxy k21 k2 2r 2 + 3Jxxz k21 k3 + 3Jyyx k1 k22 + 3Jyyz k22 k3 + 3Jxzz k21 k23 + 3Jyzz k2 k23 + 6Jxyz k1 k2 k3 ]. +
In a similar form we obtain the next U (i) in the series. References [1] J.A. Cavas, A. Vigueras, Stability of certain rotations of a gyrostat analogous to that of Lagrange and Poisson in Newtonian force field, in: Proceedings of the Fourth International Workshop on Positional Astronomy and Celestial Mechanics, 1996, pp. 129–136. [2] J.A. Cavas, A. Vigueras, An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field, Celestial Mech. Dynam. Astronom. 60 (1994) 317–330. [3] J.E. Cochran, P.H. Shu, S.D. Rews, Attitude motion of asymmetric dual-spin spacecraft, J. Guid. Control Dyn. 5 (1982) 37–42. [4] A. Elipe, V. Lanchares, Two equivalent problems: gyrostats in free motion and parametric Hamiltonians, Mech. Res. Commun. 24 (1997) 583–590. [5] J.L.G. Guirao, J.A. Vera, Lagrangian relative equilibria for a gyrostat in the three body problem, J. Phys. A 43 (19) (2010) 1–16. [6] J.L.G. Guirao, J.A. Vera, Sufficient conditions for a nondegenerate Hopf bifurcation in a generalized Lagrange–Poisson problem, J. Math. Phys. 52 (2011) 032701. http://dx.doi.org/10.1063/1.3559064 (p. 11). [7] D.D. Holm, T. Schmah, C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press, 2009. [8] T.J. Kalvouridis, Stationary solutions of a small gyrostat in the Newtonian field of two bodies with equal masses, Nonlinear Dynam. 61 (3) (2010) 373–381. [9] T.J. Kalvouridis, V. Tsogas, Rigid body dynamics in the restricted ring problem of n + 1 bodies, Astrophys. Space Sci. 282 (4) (2002) 751–765. [10] E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer Verlag, Berlin, 1965. [11] J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, in: Texts in Applied Mathematics, vol. 17, Springer Verlag, 1999. [12] A. Marsden, T. Ratiu, A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984) 147–177. [13] J.P. Ortega, T.S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity 12 (3) (1999) 693–720. [14] V.V. Rumiantsev, On the stability of motion of gyrostats, J. Appl. Math. Mech. (1961) 9–19. [15] V.V. Rumiantsev, On the stability of motion of certain types of gyrostats, J. Appl. Math. Mech. (1961) 1158–1169.
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[16] V. Tsogas, T.J. Kalvouridis, A.G. Mavraganis, Equilibrium states of a gyrostat satellite moving in the gravitational field of an annular configuration of n big bodies, Acta Mech. 175 (1–4) (2005) 181–195. [17] J.A. Vera, Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria, Nonlinear Dynam. 55 (1–2) (2009) 191–201. [18] J.A. Vera, A. Vigueras, Hamiltonian dynamics of a gyrostat in the n-body problem: relative equilibria, Celestial Mech. Dynam. Astronom. 94 (3) (2006) 289–315. [19] J.A. Vera, A. Vigueras, Soluciones de equilibrio de un problema generalizado de Lagrange–Poisson: condiciones necesarias y suficientes de estabilidad, Monogr. Real Acad. Cienc. Zaragoza 22 (2003) 141–150. [20] V. Volterra, Sur la théorie des variations des latitudes, Acta Math. 22 (1899) 201–357.