A Hamiltonian Study of the Stability and Bifurcations for the Satellite Problem

Abstract

We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometric framework of Wang et al. Novelties of our work are the use the reduced energy-momentum for the stability analysis and the treatment of axisymmetric bodies. We explicitly show the existence of new relative equilibria and study their stability and bifurcation patterns.

Appendices

Appendix 1: Hamiltonian Relative Equilibria

Let $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle )$ be a Riemannian manifold (the configuration manifold), G a compact Lie group that acts by isometries on Q (the symmetry group) and $V\in C^\infty (Q)$ a G-invariant function (the potential energy). With these data, we can construct a symmetric Hamiltonian system on $T^*Q$, equipped with its canonical symplectic form $\omega =-\mathbf{d}\theta$, in the following way: The potential energy V can be lifted to $T^*Q$, and we will denote this lifted function also by V. The Riemannian metric on Q induces an inner product on each cotangent fiber $T^*_qQ$, $q\in Q$. Then, the Hamiltonian is defined as

$$\begin{aligned} h(p_q)=\frac{1}{2} \Vert p_q\Vert ^2+V(q),\quad p_q\in T_q^*M. \end{aligned}$$

The G-action on Q induces a cotangent-lifted Hamiltonian action on $T^*Q$ with associated equivariant momentum map $\mathbf{J}:T^*M\rightarrow {\mathfrak {g}}^*$ defined by

$$\begin{aligned} \langle \mathbf{J}(p_q),\xi \rangle =\langle p_q,\xi _Q(q)\rangle \quad \forall \,\xi \in {\mathfrak {g}}\end{aligned}$$

where $\xi _Q$ is the fundamental vector field on Q associated with the generator $\xi \in {\mathfrak {g}}$ and is defined by

$$\begin{aligned} \xi _Q(q)=\frac{d}{dt}\bigg |_{t=0}e^{t\xi }\cdot q. \end{aligned}$$

Analogously, $\xi _{T^*Q}$ is the fundamental vector field of the cotangent-lifted action on $T^*Q$. This momentum map is $\text {Ad}^*$-equivariant in the sense that $\mathbf{J}(g\cdot p_q)=\mathrm{Ad}_{g^{-1}}^*(\mathbf{J}(p_q))$ for every $p_q\in T_q^*M$, $g\in G$.

The Hamiltonian h is G-invariant for this lifted action (this follows from the invariance of the metric and of V). Therefore, due to Noether’s theorem, $\mathbf{J}$ is conserved for the Hamiltonian dynamics associated with h. The quadruple $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle ,G,V)$ is called a symmetric simple mechanical system.

Let $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle ,G,V)$ be a symmetric simple mechanical system. A relative equilibrium is a point in phase space $p_q\in T^*Q$ such that its dynamical evolution lies inside a group orbit for the cotangent-lifted action. This amounts to the existence of a generator $\xi \in {\mathfrak {g}}$ such that the evolution of $p_q$ is given by $t\mapsto e^{t\xi }\cdot p_q$. The element $\xi$ is called the angular velocity of the relative equilibrium. A useful characterization of relative equilibria in simple mechanical systems is given by the following result.

Theorem 5.1

(Marsden 1992; Marsden and Ratiu 2002) A point $p_q\in T^*Q$ of a symmetric simple mechanical system $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle ,G,V)$ is a relative equilibrium with velocity $\xi \in {\mathfrak {g}}$ if and only if the following conditions are verified

• 1. $p_q = \langle \! \langle \xi _Q(q) , \cdot \rangle \! \rangle$.

• 2. q is a critical point of ,

where is defined by . Moreover, the momentum $\mu =\mathbf{J}(p_q)\in {\mathfrak {g}}^*$ of the relative equilibrium is given by

That is, any relative equilibrium is determined by a configuration-velocity pair $(q,\xi )\in Q\times {\mathfrak {g}}$ satisfying $\mathbf{d}V_\xi (q)=0$. The map is called the locked inertia tensor, while the function $V_\xi$ is called the augmented potential.

Note that if $(q,\xi )$ is a relative equilibrium pair, by G-invariance of the Hamiltonian vector field $t\mapsto g\cdot (e^{t\xi }\cdot p_q)=e^{t\mathrm{Ad}_g\xi }\cdot (g\cdot p_q)$ is also an integral curve; that is, for each $g\in G$, if $(q,\xi )$ is a relative equilibrium pair, then $(g\cdot q,\mathrm{Ad}_g\xi )$ is also a relative equilibrium pair.

1.1 The Reduced Energy-Momentum Method

The generally adopted notion of stability for relative equilibria of symmetric Hamiltonian systems is that of $G_\mu$-stability, introduced in Patrick (1992), and that we now review in the context of symmetric simple mechanical systems. This notion is closely related to the one of the Lyapunov stability of the induced Hamiltonian system on the reduced phase space.

Definition 5.2

Let $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle ,G,V)$ be a symmetric simple mechanical system and $p_q\in T^*Q$ a relative equilibrium with momentum value $\mu =\mathbf{J}(p_q)$. Let $G_\mu$ be the stabilizer of $\mu \in {\mathfrak {g}}^*$ for the coadjoint action of G in ${\mathfrak {g}}^*$. We say that $p_q$ is $G_\mu$-stable if for every $G_\mu$-invariant neighborhood $U\subset T^*Q$ of the orbit $G_\mu \cdot p_q$ there exists a neighborhood O of $p_q$ such that the Hamiltonian evolution of O lies in U for all time.

In this section, we outline the implementation of the reduced energy-momentum method following Simo et al. (1991). This method is designed for simple mechanical systems since it incorporates all of their distinguishing characteristics with respect to general Hamiltonian systems. Relative equilibria pairs $(q,\xi )$ besides being critical points of the augmented potential can also be described as relative equilibria position-momentum pairs $(q,\mu )\in Q\times {\mathfrak {g}}^*$, where . These pairs $(q,\mu )$ are precisely the critical points of the amended potential

The study of the augmented potential $V_\xi$ can only give rough stability conditions, and a better function to study in order to determine the stability of a relative equilibrium is the amended potential $V_\mu$. From now on, we will need to assume that the isotropy, or stabilizer, $G_q$ of the base point of the relative equilibrium is discrete.

Proposition 5.3

(Patrick 1992) Given a relative equilibrium $p_q$ with momentum $\mu$ and $G_q$ discrete, definiteness of the bilinear form $\mathbf{d}^2 V_\mu (q)\bigg |_{\mathcal {V}}$ where $\mathcal V=\{v\in T_qQ\ |\ \langle \! \langle v,\eta _Q\rangle \! \rangle =0 \quad \forall \eta \in {\mathfrak {g}}_\mu \}$ implies $G_\mu$-stability.

An additional benefit of the reduced energy-momentum method is to further exploit the symmetry properties of $V_\mu$ and decompose the admissible variation space $\mathcal V$ into two subspaces

$$\begin{aligned} \mathcal V= \mathcal {V}_\mathrm{RIG}\oplus \mathcal {V}_\mathrm{INT} \end{aligned}$$

(5.1)

in such a way that $\mathbf{d}^2V_\mu$ block-diagonalizes.

Let ${\mathfrak {g}}_\mu ^\perp \subset {\mathfrak {g}}$ be the orthogonal complement of ${\mathfrak {g}}_\mu$ with respect to . We define

(5.2)

(an interpretation of these spaces can be found in Marsden 1992). The Arnold form $\mathbf{Ar}:{\mathfrak {g}}_\mu ^\perp \times {\mathfrak {g}}_\mu ^\perp \rightarrow \mathbb {R}$ is the bilinear form defined as the restriction of the second variation $\mathbf{d}^2 V_\mu$ to $\mathcal {V}_\mathrm{RIG}\cong {\mathfrak {g}}_\mu ^\perp$. It can be shown that the Arnold form is independent of the potential and can be written as

(5.3)

If the Arnold form is non-degenerate, it can be shown that $\mathcal V=\mathcal {V}_\mathrm{RIG}\oplus \mathcal {V}_\mathrm{INT}$. In that case, sufficient conditions for stability are given by the following result.

Proposition 5.4

(reduced energy-momentum method Simo et al. 1991) Given a relative equilibrium $p_q$ with $G_q$ discrete, then

• If $\dim G < \dim Q$, then positive definiteness of both $\mathbf{Sm}:=\mathbf{d}^2 V_\mu \bigg |_{\mathcal {V}_\mathrm{INT}}$ and $\mathbf{Ar}$ implies $G_\mu$-stability of the relative equilibrium.

• If $\dim G=\dim Q$, then definiteness (positive or negative) of $\mathbf{Ar}$ implies $G_\mu$-stability of the relative equilibrium.

The bilinear form $\mathbf{Sm}:\mathcal {V}_\mathrm{INT}\times \mathcal {V}_\mathrm{INT}\rightarrow \mathbb {R}$ is known in the literature as the Smale form. The reduced energy-momentum method can also be used to study the linearized dynamics near a relative equilibrium, thereby offering also instability conditions. Our main instability result will be the following.

Proposition 5.5

Consider a relative equilibrium $p_q$. If the total number of negative eigenvalues of $\mathbf{Sm}$ and $\mathbf{Ar}$ is odd, then the relative equilibrium is unstable.

Proof

We will combine symplectic eigenvalue analysis and the reduced energy-momentum splitting. The linearized vector field with respect to a symplectic slice (see Patrick 1995) is an infinitesimally symplectic transformation with eigenvalues coming in n-tuples. This implies that if $\lambda =a+bi$ is a simple eigenvalue of this linearized field, then $\{\bar{\lambda },-\lambda ,-\bar{\lambda }\}$ are the companion eigenvalues completing a quadruple. If $a\ne 0$ and $b\ne 0$, all the four eigenvalues are different and their product is $|\lambda |^4$. If $a=0$, then the eigenvalue pair is $\{\lambda ,\bar{\lambda }\}$ and their product is $|\lambda |^2$, but if $b=0$, then the eigenvalue pair is $\{\lambda ,-\lambda \}$ and the product is $-|\lambda |^2$. Therefore, if the linearized vector field has negative determinant, then it must have a real eigenvalue implying instability. Additionally, it can be shown (see Marsden 1992) that the splitting (5.1) block-diagonalizes the symplectic form. Moreover, the sign of the determinant of the linearized field has to be equal to the sign of the determinant of $\mathbf{d}^2 V_\mu \bigg |_{\mathcal {V}}$. Thus, if $\mathbf{d}^2 V_\mu \bigg |_{\mathcal {V}}$ has negative determinant, then the relative equilibrium is unstable. But, the reduced energy-momentum method block-diagonalizes $\mathbf{d}^2 V_\mu \bigg |_{\mathcal {V}}$ into $\mathbf{Sm}$ and $\mathbf{Ar}$ and the hypothesis implies that the determinant of $\mathbf{d}^2 V_\mu \bigg |_{\mathcal {V}}$ being negative is equivalent to the total number of negative eigenvalues of $\mathbf{Sm}$ and $\mathbf{Ar}$ being odd. $\square$

Bifurcations of relative equilibria. By a family of relative equilibria of a symmetric simple mechanical system, we will mean the image of a map $\psi :\mathbb {R}^k\rightarrow T^*Q$ such that for each $x\in \mathbb {R}^k$ the point $\psi (x)$ is a relative equilibria and if $x\ne y$ then $\psi (x)$ and $\psi (y)$ are not G-related.

Combining the reduced energy-momentum and the results of Patrick (1995), it can be shown that if $p_q\in T^*Q$ is a relative equilibrium with discrete stabilizer such that both the Arnold form and the Smale form are non-degenerate, then there is a family of dimension $\dim G_\mu$ of relative equilibria containing $p_q$ and all the relative equilibria near $p_q$ lie in that family.

Two families $\psi _1,\psi _2$ are different if there are not x, y such that $\psi _1(x)$ and $\psi _2(y)$ are G-related. The previous result implies that at a bifurcation point, the Arnold form or the Smale form must be degenerate. Therefore, the points which are candidates for the existence of a bifurcation are those points at which at least one of these forms becomes degenerate. However, in order to decide whether actually a bifurcation happens at one of these points a local study around each candidate by other methods is needed.

Singular Reduced Energy-Momentum Method (case of Q -isotropy only). As the action of $G^\mathrm{axi}$ on the isolated equilibria points of the axisymmetric problem is not locally discrete, we cannot apply the reduced energy-momentum method described before. We will use the singular reduced energy-momentum method developed in Rodríguez-Olmos (2006) to cover that case. However, for our purposes, we will only need a particular version of the method that we will briefly outline. Let $(Q,\langle \! \langle ,\rangle \! \rangle ,G,V)$ be a symmetric simple mechanical system and $p_q\in T^*Q$ a relative equilibrium with ${\mathfrak {g}}_{p_q}=\{0\}$, and ${\mathfrak {g}}_q\ne \{0\}$. Define the subspaces

• ${\mathfrak {t}}\subset {\mathfrak {g}}$ such that ${\mathfrak {g}}={\mathfrak {g}}_q\oplus {\mathfrak {g}}_\mu \oplus {\mathfrak {t}}$ and ${\mathfrak {g}}_\mu \perp {\mathfrak {t}}$ with respect to .

• $\mathbf{S}=({\mathfrak {g}}\cdot q)^\perp \subset T_qQ$ with respect to $\langle \! \langle \cdot ,\cdot \rangle \! \rangle _q$.

• ${\mathfrak {q}}^\mu :=\{\gamma \in {\mathfrak {t}}\mid \langle \mathrm{ad}^*_\gamma \mu ,\xi \rangle =0,\quad \forall \xi \in {\mathfrak {g}}_q\}$

• where $\mathbb {P}:{\mathfrak {g}}^*\rightarrow {\mathfrak {g}}_\mu ^*\oplus {\mathfrak {t}}^*$ is the linear projection and is the directional derivative of with direction $v\in T_qQ$.

• $\Sigma :=\{\xi _Q(q)+a\mid \xi \in {\mathfrak {q}}^\mu ,\quad a\in \mathbf{S}\}$.

With these definitions, the analogue of Proposition 5.3 is

Proposition 5.6

(Theorem 4.1 of Rodríguez-Olmos 2006) Let $p_q\in T^*Q$ be a relative equilibrium with configuration-velocity pair $(q,\xi )\in Q\times {\mathfrak {g}}$. Assume that ${\mathfrak {g}}_{p_q}=\{0\}$ and $\dim (G\cdot q)<\dim (M)$.

If $(\mathbf{d}^2_q V_\xi +\mathrm{corr}_\xi (q))\bigg |_{\Sigma }$ is positive definite, then the relative equilibrium is $G_\mu$-stable.

Appendix 2: Numerical Linearization Around a Relative Equilibrium

Let $p_q\in T^*Q$ be a relative equilibrium of a symmetric simple mechanical system $(Q,\langle \! \langle \cdot ,\cdot \rangle \! \rangle ,G,V)$. Fix a $G_{p_q}$-invariant splitting $\ker T_{p_q}\mathbf{J}=T_{p_q}(G_\mu \cdot p_q)\oplus N$. The subspace N is a symplectic vector subspace of $T_{p_q}(T^*Q)$, usually called a symplectic slice at $p_q$. The linearization of the Hamiltonian system (see Patrick 1995) at the relative equilibrium $p_q$ is given by

$$\begin{aligned} L:=\Omega _N^{-1}\mathbf{d}^2 h_{\xi }\bigg |_{N} \in \mathfrak {sp}(N), \end{aligned}$$

where $\Omega _N$ is the symplectic form of the symplectic slice N. The linearization L provides a simple test for instability: If L has an eigenvalue with positive real part, then $p_q$ is an unstable relative equilibrium.

We will use a numeric approximation to the linearization in two cases where the reduced energy-momentum method was inconclusive or yielded intractable algebraic expression.

1.1 Parallel Equilibria

Fix the point (3.14). Identifying $T_{p_q} (T^*Q)$ with $\mathbb {R}^{6}\times \mathbb {R}^6\cong \mathbb {R}^{12}$, we can choose the following base for the symplectic slice N

$$\begin{aligned} \left\{ \mathbf{e}_4,\mathbf{e}_5,\mathbf{e}_6,\mathbf{e}_{10},\mathbf{e}_{11},\mathbf{e}_{12},\mathbf{e}_3+I_2\xi \mathbf{e}_7,\mathbf{e}_1-I_2\xi \mathbf{e}_9\right\} . \end{aligned}$$

With respect to this basis, the matrix expression of the linearization is

$$\begin{aligned}L=\begin{bmatrix} 0&0&\xi&-{\frac{2 I_{3}-3+9 I_{2}}{2I_{3}}}&0&0&0&{\frac{I_{2}\xi \sqrt{18I_2-6}}{2I_{3}}}\\ 0&0&0&0&-1&0&0&0\\ -\xi&0&0&0&0&- {\frac{2 I_{1}-3+9 I_{2}}{2I_{1}}}&{\frac{I_{2}\xi \sqrt{18I_2-6}}{2I_{1}}}&0\\ \frac{8}{3} {\frac{I_{1}-I_{2}}{(18I_2-6)^\frac{5}{2}}}&0&0&0&0&\xi&0&0\\ 0&\frac{8}{3(18I_2-6)^{\frac{3}{2}}}&0&0&0&0&0&0\\ 0&0&-\frac{8}{3} {\frac{-I_{3}+I_{2}}{(18I_2-6)^\frac{5}{2}}}&-\xi&0&0&0&0\\ 0&0&0&- {\frac{\sqrt{18I_2-6}}{2I_{3}}}&0&0&0&{\frac{\xi \left( -I_{3} +I_{2} \right) }{I_{3}}}\\ 0&0&0&0&0&{\frac{\sqrt{18I_2-6}}{2I_{1}}}&{\frac{\xi \left( I_{1}-I_{2} \right) }{I_{1}}}&0 \end{bmatrix}. \end{aligned}$$

Using a computer algebra system, one can compute the eigenvalues of this matrix for any value of the parameters $I_1,I_2,I_3,\xi$. For example, for $I_1=0.2$, $I_2=0.35$, $I_3=0.45$, and $\xi =0.5$, L has two real eigenvalues with positive real part. Therefore, for those values, the parallel equilibrium is unstable. We believe that this is true for all values of the parameters, but we do not have a formal proof of this statement.

1.2 Asymmetric Conical Equilibria

In this case, the computations are more involved. As a concrete case, we fix the conical relative equilibrium given by the point $\mathbf{R}=(0.16,0.11,0)$ with angular velocity $\xi =(-3.61,-48.03,0)$. Using a computer algebra system, we can check that at that point we can choose as a basis for N the columns of the following matrix

$$\begin{aligned} \begin{bmatrix} 0&0&0&0&0&0&0&0 \\ 1&0&0&0&0&0&0&0 \\ 0&1&0&0&0&0&0&0 \\ 0&-18.4&-0.&-22.0&0&0&0&-.0595 \\ 0&.34&22.0&0&0&0&-0.&.0803 \\ -.34&0&0&0&0&.0595&-.0803&0 \\ 0&0&1&0&0&0&0&0 \\ 0&0&0&1&0&0&0&0 \\ 0&0&0&0&1&0&0&0 \\ 0&0&0&0&0&1&0&0 \\ 0&0&0&0&0&0&1&0 \\ 0&0&0&0&0&0&0&1 \end{bmatrix}. \end{aligned}$$

Then, the matrix expression for the linearization is

$$\begin{aligned} L= \begin{bmatrix} 0&-1.18\cdot 10^{4}&3.67\cdot 10^{2}&3.70\cdot 10^{3}&0&0&0&1.13\cdot 10^{1} \\ 4.61&0&0&0&0&9.60\cdot 10^{-2}&-1.30\cdot 10^{-1}&0 \\ -3.29\cdot 10^{-2}&0&0&0&2.78\cdot 10^{2}&1.01&-7.71\cdot 10^{-3}&0 \\ 4.44\cdot 10^{-2}&0&0&0&-5.17&-7.71\cdot 10^{-3}&1.01&0 \\ 0&3.89&-2.48\cdot 10^{2}&9.26&0&0&0&1.12 \\ 0&-1.26\cdot 10^{2}&2.49\cdot 10^{5}&6.06\cdot 10^{4}&0&0&0&2.48\cdot 10^{2} \\ 0&-1.27\cdot 10^{3}&6.06\cdot 10^{4}&-9.29\cdot 10^{4}&0&0&0&-9.26 \\ 0&0&0&0&-1.66\cdot 10^{5}&-2.78\cdot 10^{2}&5.17&0 \end{bmatrix}. \end{aligned}$$

This matrix has one real eigenvalue with positive real part, and hence, the relative equilibrium is unstable.