Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$-Body Problem via Index Theory

Abstract

It is well known that a planar central configuration of the $n$-body problem gives rise to a solution where each particle moves in a Keplerian orbit with a common eccentricity $\mathfrak{e}\in[0,1)$. We call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it is natural to regard it as a planar $n$-body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial $n$-body problem (i. e., each particle belongs to $\mathbb{R}^{3}$). In this paper, as a spatial $n$-body problem, we first decompose the linear system of ERE into two parts, the planar and the spatial part. Following the Meyer – Schmidt coordinate [19], we give an expression for the spatial part and further obtain a rigorous analytical method to study the linear stability of the spatial part by the Maslov-type index theory. As an application, we obtain stability results for some classical ERE, including the elliptic Lagrangian solution, the Euler solution and the $1+n$-gon solution.

APPENDIX

We start this subsection with a brief review of the Maslov index theory [2, 6]. A Lagrangian subspace $V$ is an $n$-dimensional subspace with $V|_{\Omega}=0$. We let $\mathrm{Lag}(2n)$ denote the Lagrangian Grassmanian, i. e., the set of Lagrangian subspaces of $(\mathbb{R}^{2n},\Omega)$. For two continuous paths $L_{1}(t)$ and $L_{2}(t)$, $t\in[a,b]$ in $\mathrm{Lag}(2n)$, the Maslov index $\mu(L_{1},L_{2})$ is an integer invariant. Here we use the definition from [6]. We list several properties of the Maslov index. Details can be found in [6].

Property I (nullity). For $L_{1},L_{2}\in C\big{(}[a,b],\mathrm{Lag}(2n)\big{)}$ such that $\dim L_{1}(t)\cap L_{2}(t)$ is independent of $t$, then

$$\mu\big{(}L_{1}(t),L_{2}(t)\big{)}=0.$$

Property II (homotopy invariant). For two continuous families of the Lagrangian path $L_{1}(s,t)$, $L_{2}(s,t)$, $0\leqslant s\leqslant 1$, $a(s)\leqslant t\leqslant b(s)$, and satisfies $\dim L_{1}\big{(}s,a(s)\big{)}\cap L_{2}\big{(}s,a(s)\big{)}$ and $\dim L_{1}\big{(}s,b(s)\big{)}\cap L_{2}\big{(}s,b(s)\big{)}$ is constant, then

$$\mu\big{(}L_{1}(0,t),L_{2}(0,t)\big{)}=\mu\big{(}L_{1}(1,t),L_{2}(1,t)\big{)}.$$

Property III (path additivity). If $a<c<b$, then

$$\mu\big{(}L_{1}(t),L_{2}(t)\big{)}=\mu(L_{1}(t),L_{2}(t)|_{[a,c]})+\mu(L_{1}(t),L_{2}(t)|_{[c,b]}).$$

Property IV (symplectic invariance) Let $\gamma(t)$, $t\in[a,b]$ be a continuous path in $\mathrm{Sp}(2n)$, then

$$\mu\big{(}L_{1}(t),L_{2}(t)\big{)}=\mu\big{(}\gamma(t)L_{1}(t),\gamma(t)L_{2}(t)\big{)}.$$

Property V (symplectic additivity). Let $L_{1},L_{2}\in C\big{(}[a,b],\mathrm{Lag}(W_{1})\big{)}$, $\hat{L}_{1},\hat{L}_{2}\in C\big{(}[a,b], \mathrm{Lag}(W_{2})\big{)}$, where $W_{i}$, $i=1,2$ are symplectic spaces, then

$$\mu\big{(}L_{1}(t)\oplus\hat{L}_{1}(t),L_{2}(t)\oplus\hat{L}_{2}(t)\big{)}=\mu\big{(}L_{1}(t),L_{2}(t)\big{)}+\mu\big{(}\hat{L}_{1}(t),\hat{L}_{2}(t)\big{)}.$$

The Maslov index is easily explained for $n=1$. Obviously, $\mathrm{Lag}(2)$ consists of a straight line passing through the origin. We define a map $\mathcal{F}:\mathrm{Lag}(2)\to\mathbb{U}$ as follows. For $V=\mathbb{R}\big{(}\cos(\theta),\sin(\theta)\big{)}^{T}$, we let $\mathcal{F}(V)=\exp(2i\theta)$. For two paths $V_{0},V_{1}\in C\big{(}[a,b],\mathrm{Lag}(2)\big{)}$, then $\mathcal{F}(V_{0})^{-1}\mathcal{F}(V_{1})(t)\in\mathbb{U}$. Let $x(t)$ be a continuous path which satisfies

$$\exp\big{(}2\pi ix(t)\big{)}=\mathcal{F}(V_{0})^{-1}\mathcal{F}(V_{1})(t),t\in[a,b].$$

Set $[x]:=\sup\{z\in\mathbb{Z},z\leqslant x\}$. The Maslov index

$$\mu(V_{0},V_{1};[a,b])=[x(b)-\epsilon]-[x(a)-\epsilon]$$

for $\epsilon$ small enough.

In order to study the case for $\mathfrak{e}\to 1$, we need the index theory for heteroclinic orbits [11, 13]. We consider the Hamiltonian flow induced by

$$\dot{z}=JB(\tau)z,\ \ \tau\in\mathbb{R}.$$

(A.1)

We assume

(H1) $JB(\pm\infty)=\lim_{\tau\rightarrow\pm\infty}JB(\tau)$ is hyperbolic, that is, there are no eigenvalues of $JB(\pm\infty)$ on the imaginary line. It follows that $\mathbb{R}^{2n}=V^{\pm}_{s}\oplus V^{\pm}_{u}$, where $V^{\pm}_{s}(V^{\pm}_{u})$ is the stable subspace(resp. unstable subspace) of the equilibria which is spanned by the generalized eigenvector of the eigenvalue with the negative real part (positive real part) of $JB(\pm\infty)$. Both the stable subspace $V^{\pm}_{s}$ and the unstable subspace $V^{\pm}_{u}$ are Lagrangian subspaces of $(\mathbb{R}^{2n},\Omega)$.

Under the condition of (H1), $\mathcal{A}:=-J\frac{d}{d\tau}-B$ is a Fredholm operator on $L^{2}(\mathbb{R},\mathbb{R}^{2n})$ with domain $W^{1,2}(\mathbb{R},\mathbb{R}^{2n})$. In some cases, we need to consider the solution $z$ of (A.1) defined on $(-\infty,0]$ and satisfying $\lim_{\tau\to-\infty}z(\tau)=0$, $z(0)\in\Lambda$ for $\Lambda\in \mathrm{Lag}(2n)$. Let

$$D(\Lambda,\mathbb{R}^{-}):=\{x\in W^{1,2}(\mathbb{R}^{-},\mathbb{R}^{2n}),x(0)\in\Lambda\},$$

then $\mathcal{A}|_{D(\Lambda,\mathbb{R}^{-})}$ is a Fredholm self-adjoint operator on $L^{2}(\mathbb{R},\mathbb{R}^{2n})$.

Let $\gamma(t,\tau)$ satisfy (A.1) with $\gamma(\tau,\tau)=I_{2n}$. In what follows we set $\gamma(t):=\gamma(t,0)$. Clearly, $\gamma$ satisfies a semigroup property, that is, $\gamma(t,\nu)\gamma(\nu,\tau)=\gamma(t,\tau)$. For $\tau\in\mathbb{R},$ define

$$V_{s}(\tau)=\{\zeta\ \ |\ \ \zeta\in\mathbb{R}^{2n}\ \ \text{and}\ \ \lim_{t\rightarrow\infty}\gamma(t,\tau)\zeta=0\},\quad V_{u}(\tau)=\{\zeta\ \ |\ \ \zeta\in\mathbb{R}^{2n}\ \ \text{and}\ \ \lim_{t\rightarrow-\infty}\gamma(t,\tau)\zeta=0\}.$$

We remark that

$$\lim_{\nu\rightarrow\infty}V_{s}(\tau)=V^{+}_{s}\ \ \text{and}\ \ \lim_{\nu\rightarrow-\infty}V_{u}(\tau)=V^{-}_{u}.$$

It is well known that both $V_{s}(\tau)$ and $V_{u}(\tau)$ are Lagrangian subspaces of $(\mathbb{R}^{2n},\Omega)$. An important property from [1] is the following: if $V$ is transversal to $V_{s}(0)$, then

$$\lim_{t\rightarrow+\infty}\gamma(t,0)V=V^{+}_{u}.$$

Similarly, if is $V$ transversal to $V_{u}(0)$, then

$$\lim_{t\rightarrow-\infty}\gamma(t,0)V=V^{-}_{s}.$$

For $\Lambda\in \mathrm{Lag}(2n)$, $V_{u}$ is the unstable path of (A.1), the index for the past half-clinic orbit is defined by

$$\iota_{-}(\Lambda,B)=\mu(\Lambda,V_{u}(\tau),\tau\in(-\infty,0]),$$

which is well defined since $V_{u}(-\infty)$ exist. Note that

$$\mu(\Lambda,V_{u}(\tau),\tau\in(-\infty,0])=-\mu\big{(}\Lambda,V_{u}(-\tau);\tau\in[0,+\infty)\big{)},$$

so $\iota_{-}(\Lambda,B)$ is just the definition of the geometrical index given in [13]. We also denote $\nu(\mathcal{A})=\dim\ker(\mathcal{A}),$ then

$$\nu(\mathcal{A})=\dim V_{s}(0)\cap V_{u}(0)\leqslant n.$$

Similarly,

$$\nu(\mathcal{A}|_{D(\Lambda,\mathbb{R}^{-})})=\dim\Lambda\cap V_{u}(0)\leqslant n.$$

Now we consider the family of the linear Hamiltonian system

$$\dot{z}=JB_{\lambda}(\tau)z\quad\tau\in\mathbb{R}\quad\lambda\in[a,b].$$

We assume that $B_{\lambda}(\tau)$ converges to $B_{\lambda}(\pm\infty)$ uniformly with respect to $\lambda$, and that $JB_{\lambda}(\pm\infty)$ is hyperbolic. It is well known that $\mathcal{A}_{\lambda}:=-J\frac{d}{d\tau}-B_{\lambda}$ and $\mathcal{A}_{\lambda}|_{D(\Lambda,\mathbb{R}^{-})}$ is a continuous path of Fredholm operators, so its spectral flow is well defined. The spectral flow was first introduced by Atiyah, Patodi and Singer [5] in their study of index theory on manifolds with boundary. Let $\{A(\lambda),\lambda\in[0,1]\}$ be a continuous path of self-adjoint Fredholm operators on a Hilbert space $E$. Roughly speaking, the spectral flow of path $\{A(\lambda),\lambda\in[0,1]\}$ counts the net change in the number of negative eigenvalues of $A(\lambda)$ as $\lambda$ goes from $0$ to $1$, where the enumeration follows from the rule that each negative eigenvalue crossing the positive axis contributes $+1$ and each positive eigenvalue crossing the negative axis contributes $-1$, and for each crossing the multiplicity of the eigenvalue is counted. We refer the reader to [13] for a brief introduction.

From [13], we have the index theorem

Theorem 8

Under the above notation,

$$-sf(\mathcal{A}_{\lambda}|_{D(\Lambda,\mathbb{R}^{-})})=\iota_{-}(\Lambda,B_{b})-\iota_{-}(\Lambda,B_{a})+\mu(\Lambda,V^{\lambda}_{u}(-\infty);[a,b]).$$

(A.2)

The Maslov-type index for a symplectic path is a very useful tool for the study of the stability problem of periodic orbits. In this section, we give only a brief review. Details can be found in [17]. For $\tau>0$ we are interested in paths in $\mathrm{Sp}(2n)$:

$$\mathcal{P}_{\tau}(2n)=\left\{\gamma\in C\big{(}[0,\tau],\operatorname{Sp}(2n)\big{)}\mid\gamma(0)=I_{2n}\right\}.$$

For any $\omega\in\mathbb{U}$(the unit circle in $\mathbb{C}$), the following $\omega$-degenerate hypersurface of codimension one in $\mathrm{Sp}(2n)$ is defined [17]:

$$\mathrm{Sp}(2n)_{\omega}^{0}=\{M\in \mathrm{Sp}(2n)|\det(M-\omega I_{2n})=0\}.$$

Moreover, the $\omega$-regular set of $\mathrm{Sp}(2n)$ is defined by $\mathrm{Sp}(2n)^{*}_{2n}=\mathrm{Sp}(2n)\setminus \mathrm{Sp}(2n)_{\omega}^{0}$.

For $M\in \mathrm{Sp}(2n)_{\omega}^{0}$, we define a coorientation of $\mathrm{Sp}(2n)_{\omega}^{0}$ at $M$ by the positive direction $\frac{d}{dt}Me^{tJ}|_{t=0}$ of the path $Me^{tJ}$ with $|t|$ sufficiently small. Now we will give a definition of the $\omega$-index [17].

Definition 1

For $\omega\in\mathbb{U}$, $\gamma\in\mathcal{P}_{\tau}$, the $\omega$-index of $\gamma$ can be defined as

$$i_{\omega}(\gamma)=\begin{cases}[e^{-\epsilon J}\gamma:\mathrm{Sp}(2n)_{\omega}^{0}]-n&\text{ if }\omega=1\\ [e^{-\epsilon J}\gamma:\mathrm{Sp}(2n)_{\omega}^{0}] &\text{ if }\omega\neq 1.\end{cases}$$

For $\epsilon>0$ small enough, where $[\cdot:\cdot]$ is the intersection number, we also denote

$$\nu_{\omega}(\gamma)=\dim_{\mathbb{C}}\ker_{\mathbb{C}}(\gamma(\tau)-\omega I).$$

The Maslov-type index satisfies the homotopy invariant property, that is, for a family of $\gamma_{s}\in\mathcal{P}_{\tau}(2n)$ continuously depending on $s\in[0,1]$, we have that, if $\nu_{\omega}(\gamma_{s})$ is constant, then $i_{\omega}(\gamma_{0})=i_{\omega}(\gamma_{1})$. Note that, for $M\in \mathrm{Sp}(2n)$, $\mathrm{Gr}(M):=\{(x,Mx)|x\in\mathbb{R}^{2n}\}$ is a Lagrangian subspace of the symplectic vector space $(\mathbb{R}^{2n}\oplus\mathbb{R}^{2n},-\Omega\oplus\Omega)$. For $\gamma\in\mathcal{P}_{\tau}(2n)$, we have

$$i_{1}(\gamma)+n=\mu \Big{(}\mathrm{Gr}(I),\mathrm{Gr}\big{(}\gamma(t)\big{)}\Big{)},$$

and

$$i_{-1}(\gamma)=\mu\Big{(}\mathrm{Gr}(-I),\mathrm{Gr}\big{(}\gamma(t)\big{)}\Big{)}.$$

The Maslov-type index is essentially equal to the Morse index. Consider a second-order system $\ddot{x}=D(t)x$, $t\in[0,\tau]$. Let $A=-\frac{d^{2}}{dt^{2}}+D$, which is a self-adjoint operator on $L^{2}([0,\tau],\mathbb{C}^{n})$ with domain

$$\bar{D}(\omega,\tau)=\{W^{2,2}([0,\tau],\mathbb{C}^{n})|y(\tau)=\omega y(0),\dot{y}(\tau)=\omega\dot{y}(0)\}.$$

We set $v_{\omega}(A)=\dim\ker(A)$ and denote by $\phi_{\omega}(A)$ its Morse index, then from [17], we have the fact

$$\phi_{\omega}(A)=i_{\omega}(\gamma),\ \ v_{\omega}(A)=\nu_{\omega}(\gamma),$$

(A.3)

where $\gamma$ is the fundamental solution of the corresponding Hamiltonian systems, that is, $\dot{\gamma}=J\left(\begin{array}[]{cc}I_{n}&0_{n}\\ 0_{n}&-D(t)\end{array}\right)\gamma,$ with $\gamma(0)=I$. For a formula for general boundary conditions, see [15].

For $M_{0},M_{1}\in \mathrm{Sp}(2n)$, we write $M_{0}\approx M_{1}$ if they are symplectically similar. For $M\in \mathrm{Sp}(2)$, it is symplectically similar to one of the following normal forms:

$$\begin{array}[]{c}D(\lambda)=\left(\begin{array}[]{cc}\lambda&0\\ 0&\lambda^{-1}\end{array}\right),\quad\lambda\in\mathbb{R}\setminus\{0\}\\ N_{1}(\lambda,a)=\left(\begin{array}[]{cc}\lambda&a\\ 0&\lambda\end{array}\right),\quad\lambda=\pm 1,a=\pm 1,0\\ R(\theta)=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right),\quad\theta\in(0,\pi)\cup(\pi,2\pi).\end{array}$$

For $\gamma\in\mathcal{P}_{\tau}(2)$, Y. Long has given a nice picture to explain the index [17]. In this case, we can get the normal form of $\gamma(\tau)$ by its index $i_{\omega}(\gamma)$.