Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

Abstract

We study solutions of the Newtonian n-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t\rightarrow +\infty$ or as $t\rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold “at infinity”. We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy’s classical asymptotic formulas. We also address the scattering problem, namely: for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering “near infinity”, that is, when the bodies stay far apart and interact only weakly.

Appendix A. Analytic Linearizations

Here we prove the analytic linearization theorem, Theorem 3.1 as a corollary to

Theorem A.1

Let X be a real analytic vector field defined on an open subset of ${\mathbb {V}}= {\mathbb {R}}^m \times {\mathbb {R}}^k$ and vanishing on ${\mathcal {E}}= U \times 0$ where U is an open subset of ${\mathbb {R}}^m$. Suppose that the linearization $L(p) = DX_p: {\mathbb {V}}\rightarrow {\mathbb {V}}$ of X at each $p \in {\mathcal {E}}$ has exactly one nonzero eigenvalue $\lambda (p)$ (necessarily real since X is real) of algebraic multiplicity k. Set $N_p = L(p) ({\mathbb {V}}) \subset {\mathbb {V}}$ and write $N \rightarrow {\mathcal {E}}$ for the rank k analytic vector bundle over ${\mathcal {E}}$ whose fiber over p is $N_p$. Let $X_N: N \rightarrow N$ be the fiber-linear vector field given by $X_N (p, v) = (0, L(p) v)$. Then there is an analytic diffeomorphism $\Phi$ such that $\Phi ^* X = X_N$ where $\Phi$ maps a neighborhood of the zero section in N to a neighborhood of ${\mathcal {E}}$ in ${\mathbb {V}}$. Finally, if we define $i: N \rightarrow {\mathbb {V}}$ by $i(p,v) = p+ v$, then, upon restriction to a neighborhood of the zero section, i is an analytic diffeomorphism onto a neighborhood of ${\mathcal {E}}$ and $\Phi$ agrees with i to 1st order along the zero section.

Remark A.1

The kernel of L(p) equals ${\mathbb {R}}^m \times 0$. This fact follows from the assumptions that U is open and that the multiplicity of $\lambda (p)$ is k. $N_p$ is a complementary subspace to ${\mathbb {R}}^m \times 0$ but need not be equal to $0 \times {\mathbb {R}}^k$. L(p) restricted to $N_p$ is a rank k linear map. It is essential for our application, namely theorem  3.1, that this map be allowed to have nonzero Jordan blocks.

Let us see see how theorem 3.1 follows from theorem A.1.

Proof of theorem 3.1

Let $X^{newt}$ be our N-body vector field, extended to infinity, as per (6). We saw in the discussion of the linearization structure, immediately following (11) and also remark 3.3, that $X^{newt}$ is the restriction to a codimension 2 analytic subvariety of an analytic vector field X defined on an open subset of ${\mathbb {R}}\times {\mathbb {E}}\times {\mathbb {R}}\times {\mathbb {E}}= {\mathbb {V}}$ and whose variables we wrote as $(\rho , s, v, w)$ in eq (6). Rearranging these coordinates in the order $(s, v, \rho , w)$, viewing $(s,v) \in {\mathbb {R}}^m, (\rho , w) \in {\mathbb {R}}^k$ (with $k = m$), and referring to the discussion of the linearization structure of X following (11), we see that X satisfies the hypothesis of Theorem A.1 with $U = \{(s,v) \in {\mathbb {E}}\times {\mathbb {R}}: s \notin \Delta , v \ne 0 \}$, and with $\lambda (s, v) = -v$. Theorem A.1 now supplies the analytic conjugation to $X_N$ for X. But theorem 3.1 claims the analytic conjugation for $X^{newt}$ not for X. To finish off, observe that the extended phase space P on which $X^{newt}$ lives is invariant under the flow of X, and hence $\Phi ^{-1} (P)$ is invariant under the flow of $X_N$. Restricting $X_N$ and $\Phi$ to $\Phi ^{-1}(P)$ yields the claimed conjugacy. $\quad \square$

Proof of A.1

Decompose ${\mathcal {E}}$ into ${\mathcal {E}}_+$ and ${\mathcal {E}}_-$ with ${\mathcal {E}}_- = \{ p \in {\mathcal {E}}: \lambda (p) < 0 \}$ and ${\mathcal {E}}_+ = \{ p \in {\mathcal {E}}: \lambda (p) > 0 \}$. (The $+$ and − subscripts are for forward-time attracting and backward time attracting.) We give the proof for the open component ${\mathcal {E}}_+$ of ${\mathcal {E}}$. The proof for ${\mathcal {E}}_-$ follows a symmetrical proof with stable manifold replaced by unstable and $t \rightarrow + \infty$ by $t \rightarrow - \infty$.

The proof proceeds in three steps. Step 1 Straighten out the local stable manifolds so they are aligned with the fibers of N. Step 2 Work within each fiber $N_p$ and linearize the vector field there insuring that the linearization is also analytic in p. Step 3 Use uniqueness of the linearization from step 1 and 2 to insure that the linearizing transformation is defined in a whole neighborhood of ${\mathcal {E}}_+$.

Step 1 [Straightening out the stable manifolds.] The generalized eigenspace for $\lambda (p)$ is $N_p = L(p)( {\mathbb {V}})$ and must be the tangent space at p to the local stable manifold $W(p): = W^s _{loc} (p)$ passing through $p \in {\mathcal {E}}_+$. Now ${\mathbb {V}}= {\mathbb {R}}^m \oplus N_p$. According to the usual stable manifold theorem, W(p) is the graph of a smooth function $\psi _p: (N_p,0) \rightarrow {\mathbb {R}}^m$ with $\psi _p (0) = p$ and $d \psi _p (0) = 0$. Here, and in the remainder of this section, the broken arrow notation is used to indicate that the domain of the function is an open neighborhood of p and not all of $N_p$.

We need to upgrade $v \rightarrow \psi _p (v)$ so as to be analytic not just smooth, and analytic in both v and p. To this end, look at [6, Chapter 13, Theorem 4.1, also Exercise 4.11], where this work is essentially done. The Perron-style proof of the stable manifold theorem there proceeds by constructing a nonlinear integral operator $F= F_{p, v}$ acting on paths. The path space it acts on is the space of smooth paths $\gamma : [0, \infty ) \rightarrow {\mathbb {V}}$ which tend to p as $t \rightarrow \infty$ and have initial condition $\gamma (0)$ projecting onto $v \in N_p$. One proves that for |v| small F is a contraction mapping. Its unique fixed point $\gamma (t; p, v)$ lies in W(p). One has, by definition, that $\gamma (0; p, v) = p + \psi _p (v) + v$ - yielding the map $\psi _p: {\mathbb {V}}\rightarrow {\mathbb {R}}^m$. To get $\psi$ analytic in both p and v, complexify both vectors so that $p \in {\mathbb {C}}^m$ , $v \in {\mathbb {N}}_p ^{{\mathbb {C}}} = N_p \otimes {\mathbb {C}}$, and $p + v \in {\mathbb {V}}^{{\mathbb {C}}}$. Write $Re(p), Im(p) \in {\mathbb {R}}^m, Re(v), Im(v) \in N_p$ for their real and imaginary parts. One verifies that all the properties of F are retained in this complexified setting provided $|Im(p)| + |Im(v)| < \delta$ is sufficiently small. One also verifies that the iteration scheme $\gamma ^j (t; p, v) = F_{(p, v)} ^j (\gamma _0 (t)) \in {\mathbb {V}}^{{\mathbb {C}}}$ satisfies uniform $C^0$ bounds in this thickened neighborhood with the $\gamma ^j$ analytic in t, p, v at each step. Since the uniform limit of complex analytic functions is complex analytic, we get that the endpoints, the $\gamma (0; p, v) = p + \psi _p (v)+ v$ are complex analytic in p, v in this thickened strip, so, upon restriction to the real parts, are analytic. We have our analytic stable manifold, depending analytically on p.

Now write $\phi _p (v) = p + v + \psi _p (v)$ for $v \in N_p$. Because W(p) is the graph of $\psi _p$ we have that $\phi _p$ maps a neighborhood of 0 in $N_p$ to a neighborhood of p in W(p). Then $\Phi : N - \rightarrow {\mathbb {V}}$ by $\Phi (p, v) = \phi _p (v)$ is our desired analytic straightening diffeomorphism, with domain a neighborhood of the zero section of N. Recall that $N_p$ is tangent to W(p) at p, and that $\psi _p (0) = 0, d \psi _p (0) = 0$ to see that the derivative of $\Phi$ along the zero section (p, 0) of N can be identified with the “identity” $(h, v) \mapsto h+v$ from ${\mathbb {R}}^m \oplus N_p \rightarrow {\mathbb {V}}$. In other words, the derivative is invertible and $\Phi$ agrees with i to first order along the zero section. ( By the inverse function theorem, $\Phi$ and i are invertible in a neighborhood of the zero section.) Now the vector field X is everywhere tangent to the foliation of a neighborhood of ${\mathcal {E}}_+$ by the stable manifolds, the W(p)’s, so $\Phi ^* X$ must be everywhere tangent to the inverse image of this foliation by $\Phi$, which is to say, tangent to the fibers of $N \rightarrow {\mathcal {E}}_+$. This means that $\Phi ^* X$ is a vertical vector field. Finally, due to the nature of the linearization of $\Phi$ along the zero section, the linearization of $\Phi ^* X$ and of X both agree along the zero section, which means that

$$\begin{aligned} \Phi ^* X (p, v) = (0 , L(p) v + g(p, v)), g(p, v) = O(|v|^2), \end{aligned}$$

with g analytic. $\quad \square$

Step 2 (Linearizing fiber-by-fiber). Invoke the theorem of Brushlinskaya.

Theorem A.2

(Brushlinskaya [4]) Consider a family of analytic vector fields $X_p$ on ${\mathbb {R}}^k$, analytically depending on the parameters $p\in \Omega \subset {\mathbb {R}}^m$, $\Omega$ open. Assume that at 0 and $p=p_0 \in \Omega$ there is an equilibrium with eigenvalues whose real parts all have the same sign. Then, with the aid of a near identity transformation $\Phi$, analytic in a neighborhood of 0 and depending analytically on the parameters p within a small neighborhood of $p_0$, one can reduce the family $X_p$ to “resonant polynomial form”: ${{\hat{X}}}_p = \Phi ^* X_p$ is a family of polynomial vector fields whose coefficients depend analytically on the parameters p and for which the only monomials occuring in these polynomials are the resonant monomials in the sense of the Poincaré–Dulac theory.

To use this theorem in our situation we will need to understand the terms “resonant monomials” and “near-identity”. Fix p. Use the notation $v^Q = v_1^{q_1}\ldots v_k^{q_k}, Q=(q_1,\ldots ,q_k)\in {\mathbb {N}}^k$ to describe monomials on ${\mathbb {R}}^k$. Let $\lambda = (\lambda _1,\ldots ,\lambda _k)$ be the eigenvalues of the linearization of $X_p$ at 0.

Definition A.1

A resonant monomial $v^Q$ for $X_p$ is a monomial of degree 2 or higher for which $Q\cdot \lambda = \lambda _i$ for some $i = 1,\ldots ,k$.

Observe “degree 2” or higher is equivalent to $|Q| : = q_1+\cdots + q_k \ge 2$.

Definition A.2

A near-identity analytic transformation $\phi$ of ${\mathbb {R}}^k$ is an analytic transformation defined near 0 and having convergent power series expansion $v \mapsto v + \phi _2 (v) + \phi _3 (v) + \ldots$, with the $\phi _i (v)$ being homogeneous vector-valued polynomials of degree i.

In the Poincaré–Dulac method, as explained for example in [2], one tries to successively kill all the monomials of degree 2 or higher arising in $X_p$ by appropriately choosing the $q_i$ of $\phi$. All non-resonant monomials can be killed. The non-resonant ones, being in the kernel of the “cohomological operator” associated to the process remain. (See step 3 below for this cohomological operator.) The essence of the Brushlinskaya theorem then is that this process leads to a convergent power series for the map $\phi$ with coefficents depending analytically on p.

What does that mean for our specific family? We have that the vector of eigenvectors $\lambda$ is $\lambda = \lambda (p)(1, 1, \ldots , 1)$ at each point p. The resonance condition of definition A.1 then reads $|Q| \lambda (p) = \lambda (p)$ which is impossible since $|Q| \ge 2$. There are no resonant monomials! The normal form ${\hat{X}}_p$ is simply the linearization $v \mapsto L(p) v$ of ${\tilde{X}}_p$ for each p. We now have that X is locally analytically conjugate to its linearizations $X_N$ in neighborhoods $U_p$ of each $p \in {\mathcal {E}}_+$.

Step 3 [Patching together] We must make sure that all these local analytic conjugacies guaranteed by the last two steps piece together to form a single analytic conjugacy defined in a neighbhorhood of all of ${\mathcal {E}}_+$.

To this end, suppose that $\Psi _1, \Psi _2$ are two linearizations, each defined on its own open set ${\mathcal {U}}_i \subset N$, this open set containing neighborhoods $U_i \subset {\mathcal {E}}$. For simplicity we will say “$\Psi _i$ is defined over $U_i$” to encode this information. Then $\Psi _1 ^* X = X_N$ holds over $U_1$ while $\Psi _2 ^* X = X_N$ holds over $U_2$. The map $\Phi = \Psi _2 ^{-1} \circ \Psi _1$ is then a near identity transformation defined over $U_1 \cap U_2$ and satisfying $\Phi ^* X_N = X_N$ over $U_1 \cap U_2$. $\Phi$ must preserve the fibers of $N \rightarrow {\mathcal {E}}_+$, for otherwise upon pull-back it would add “tangential terms” to $X_N$. Thus we have $\Phi (p,v) = (p, v + h(p, v)))$. Because $\Psi _1$ and $\Psi _2$ are both near-identity, so is $\Phi$ which means that when expanded as a power series h(p, v) consists entirely of terms quadratic and higher in v. Write out its convergent Taylor series

$$\begin{aligned} h=h_2+h_3+\cdots , \end{aligned}$$

with $h_i (p, \cdot ) : N_p \rightarrow N_p$ a homogeneous degree i vector-valued polynomial. At the heart of the Poincaré–Dulac method is the fact that $\Phi ^* X_N = X_N$ is equivalent to the (infinite) system of “cohomological equations” $[L,h_i]= 0$. See [2, Chapter 5] for details. However the kernel of this cohomological operator $h \mapsto [h, L]$ is precisely the linear span of the resonant monomials. (To be precise, what we mean by a ‘monomial’ h is any h of the form $h(v) = v^Q e_i$ with $e_i$ one of the standard basis elements for ${\mathbb {R}}^k$.) We have seen that in our case there are no resonant monomials: this kernel is zero. Thus $h_i = 0$, $i =2,3,\ldots$ and hence $\Phi = Id$ so that $\Psi _1 = \Psi _2$ over $U_1 \cap U_2$. Thus there is a single $\Psi : N - \rightarrow {\mathbb {V}}$, defined over ${\mathcal {E}}_+$, whose restrictions to the various open sets $U_p \subset {\mathcal {E}}$ as per step 2 are the analytic near-identity diffeomorphisms guaranteed by Brushlinskaya.