Compactification of the Energy Surfaces for $n$ Bodies

Abstract

For $n$ bodies moving in Euclidean $d$-space under the influence of a homogeneous pair interaction we compactify every center of mass energy surface, obtaining a $\big{(}2d(n-1)-1\big{)}$-dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and nontrivial on the boundary.

APPENDIX. MANIFOLDS WITH CORNERS

We follow [2] and [23] in our presentation. Manifolds with corners are modeled on the $m$-dimensional cylinders

$${\mathbb{R}}^{m}_{k}:=[0,\infty)^{k}\times{\mathbb{R}}^{m-k}\subseteq{\mathbb{R}}^{m}\qquad(k\leqslant m\in{\mathbb{N}}_{0}).$$

Their subsets

$$L_{I}:=\{x=(x_{1},\ldots,x_{m})\in{\mathbb{R}}^{m}_{k}\mid x_{i}=0\mbox{ if }i\in I\}\qquad(I\subseteq\{1,\ldots,m\})$$

(A.1)

will be used to define submanifolds.

Definition 3

Let $U\subseteq{\mathbb{R}}^{m}_{k}$ and $V\subseteq{\mathbb{R}}^{m^{\prime}}_{k^{\prime}}$ be open, and $f:U\to V$.

• $f$ is called smooth if for some open neighborhood $\tilde{U}\subseteq{\mathbb{R}}^{m}$ of $U$ there exists $\tilde{f}\in C^{\infty}\big{(}\tilde{U},{\mathbb{R}}^{m^{\prime}}\big{)}$ with $\tilde{f}|_{U}=f$.

• $f$ is called a diffeomorphism if it is a smooth bijection with $f^{-1}$ smooth.

Definition 4 (manifolds with corners)

Let $X$ be a Hausdorff space.

• An $($ $m$ -dimensional $)$ corner chart $(U,\phi)$ on $X$ is a homeomorphism $\phi:U\to V$ , with $V$ open in ${\mathbb{R}}^{m}_{k}$ .

• Corner charts $(U_{1},\phi_{1})$ and $(U_{2},\phi_{2})$ on $X$ are compatible if for $U:=U_{1}\cap U_{2}$

  $$\phi_{2}\circ\phi_{1}^{-1}:\phi_{1}(U)\to\phi_{2}(U)$$

  is a diffeomorphism.

• A (corner) atlas $\{(U_{i},\phi_{i})\mid i\in I\}$ on $X$ is a family of pairwise compatible charts $(U_{i},\phi_{i})$ on $X$ of equal dimension with $\bigcup_{i\in I}U_{i}=X$ .

• Corner atlases on $X$ are equivalent if their union is a corner atlas on $X$ . A corner structure on $X$ is an equivalence class of corner atlases of $X$ .

• A paracompact Hausdorff space $X$ with a corner structure consisting of $m$ -dimensional corner charts is an ( $m$ -dimensional) manifold with corners .

• For $\partial_{\ell}{\mathbb{R}}^{m}_{k}:=\{x\in{\mathbb{R}}^{m}_{k}\mid\mbox{of }x_{1},\ldots,x_{k},\mbox{ exactly }\ell\mbox{ vanish}\}$ ,

  $$\partial_{\ell}X:=\{p\in X\mid\mbox{coordinates at }p\mbox{ map to }\partial_{\ell}{\mathbb{R}}^{m}_{k}\}$$

  and the boundary $\partial X:=\partial^{1}X$ of $X$ for $\partial^{\ell}X:=\overline{\partial_{\ell}X}$ .

Unlike for manifolds with boundary, the Cartesian product of two manifolds with corners is naturally a manifold with corners.

Definition 5 (submanifolds of manifolds with corners)

• A subset $S\subseteq X$ of an $m$ -dimensional manifold with corners is a weak submanifold if for every $x\in S$ there exist $k\in\{1,\ldots,m\}$ and a corner chart $\phi:U\to\Omega\subseteq{\mathbb{R}}^{m}_{k}$ with $x\in U$ such that $\phi(S\cap U)$ is a submanifold of ${\mathbb{R}}^{m}$ . Then the dimension of $S$ at $x$ is $\dim\big{(}\phi(S\cap U)\big{)}$ at $\phi(x)$ .

• A weak submanifold $S\subseteq X$ is a submanifold $($ in the sense of manifolds with corners $)$ if, additionally, there are integers $m^{\prime}\leqslant m$ and $k^{\prime}\leqslant m^{\prime}$ , and a matrix $G\in{\rm GL}(m,{\mathbb{R}})$ such that

  • (a) $G\cdot\big{(}{\mathbb{R}}^{m^{\prime}}_{k^{\prime}}\times\{0\}\big{)}\subseteq{\mathbb{R}}^{m}_{k}$ ;

  • (b) the chart $\phi$ maps $S\cap U$ bijectively to the intersection of this linear submanifold with $\Omega$ , in other words, $\phi(S\cap U)=G\cdot\big{(}{\mathbb{R}}^{m^{\prime}}_{k^{\prime}}\times\{0\}\big{)}\cap\Omega$ .

• A submanifold $S\subseteq X$ is a $p$-submanifold if for $x\in X$ there exists a corner chart $(U,\phi)$ at $x$ and $I\subseteq\{1,\ldots,m\}$ with (see Definition (A.1))

  $$\phi(S\cap U)=L_{I}\cap\phi(U).$$

  Then $|I|$ is the codimension of $S$ at $x$ and $|I\cap\{1,\ldots,k\}|$ is the boundary depth of $S$ at $x$.

So a $p$-submanifold $S$ of $X$ is a closed submanifold that has a tubular neighborhood: $S\subseteq U\subseteq X$ that is locally of product form.