The spatial N-centre problem: scattering at positive energies

Abstract

For the spatial generalized N-centre problem

$$\begin{aligned} \ddot{x} = -\sum _{i=1}^{N} \frac{m_i (x - c_i)}{\vert x - c_i \vert ^{\alpha +2}},\quad x \in \mathbb {R}^3 {\setminus } \{c_1,\ldots ,c_N \}, \end{aligned}$$

where $m_i > 0$ and $\alpha \in [1,2)$, we prove the existence of positive energy entire solutions with prescribed scattering angle. The proof relies on variational arguments, within an approximation procedure via (free-time) boundary value problems. A self-contained appendix describing a general strategy to rule out the occurrence of collisions is also included.

Appendix

In this Appendix, we describe a strategy to investigate the behavior of “generalized solutions” to (1) (when $\alpha = 1$, this being the most delicate case), so as to eventually rule out the occurrence of collision. We do not claim any originality in the forthcoming results, which are probably well known by experts in Celestial Mechanics; however, we hope it can be of some interest to collect them in the present form, since no appropriate reference in the literature seems to exist.

Throughout this section, we deal with the perturbed Kepler equation

$$\begin{aligned} \ddot{q} = - \frac{\mu q}{\vert q \vert ^3} + \nabla U(q), \end{aligned}$$

(46)

where $\mu > 0$ and U is a $\mathcal {C}^\infty$-function defined on some open set $\Omega \subset \mathbb {R}^3$ containing the origin; we will be interested in solutions to (46) possibly taking the value $q = 0$. Notice that (1) can be written in the above form, setting (for some $i=1,\ldots ,N$) $q = x - c_i,$ $\mu =m_i$ and $\Omega = \mathbb {R}^3 {\setminus } \cup _{j\ne i}\{c_j - c_i\}$; of course, such a choice leads to investigations about solutions colliding with the centre $c_i$.

Following [2] we call generalized solution to (46) a continuous function $q: I \subset \mathbb {R} \rightarrow \Omega$ (with $I \subset \mathbb {R}$ an interval) such that:

• the set $Z := q^{-1}(0)$ of collisions has zero measure,

• on $I {\setminus } Z$, the function q is of class $\mathcal {C}^\infty$ and solves Eq. (46) therein,

• the energy is preserved through collisions, i.e., there exists $h \in \mathbb {R}$ such that

  $$\begin{aligned} \frac{1}{2}\vert \dot{q}(t) \vert ^2 - \frac{\mu }{\vert q(t) \vert } - U(q(t)) = h \end{aligned}$$

  for any $t \in I {\setminus } Z$.

This is a very weak notion of solution; in order to restrict the attention to “physically meaningful” solutions, the incoming and outgoing collision directions at $t_0 \in Z$, namely

$$\begin{aligned} \lim _{t \rightarrow t_0^-} \frac{q(t)}{\vert q(t) \vert } \quad \text{ and } \quad \lim _{t \rightarrow t_0^+} \frac{q(t)}{\vert q(t) \vert }, \end{aligned}$$

play a role. Indeed, roughly speaking, a solution can be considered physically meaningful (that is, from a mathematical point of view, converted to a solution of a suitable regularized equation) if and only if the collision directions coincide (compare with [9], where the more general situation of a time-dependent perturbation U(t, q) is also discussed, and the equality between the collision directions is indeed incorporated in the definition of generalized solution). Actually, for such solutions the behavior is very simple: they are just reflected back after collision. We give here below a proof of this fact; it is worth mentioning that our arguments just rely on the classical Sperling estimates [18], thus avoiding typical three-dimensional regularization techniques (like Kustaanheimo-Stiefel one, see for instance [22]).

Proposition 3.2

Let $q: (-\varepsilon ,\varepsilon ) \rightarrow \Omega$ be a generalized solution to (46) with $q^{-1}(0) = \{0\}$. Assume further that the limit

$$\begin{aligned} \lim _{t \rightarrow 0} \frac{q(t)}{\vert q(t) \vert } \end{aligned}$$

exists. Then, q is a collision-reflection solution to (46), i.e.,

$$\begin{aligned} q(t) = q(-t), \quad \text{ for } \text{ every } t \in (0,\varepsilon ). \end{aligned}$$

Proof

As proved in [18], it holds that

$$\begin{aligned} q(t) = \left( \frac{9}{2} \mu \right) ^{1/3} \vert t \vert ^{2/3} \,\xi + \mathcal {O}\left( \vert t \vert ^{4/3}\right) , \quad t \rightarrow 0, \end{aligned}$$

(47)

and

$$\begin{aligned} \dot{q}(t) = \frac{2}{3} \left( \frac{9}{2} \mu \right) ^{1/3} t ^{-1/3} \, \xi + \mathcal {O}\left( \vert t \vert ^{1/3}\right) , \quad t \rightarrow 0, \end{aligned}$$

(48)

where $\xi = \lim _{t \rightarrow 0}\frac{q(t)}{\vert q(t) \vert }$. Based on this, we first define the Sundman integral

$$\begin{aligned} s(t) = \int _0^t \frac{d\tau }{\vert q(\tau ) \vert }, \quad t \in (-\varepsilon ,\varepsilon ), \end{aligned}$$

and we set, for s in a neighborhood of zero,

$$\begin{aligned} u(s) = q(t(s)), \end{aligned}$$

where t(s) denotes as usual the inverse of s(t); incidentally, notice that

$$\begin{aligned} t(s) = \int _0^s \vert u(\sigma ) \vert \,d\sigma . \end{aligned}$$

(49)

Then, we further set

$$\begin{aligned} v(s)&= (\vert q \vert \dot{q}) \circ t (s)\\ w(s)&= \left( - \frac{\mu q}{\vert q \vert } + \langle q, \dot{q} \rangle \dot{q} \right) \circ t (s). \end{aligned}$$

Notice that the function $z(s) = (u(s),v(s),w(s))$ is defined on a punctured neighborhood of zero and is smooth therein. Elementary computations, using the differential equation and the energy relation, show that, for any $s \ne 0$,

$$\begin{aligned} u'(s)&= (\vert q \vert \dot{q}) \circ t(s)\\ v'(s)&= \left( - \frac{\mu q}{\vert q \vert } + \langle q, \dot{q} \rangle \dot{q} + \vert q \vert ^2 \nabla U(q)\right) \circ t (s) \\ w'(s)&= \Big [\langle q, \dot{q} \rangle \vert q \vert \nabla U(q) + \big (2h + 2U(q) + \langle q, \nabla U(q) \rangle \big ) \vert q \vert \dot{q} \Big ] \circ t(s). \end{aligned}$$

Writing the right-hand sides in terms of (u, v, w), we thus see that z(s) satisfies the differential equation

$$\begin{aligned} z'(s) = F(z(s)),\quad s \ne 0, \end{aligned}$$

where the vector field $F = (F_1,F_2,F_3)$ is given by

$$\begin{aligned} F_1(z)&= v\\ F_2(z)&= w + \vert u \vert ^2 \nabla U(u)\\ F_3(z)&= \langle u, v \rangle \nabla U(u) + \big (2h + 2U(u)+ \langle u, \nabla U(u) \rangle \big ) v. \end{aligned}$$

On the other hand, (47) and (48) readily imply that z(s) can be continuously extended to $s = 0$, with

$$\begin{aligned} z(0) = \left( 0,0,\mu \xi \right) =: z_0. \end{aligned}$$

Therefore, z(s) turns out to be a local solution of the Cauchy problem

$$\begin{aligned} z' = F(z), \quad z(0) = z_0; \end{aligned}$$

since the vector field F satisfies

$$\begin{aligned} F_1(u,-v,w)&= -F_1(u,v,w)\\ F_2(u,-v,w)&= F_2(u,v,w)\\ F_3(u,-v,w)&= -F_3(u,v,w) \end{aligned}$$

we see that it has to be $u(s) = u(-s)$ for any s. Recalling (49), we obtain $t(s) = -t(s)$, finally implying the conclusion. $\square$

In view of Proposition 3.2, it becomes of interest to investigate under which conditions generalized solutions actually have coincident incoming and outgoing collision direction. To present our results in this direction, let $I \subset \mathbb {R}$ be a bounded interval and consider a sequence of classical solutions $q_n: I \rightarrow \Omega$ of the equation

$$\begin{aligned} \ddot{q}_n = - \frac{\mu q_n}{\vert q_n \vert ^3} - \frac{2 \varepsilon _n \mu q_n}{\vert q_n \vert ^4} + \nabla U(q_n), \end{aligned}$$

(50)

where $\varepsilon _n \ge 0$ and $\varepsilon _n \rightarrow 0^+$; we also assume that the associated energy is independent of n, namely, there exists $h \in \mathbb {R}$ such that

$$\begin{aligned} \frac{1}{2}\vert \dot{q}_n(t) \vert ^2 - \frac{\mu }{\vert q_n(t) \vert } - \frac{\varepsilon _n \mu }{\vert q_n(t) \vert ^2} - U(q_n(t)) \equiv h, \quad \text{ for } \text{ any } n \ge 1. \end{aligned}$$

(51)

Of course, for $\varepsilon _n = 0$ we are simply considering a family of classical solutions of the perturbed Kepler problem (46). On the other hand, the choice $\varepsilon _n > 0$ allows us to deal with the solutions given by Proposition 2.2 (notice indeed that $q_n := x_{\beta _n} - c_i$ satisfies an equation like (50) in a sufficiently small neighborhoof of the origin). From now on, we will actually consider generalized solutions to (46) arising as limits (in a suitable topology) of the above solutions $q_n$. We start with the following preliminary result.

Lemma 3.3

Let $q_\infty : I \rightarrow \Omega$ be an $H^1$-function such that $q_n \rightarrow q_\infty$ weakly in $H^1(I)$. Then, $q_\infty$ is a generalized solution to (46) and $q_\infty ^{-1}(0)$ is a finite set.

Proof

To prove that $q_\infty$ is a generalized solution to (46) we just need to check that $q_\infty ^{-1}(0)$ has zero measure. To this end, we use the energy relation (51) together with Fatou’s lemma to write

$$\begin{aligned} \int _I \frac{\mu }{\vert q_\infty (t) \vert } \,dt&\le \liminf _{n \rightarrow +\,\infty } \int _I \frac{\mu }{\vert q_n(t) \vert } \\&\le \liminf _{n \rightarrow +\,\infty } \int _I \left( \frac{1}{2}\vert \dot{q}_n(t) \vert ^2 - \frac{\varepsilon _n \mu }{\vert q_n(t) \vert ^2} - U(q_n(t)) - h \right) \,dt. \end{aligned}$$

Since $U(q_n)$ is $L^\infty$-bounded by uniform convergence and $\int _I \vert \dot{q}_n \vert ^2$ is bounded by weak $H^1$-convergence, we see that the above quantity is finite, thus implying the conclusion.

To prove that $q_\infty ^{-1}(0)$ is a actually a finite set, we are going to show that collisions are isolated, namely, if $q_\infty (t_0) = 0$ for some $t_0 \in I$ then $q_\infty (t) \ne 0$ in a suitable punctured neighborhood of $t_0$. To this end, we preliminarily fix $r_0 > 0$ such that

$$\begin{aligned} \frac{\mu }{\vert q \vert } + 2U(q) + \langle q, \nabla U(q) \rangle + 2h > 0, \quad \text{ for } \text{ every } \vert q \vert \le r_0. \end{aligned}$$

In view of the uniform convergence of $q_n$ to $q_\infty$, there exists an interval $J \subset I$ with $t_0 \in J$ such that $\min _{t \in J} \vert q_n(t) \vert \le r_0$ for any n large enough. An easy computation shows that

$$\begin{aligned} \frac{d^2}{dt^2} \frac{1}{2} \vert q_n(t) \vert ^2 = \frac{\mu }{\vert q_n(t) \vert } + 2U(q_n(t)) + \langle q_n(t), \nabla U(q_n(t)) \rangle + 2h \end{aligned}$$

for any n large enough and for any $t \in J$, implying that the function $t \mapsto \vert q_n(t) \vert ^2$ is strictly convex on J for large n. Again by uniform convergence, $t \mapsto \vert q_\infty (t) \vert ^2$ is convex on J, showing that either $q_\infty \equiv 0$ on a neighborhood of $t_0$ or $q_\infty (t) \ne 0$ for every $t \in J {\setminus } \{t_0\}$. Since the first case cannot occur because the set of collisions has zero measure, we are done. $\square$

Remark 3.4

It is worth mentioning that the choice of exponent $\beta = 2$ for the penalization term $\varepsilon _n \vert q_n \vert ^{-\beta }$ is crucial, this being the unique value such that both a strong force assumption is satisfied (this being needed for the min–max argument of Sect. 2) and the above Lagrange–Jacobi argument is possible.

In view of Lemma 3.3, we can perform a local analysis around each singularity. The next result is essentially proved in [19] and provides an estimate for the collision directions of a generalized solution $q_\infty$ in terms of the sequence $q_n$.

Proposition 3.5

Let $q_\infty : I \rightarrow \Omega$ be an $H^1$-function with $q_\infty ^{-1}(0) = \{0\}$ and such that $q_n \rightarrow q_\infty$ weakly in $H^1(I)$. Then, if the limit

$$\begin{aligned} d:= \lim _{n \rightarrow +\,\infty } \frac{\varepsilon _n }{\mu ^{1/3} \min _t \vert q_n(t) \vert } \end{aligned}$$

exists finite, the angle between

$$\begin{aligned} \lim _{t \rightarrow 0^-} \frac{q_\infty (t)}{\vert q_\infty (t) \vert } \quad \text{ and } \quad \lim _{t \rightarrow 0^+} \frac{q_\infty (t)}{\vert q_\infty (t) \vert } \end{aligned}$$

is equal to $2\pi \sqrt{1+ d}$ (modulo $2\pi$).

Sketch of the proof

We argue as in the proof of [19, Theorem 0.1 (ii)], using a blow-up technique. Assuming $\delta _n := \min _t \vert q_n(t) \vert = \vert q_n(t_n) \vert$ with $t_n \rightarrow 0$, we set

$$\begin{aligned} y_n(t) = \frac{1}{\delta _n} q_n\left( \delta _n^{3/2} \,t + t_n \right) ; \end{aligned}$$

then it is not difficult to see that $y_n$ converges uniformly on compact sets to a zero-energy solution $y_{\infty ,d}$ of the equation

$$\begin{aligned} \ddot{y}_{\infty ,d} = - \frac{\mu y_{\infty ,d}}{\vert y_{\infty ,d} \vert ^3} - \frac{2d \mu ^{4/3} y_{\infty ,d} }{\vert y_{\infty ,d} \vert ^4}. \end{aligned}$$

Via some delicate angular momentum estimates, it is possible to show that the collision directions of $q_\infty$ at $t = 0$ can be related to the asymptotic directions of $y_{\infty ,d}$ (see [19, Proposition 1.2]), which in turn are easily computed (see [19, Proposition 1.1 (iii)]). Notice that all the assumptions in [19] are satisfied in our case, at least in a neighborhood of $q = 0$; even more, some simplifications are here possible with respect to the proof given therein, since we deal with fixed-energy solutions of an autonomous problem. However, the complete argument is still very long and we omit it. $\square$

Combining Propositions 3.2 and 3.5 clearly suggests a strategy to exclude the occurrence of collisions for a generalized solution: indeed, whenever $d = 0$ the incoming and outgoing collisions directions must coincide, so that $q_\infty$ is just a collision-reflection solution (a case which is typically ruled out for some other reasons).

We describe below two situations in which this procedure works. In the first one, we simply deal with perturbations of the Kepler problem: that is, $\varepsilon _n = 0$, clearly implying $d = 0$. In the second one, the key role is played by a Morse index assumption (by Morse index $j (q_n)$ of the solution $q_n$ we will mean the Morse index of $q_n$ when regarded as a critical point of the action functional on the space of $H^1$-paths with fixed ends): indeed, by [19, Proposition 1.1 (iv)], it holds that

$$\begin{aligned} d > 0 \quad \Longrightarrow \quad j (q_n) \ge 2, \; \hbox { for sufficiently large}\ n. \end{aligned}$$

Actually, some care is needed to obtain the above implication, since in [19] a periodic boundary value problem is considered; however, the constructed variations have compact support so that the argument fits with our setting as well.

We can summarize the above discussion in the following corollary.

Corollary 3.6

Let $q_\infty : I \rightarrow \Omega$ be an $H^1$-function with $q_\infty ^{-1}(0) = \{0\}$ and such that $q_n \rightarrow q_\infty$ weakly in $H^1(I)$. Then:

• if $\varepsilon _n = 0$ for every n, then $q_\infty$ is a collision-reflection solution,

• if $j (q_n) \le 1$ for every n, then $q_\infty$ is a collision-reflection solution.

Both the results of the above Corollary will be used in the paper. Precisely, the first one is used at the very end of Sect. 3.2, when constructing the entire hyperbolic solution as limit for $R \rightarrow +\,\infty$ of the fixed-endpoints problems (27), for $\alpha = 1$. The second one is used in the proof of Theorem 2.1, when passing to the limit for $\beta \rightarrow 0^+$.