*(English)*Zbl 1073.70011

The paper approaches in an elaborate manner the $n$-body problem from celestial mechanics with the tools of the mathematical theory of optimal processes, namely the Pontryagin maximum principle. The setup of the problem is given by the equations $d{r}_{j}/dt={V}_{j}$, $dA/dt=\{{\sum}_{j=1}^{n}{m}_{j}{V}_{j}^{2}/2\}+G{\sum}_{1\le j<k\le n}{m}_{j}{m}_{k}/{r}_{jk}$; here, $G$ is the constant of the law of universal attraction, ${r}_{jk}$ is the distance between two point masses ${m}_{j}$, ${m}_{k}$, and $A$ is the action to be minimized (between the initial and final positions of the $n$ point masses at given initial and final times). The positions of the point masses and the action are state parameters while velocities are control parameters.

Section 2 contains a detailed presentation of the theory of optimization of deterministic dynamical systems (including relevant references). Following the author’s terminology, for a (natural) negative Pontryagin coefficient, the optimal Hamiltonian (the maximum of the Hamiltonian in the control domain) reads as ${H}^{*}=\{-{\sum}_{j=1}^{n}{p}_{j}^{2}/2{p}_{A}{m}_{j}\}+G{p}_{A}$ ${\sum}_{1\le j<k\le n}{m}_{j}{m}_{k}/{r}_{jk}$ (the sign $-$ should be added to the original formulation, in my opinion) and is obtained for ${V}_{j}=-{p}_{j}/{p}_{A}{m}_{j}$, where ${p}_{j}$ (the Pontryagin vector) is the momentum of the point mass ${m}_{j}$. The Pontryagin maximum principle shows that, if the singularities and especially collisions are avoided, the Pontryagin solutions are also solutions of the $n$-body problem.

The paper further analyzes the singularities: by showing that the minimum of the action is bounded (Subsection 4.1), $0<{A}_{f}\le K<+\infty $, and using the formula $\left|{r}_{{j}_{2}}-{r}_{{j}_{1}}\right|\le {[2K({t}_{2}-{t}_{1})/{m}_{j}]}^{1/2}$, where ${t}_{i}$, ${r}_{{j}_{i}}$ are arbitrary succesive instants, respectively positions, it is inferred that the eventual singularities (at intermediate times) are necessarily collisions; using a “reductio ad absurdum” method, a thorough analysis of binary/multiple collisions is performed (this is the core of the paper, Subsection 4.3): it can be concluded that the minimizing solutions are pure $n$-body motions with no intermediate collisions even if one or several collisions are imposed at initial and/or final times.

As a by-product, in the case of eventual binary collisions, a Lambert-type theorem is derived: there are five relations between the parameters $T$ (the time necessary to go from the initial position ${r}_{+}$ to the final position ${r}_{-}$: $T={t}_{+}-{t}_{-}$), ${\Sigma}$ (the quantity $\left|{r}_{+}+{r}_{-}\right|$), $a$ (the semimajor axis of the relative two-body orbit), $M$ (the quantity $\left|{r}_{+}-{r}_{-}\right|$), ${A}_{in}$ (the inner action: $[{m}_{j}{m}_{k}/({m}_{j}+{m}_{k})]{\int}_{{t}_{-}}^{{t}_{+}}[({V}^{2}/2)+(\mu /r)]dt$, where $V=\left|(d/dt)({r}_{k}-{r}_{j})\right|$, $\mu =G({m}_{j}+{m}_{k})$ and $r=\left|{r}_{k}-{r}_{j}\right|$), ${\Delta}$ (the difference between initial eccentric anomalies: ${E}_{+}-{E}_{-}$; ${F}_{+}-{F}_{-}$ in the hyperbolic case), $S$ (the quantity $e($sin${E}_{+}-$sin${E}_{-})$; in the hyperbolic case: $e($sinh${F}_{+}-$sinh${F}_{-})$, where $e$ is the eccentricity of the two-body orbit) and $C$ (the quantity $e($cos${E}_{+}+$cos${E}_{-})$; in the hyperbolic case: $e($cosh${F}_{+}+$cosh${F}_{-})$). The case of infinitesimal masses is discussed in Appendix 4, and an application to periodic solutions is given in Appendix 5.