# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
How the method of minimization of action avoids singularities. (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=\left\{{\sum }_{j=1}^{n}{m}_{j}{V}_{j}^{2}/2\right\}+G{\sum }_{1\le j; 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 ${H}^{*}=\left\{-{\sum }_{j=1}^{n}{p}_{j}^{2}/2{p}_{A}{m}_{j}\right\}+G{p}_{A}$ ${\sum }_{1\le j (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 $\left|{r}_{{j}_{2}}-{r}_{{j}_{1}}\right|\le {\left[2K\left({t}_{2}-{t}_{1}\right)/{m}_{j}\right]}^{1/2}$, 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: $\left[{m}_{j}{m}_{k}/\left({m}_{j}+{m}_{k}\right)\right]{\int }_{{t}_{-}}^{{t}_{+}}\left[\left({V}^{2}/2\right)+\left(\mu /r\right)\right]dt$, where $V=\left|\left(d/dt\right)\left({r}_{k}-{r}_{j}\right)\right|$, $\mu =G\left({m}_{j}+{m}_{k}\right)$ 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 $e\left($sin${E}_{+}-$sin${E}_{-}\right)$; in the hyperbolic case: $e\left($sinh${F}_{+}-$sinh${F}_{-}\right)$, where $e$ is the eccentricity of the two-body orbit) and $C$ (the quantity $e\left($cos${E}_{+}+$cos${E}_{-}\right)$; in the hyperbolic case: $e\left($cosh${F}_{+}+$cosh${F}_{-}\right)$). The case of infinitesimal masses is discussed in Appendix 4, and an application to periodic solutions is given in Appendix 5.

##### MSC:
 70F10 $n$-body problems 70F15 Celestial mechanics 70F16 Collisions in celestial mechanics, regularization 49J15 Optimal control problems with ODE (existence)