How the method of minimization of action avoids singularities
type: Item
instance of: scholarly article
title: How the method of minimization of action avoids singularities
zbMATH Open document ID: 1073.70011
DOI: 10.1023/A:1020128408706
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2003-05-06T00:00:00Z
review text: 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 \(dr_j/dt=V_j\), \(dA/dt=\{\sum_{j=1}^nm_jV_j^2/2\}+G\sum_{1\leq j<k\leq n}m_jm_k/r_{jk}\); 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 \(H^{*}=\{-\sum_{j=1}^np_j^2/2p_Am_j\}+Gp_A\) \(\sum_{1\leq j<k\leq n}m_jm_k/r_{jk}\) (the sign \(-\) should be added to the original formulation, in my opinion) and is obtained for \(V_j=-p_j/p_Am_j\), 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), \(0<A_f\leq K<+\infty \), and using the formula \(\left| r_{j_2}-r_{j_1}\right| \leq [2K(t_2-t_1)/m_j]^{1/2}\), 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 collis…
Mathematics Subject Classification ID: 70F10
Mathematics Subject Classification ID: 70F15
Mathematics Subject Classification ID: 70F16
Mathematics Subject Classification ID: 49J15
zbMATH DE Number: 1906520
zbMATH Keywords: optimal Hamiltonian
zbMATH Keywords: Pontryagin maximum principle
zbMATH Keywords: collisions
zbMATH Keywords: Lambert theorem
author: Christian Marchal
reviewed by: Octavian Mustafa
MaRDI profile type: Publication
full work available at URL: a:1020128408706
OpenAlex ID: W4246904602
description: scientific article; zbMATH DE number 1906520