How the method of minimization of action avoids singularities
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 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: \([m_jm_k/(m_j+m_k)]\int_{t_{-}}^{t_{+}}[(V^2/2)+(\mu /r)]dt\), 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.
scientific article; zbMATH DE number 1906520