Application Of Variational Methods In Mathematical Economics And Celestial Mechanics

The thesis is about the applications of variational methods to problems in mathe-matical economics and celestial mechanics, it consists of two parts.In the first part, we study time-inconsistency and sustainable development in eco-nomic growth model. We firstly investigate an infinite-horizon problems in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Motivated by the works of Chichilnisky on sustainable preferences, we introduce an additional term to the criterion of the Ramsey model, which models the concern for the well-being of future generations. We show that there are no optimal solutions. Moreover, any attempt to maximize the new criterion is failed because of each decision maker cannot commit his successors. But the non-commitment among the decision makers will greatly affect the original attempt. In other words, under the new criterion, time-inconsistency occurs. However, we prove the existence of the equilibrium strategies, i.e., there exists subgame perfect Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thorn. We use the central manifold techniques to prove the existence of solutions of the singular equations. Our analysis extends earlier works by Ekeland and Lazrak.In the second part, we study the linear stability problem of relative equilibria in celestial mechanics, we firstly consider the charged three-body problem. Following the central configuration coordinate decomposition of Meyer and Schmit, we linearize the Hamiltonian system of the charged three-body problem near an relative equilibria. We prove that it possesses the same linearized Hamiltonian system with the corresponding classical three-body problem and have a somewhat different mass parameter. Moreover, we show that the full range of the mass parameter of the charged case and the corre- sponding one of the classical case are the same. Thus we obtain the equivalence of lin-ear stabilities of elliptic triangle solutions of the planar charged and classical three-body problem. Consequently, the numerical results of Martinez-Samaa-Simo in2004-2006and analytical results of Hu-Long-Sun in2012can be applied to linear stability of such solutions of the charged three-body problem.After that, we consider the Euler elliptic solution of classical three-body problem. Using a similar central configuration coordinate decomposition, we linearize the Hamil-tonian system near the Euler solution. Then use the ω-index theory of symplectic paths and the theory of linear operators to study the linearized system. After establishing the ω-index increasing property of the solutions in mass parameter for fixed eccentricity, we obtain some linear stability properties of such Euler solution.