In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the 3-body elliptic Euler solutions. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute the ω-indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem.

In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of n-bodies splits into (n − 1) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other (n − 2) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martinéz, Samà and Simó in [13] and [14] on 3-body Euler solutions in [2004][2005][2006]. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler-Moulton solution of the 4-body problem with two small masses in the middle.

IntroductionIn economic theory, and in optimal control, it has been customary to discount future gains at a constant rate δ > 0. If an individual with utility function u (c) has the choice between several streams of consumption c (t), 0 ≤ t, he or she will choose the one which maximises the present value, given by:That future gains should be discounted is well grounded in fact. On the one hand, humans prefer to enjoy goods sooner than later (and to suffer bads later than sooner), as every child-rearing parent knows. On the other hand, it is also a reflection of our own mortality: 10 years from now, I may simply no longer be around to enjoy whatever I have been promised. These are two good reasons why people are willing to pay a little bit extra to hasten the delivery date, or will require compensation for postponement, which is the essence of discounting.On the other hand, there is no reason why the discount rate should be constant, i.e. why the discount factor should be an exponential e −δt . The practice probably arises from the compound interest formula lim ε→0 (1 − εδ) t/ε = e −δt , when a constant interest rate δ is assumed, but even in finance, interest rates vary with the horizon: long-term rates can be widely different from shortterm ones. As for economics, there is by now a huge amount of evidence that individuals use higher discount rates for the near future than for the long-term (see [18] for a review up to 2002). There

In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the linearized Poincaré map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters α, β with α ≥ β > 0 and the eccentricity e ∈ [0, 1). Via ω-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to α, β and e.Especially, two linearly stable sub-regions are found.

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Martínez, Samà and Simó ([15] in J. Diff. Equa.) of 2006 and results of Hu, Long and Sun ([6] in arXiv.org) of 2012 can be applied to these solutions of the charged three-body problem to get their linear stability.