Matched Asymptotic Expansions to the Circular Sitnikov Problem with Long Period

Abstract

The circular Sitnikov problem is revisited, using matched asymptotic expansions. In the case of large oscillation periods, approximate analytical expressions for the period and the orbit of the third body are found. The results are compared with those described in the literature and show that the movement of the third body can be well described by two analytical solutions, the inner and outer solutions.

Asymptotic approach to the one-dimensional motion of a body in modified Newtonian dynamics

“…Note that in these new scales, the differential equation (7) depends on a single parameter E GMa , 0 this greatly simplifies the analysis. In addition we see that the movement can be defined by the new potential:…”

“…If ε is a positive small parameter, the trajectory of the body is bounded but with a large amplitude, then the equation can be solved by matching between two asymptotic solutions [6][7][8].…”

“…Then, for any value of energy E, the orbit of the test particle will always be bounded. In the case of the Newtonian potential (segmented line), bounded orbits exist only in the case E<0.Integrating equation(7), the trajectory of the test particle η(τ) is given by:…”

Asymptotic approach to the Schrödinger equation in the presence of a screened Coulomb potential and a uniform field

“…If ε is a positive small parameter, the cup potential has a very short range, and the equation can be solved by matching between two asymptotic solutions using a technique similar to that performed in [13]. We divide the domain, x, into two regions: the near field where ε | | x ~, and the far field where…”

Asymptotic description of a test particle around a Schwarzschild black hole

“…In astro-dynamics, we can mention the classic problem of two fixed center-force, analyzed by Lagerstrom et al [13], and the three body problem analyzed by Nayfeh [14]. Recently, Rosales-Vera applied the matched asymptotic expansions technique to the circular Sitnikov problem [15]. In the field of quantum mechanics, a good example of the application of the matching method is the analysis of the Stark effect in the hydrogen atom [16][17][18]; in other areas of physics, good examples of its application can be found in [21][22][23].…”