About analytical ansatz to the solving procedure for Kelvin–Kirchhoff equations

Solving procedure for 3D motions near libration points in CR3BP

“…The last but not least, we should especially note that by the use of formulae (13) (in Appendix A1) in new expression (15) Also, some remarkable articles should be cited, which concern the problem under consideration, [18]- [25].…”

“…In this case, formulae (7) Then, formulae (6)-(7) yield (k = 2) With respect to the essential details about the possible physical properties of the solution (6)- (7), let us discuss first the relevance of this new solution (or solving procedure) illuminating its unexpected features. Equation (7) is known to be the Riccati ordinary differential equation, ODE [13]. We also note that due to the special character of the solutions of Riccati-type ODEs, there is the possibility for sudden jumping in the magnitude of the solution at some time t₀ [14][15][16] (in [16] see e.g.…”

“…Poisson equations, the system of Eqns. (5) has the analytical way to present the general solution [13] in regard to the time t:…”

“…Let us transform expression (2) We should also take into consideration that the plane of motion of this new planet m3 around the common centre of masses with Sun does not coincide with the plane of mutual rotation of Sun and Earth, but intersects it at a fixed angle  (see e.g. Fig.1); then additionally we should take into account that projection of any circle onto the arbitrary plane will always be the proper ellipse (see Fig.3): (14) should be changed to the mean motion n in (2)): where we should take into account the aforementioned expressions for coordinates (13). Here M Sun  1 -μ 1 and μ 1 = m 3, where μ 1 is the ratio of mass of the new planet m 3 to the total mass of all the primaries and 0 ≤ μ 1 ≤ 1/2; as for the mean motion n in the rotating coordinate system r  ={x, y, z}, see (2); n 1 is the angular velocity of the rotating coordinate system R  ={X, Y, Z} (rotation of this new planet m3 around the Sun with mean motion n 1 = 2/T1, with orbital period T1).…”

“…Here M Sun  1 -μ 1 and μ 1 = m 3, where μ 1 is the ratio of mass of the new planet m 3 to the total mass of all the primaries and 0 ≤ μ 1 ≤ 1/2; as for the mean motion n in the rotating coordinate system r  ={x, y, z}, see (2); n 1 is the angular velocity of the rotating coordinate system R  ={X, Y, Z} (rotation of this new planet m3 around the Sun with mean motion n 1 = 2/T1, with orbital period T1). The distance between the primaries is taken as the unit of distance in (13); the unit of time is chosen so that the gravitational constant is equal to 1 in (14).…”