Open Access

Amin Mohebbi

• The rotating Kepler problem is a special case of the restricted three body problem such that the mass of one primaries in the R3BP is zero. The R3BP and the RKP have many applications on classical mechanics and dynamical systems. Since the Kepler problem gives us mathematical models to explain the moving of the planets, satellites, and their Orbits, we are interested to study the dynamics of them on the space in a rotating coordinate system that is independent of time via the RKP. In this thesis, we are going to compute the ECH capacities for the RKP that by these capacities with the goal to find a sharp embedding obstruction between the symplectic 4-manifold belonging to the RKP and another symplectic 4-manifold. In the first step, we will give an introduction to symplectic manifolds and the study of the Hamiltonian of the RKP, the Hills region of the RKP, and the periodic orbits of the RKP. In chapter 4, we will see the Ligon-Schaaf symplectomorphism and the Levi-CivitaThe rotating Kepler problem is a special case of the restricted three body problem such that the mass of one primaries in the R3BP is zero. The R3BP and the RKP have many applications on classical mechanics and dynamical systems. Since the Kepler problem gives us mathematical models to explain the moving of the planets, satellites, and their Orbits, we are interested to study the dynamics of them on the space in a rotating coordinate system that is independent of time via the RKP. In this thesis, we are going to compute the ECH capacities for the RKP that by these capacities with the goal to find a sharp embedding obstruction between the symplectic 4-manifold belonging to the RKP and another symplectic 4-manifold. In the first step, we will give an introduction to symplectic manifolds and the study of the Hamiltonian of the RKP, the Hills region of the RKP, and the periodic orbits of the RKP. In chapter 4, we will see the Ligon-Schaaf symplectomorphism and the Levi-Civita regularization then, in the next chapter, by using them we will define a special concave toric domain for the RKP which is a symplectic 4-manifold and we will find the weights of the SCTD of the RKP when the energy $c \leq - \frac{3}{2}$ via the extension of a new method to computing ECH capacities of a concave toric domain with the help of a new tree which is introduced in chapter 6. In the last step, we will use those weights and compute some ECH capacities of the RKP for $c \leq - \frac{3}{2}$ and more examples in the case $c = - \frac{3}{2}$.…