Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RP...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/386358 |
| Acceso en línea: | https://hdl.handle.net/2117/386358 https://dx.doi.org/10.1016/j.aim.2022.108562 |
| Access Level: | acceso abierto |
| Palabra clave: | Dynamics Hamiltonian systems Lagrange equations Exponentially small phenomena Splitting of separatrices Celestial mechanics Coorbital motions L3 Lagrange point Dinàmica Hamilton, Sistemes de Lagrange, Equacions de Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics Àrees temàtiques de la UPC::Física |
| Sumario: | The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, , called the Lagrange points. The Lagrange point is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel [10], we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of when the ratio between the masses of the primaries µ is small. It implies that cannot have one-round homoclinic orbits. If the mass ratio µ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order . This implies that the distance between the invariant manifolds is exponentially small with respect to µ and, therefore, the classical Poincaré–Melnikov method cannot be applied. In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of . In the sequel [10] we complete the proof of the asymptotic formula for the distance between the invariant manifolds. |
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