New normal form approaches adapted to the Trojan problem

The main subject of this work is the study of the problem of the Trojan orbits from a perturbative Hamiltonian perspective. We face this problem by introducing first a novel Hamiltonian formulation, exploiting the well-differentiated temporal scales of the Trojan motion. The resulting Hamiltonian allows to separate the secular (very slow) component of the motion from the librating and fast degrees of freedom. This decompositon provides the foundation of a so-called Basic Hamiltonian model (Hb), i.e. the part of the Hamiltonian for Trojan orbits independent of all secular angles. Our study shows that, up to some extent, the model Hb successfully represents the features of the motion under more complete models, in a range of physical parameters relevant for dynamics in the Solar System or in extrasolar planetary systems. We propose, then, two novel normal form schemes in order to analytically study the model Hb. The first scheme takes into account the existence of a real singularity due to close encountersoftheTrojanbodywiththeprimary,byavoidinganypolynomialortrigonometric expansion for the librating angle. The second scheme exploits the fact that the Trojan orbits are highly asymmetric with respect to the libration center. We then analytically construct a so-called "asymmetric expansion", which extends the domain of the normal form series’ convergence with respect to the usual polynomial expansions around the stable Lagrangian points L4 or L5. Both schemes are tested in detail in the framework of the Circular and Elliptic Restricted 3-Body Problems, focusing particularly on the analytical derivation of the location of secondary resonances embedded within the libration domain. Additionaly, the second scheme provides an analytical estimation of the width of such resonances. Finally,thethesisanalysesthekeyusefulnessofthe Hb model,pointingoutthepossibility for straightforward extensions allowing to include additional bodies (Restricted Multi-Planet Problem), and/or Trojan motions in 3D space.