Throughout the evolution of a planetary system, planets, especially those newly formed, interact by several mean with a variety of the system's constituents. In particular, the influence of the most massive planets is expected to govern much of the long-term evolution of the system. In early stages of this evolution, the gas disk that provided the material from which the planets formed also acts to couple the planets to its own dynamics. In part I of this thesis, I describe a new hydrodynamic code that I have developed, tuned to study these interactions. Using this code, I explore the formation of hydrodynamic structures within the disk, such as jets and eddies, that arise from the influence of the planets on the overall flow. I show that while the formation of vortices is damped in disks with a large enough viscosity, jet formation is more robust in this sense and jet structures form even in viscous flows. I further propose that these jets may affect the amount of material transport that occurs in the flow in a manner similar to that found in the Earth's atmosphere and in the weather layers of the Jovian planets. In order to qualify this claim, I perform preliminary numerical experiments that aim to establish this relationship.
Even after the removal of the gas disk, the gravitational influence of massive planets - or stellar companions in the case of multiple systems - severely limits the range of stable orbits of the system's lesser planets. In part II of this thesis, I examine the physical mechanisms responsible for planet ejection from unstable orbits previously observed in numerical experiments. I determine the instability is due to overlap of subresonances lying within mean-motion resonances.