As part of the Large Hadron Collider high luminosity upgrade it is proposed to include crab cavities and large aperture niobium tin final focussing magnets in the lattice in order to enhance the luminosity. In this thesis the dynamics of a proposed cavity design were considered in terms of their impact upon the dynamic aperture of the machine. Taylor maps for the cavity were created and used to perform this analysis with a full assessment of their validity. A set of symplectic thin cavity models were also developed and cross checked with the Taylor maps. Finally, dynamic aperture studies were performed using these models in order to determine which components of the crab cavity dynamics are important when considering the long term stability of the beam in the LHC upgrade. It is shown that crab cavities exhibit little impact on the LHC beam stability. For the final focussing magnets a preliminary study was conducted into the importance of including their fringe fields in a model of the LHC upgrade. A technical study was carried out into developing a symplectic model which was compatible with the current magnet models use for dynamic aperture studies. A preliminary dynamic aperture study was performed with the inclusion of fringe fields for the final focussing magnets from which the fringe fields are shown to have a negative impact on the long term beam stability.