We study the dynamics of the Magellanic Clouds in a model for the Local Group whose mass is constrained using the timing argument/two-body limit of the action principle. The goal is to evaluate the role of M31 in generating the high angular momentum orbit of the Clouds, a puzzle that has only been exacerbated by the latest $HST$ proper motion measurements. We study the effects of varying the total Local Group mass, the relative mass of the Milky Way and M31, the proper motion of M31, and the proper motion of the LMC on this problem. Over a large part of this parameter-space we find that tides from M31 are insignificant. For a range of LMC proper motions approximately $3sigma$ higher than the mean and total Local Group mass $> 3.5times 10^{12} M_odot$, M31 can provide a significant torque to the LMC orbit. However, if the LMC is bound to the MW, then M31 is found to have negligible effect on its motion and the origin of the high angular momentum of the system remains a puzzle. Finally, we use the timing argument to calculate the total mass of the MW-LMC system based on the assumption that they are encountering each other for the first time, their previous perigalacticon being a Hubble time ago, obtaining $M_{rm MW} + M_{rm LMC} = (8.7 pm 0.8) times 10^{11} M_odot$.
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