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It is generally perceived that two cores cannot survive very long within the nuclear regions of a galaxy. The recent HST discovery of a double nucleus in M31 brings this question into prominence. Physical conditions in the nuclear regions of a typical galaxy help a second core survive so it can orbit for a long time, possibly for thousands of orbits. Given the nearly uniform mass density in a core, tidal forces within a core radius are compressive in all directions and help the core survive the buffeting it takes as it orbits near the center of the galaxy. We use numerical experiments to illustrate these physical principles. Modifications to the experimental method allow the full power of the experiments to be concentrated on the nuclear regions. Spatial resolution of about 0.2 parsec comfortably resolves detail within the 1.4 parsec core radius of the second, but brighter, core (P1) in M31. The same physical principles apply in other astronomical situations, such as dumbbell galaxies, galaxies orbiting near the center of a galaxy cluster, and subclustering in galaxy clusters. The experiments also illustrate that galaxy encounters and merging are quite sensitive to external tidal forces, such as those produced by the gravitational potential in a group or cluster of galaxies.
