The fully nonlinear gravitationally induced trajectories of a nearly complete set of galaxies, groups, and clusters in the Local Supercluster are constructed in a numerical action method model constrained by data from the CosmicFlows survey and various distance indicators. We add the gravity field due to inhomogeneities external to the sample sphere by making use of larger-scale peculiar flow measurements. Assignments of total masses were made to find the best overall set of mutual attractions, as determined by a goodness criterion based on present-day radial velocities, individually for the Virgo Cluster, M31, and the Milky Way (MW), and via a mass-to-light ratio relationship for other masses. The low median chi-square found indicates that the model fits the present-day velocity flow well, but a slightly high mean chi-square may indicate that some masses underwent complex orbits. The best fit, when setting the value of H_0_ to the CosmicFlows value of 75km/s/Mpc and the WMAP value for {Omega}_m_=0.244 consistent with that H_0_, occurs with the following parameters: {Omega}_orphan_=0.077+/-0.016, M/L_K_=40+/-2L_10_^0.15^M_{sun}_/L_{sun}_ (L_10_ is the K-band luminosity in units of 10^10^L_{sun}_), a Virgo mass of 6.3+/-0.8x10^14^M_{sun}_ (M/L_K_=113+/-15M_{sun}_/L_{sun}_), and a mass for the MW plus M31 of 5.15+/-0.35x10^12^M_{sun}_. The best constant mass-to-light ratio is M/L_K_=58+/-3M_{sun}_/L_{sun}_. The Virgocentric turnaround radius is 7.3+/-0.3Mpc. We explain several interesting trends in peculiar motions for various regions now that we can construct the 3D orbital histories.