We present non-convex maximal dissipation principle (NMDP), a time integration scheme for articulated bodies with simultaneous contacts. Our scheme resolves contact forces via the maximal dissipation principle (MDP). Prior MDP solvers compute contact forces via convex programming by assuming linearized dynamics integrated using the forward multistep scheme. Instead, we consider the coupled system of nonlinear Newton-Euler dynamics and MDP, which is time-integrated using the backward integration scheme. We show that the coupled system of equations can be solved efficiently using the projected gradient method with guaranteed convergence. We evaluate our method by predicting several locomotion trajectories for a quadruped robot. The results show that our NMDP scheme has several desirable properties including: (1) generalization to novel contact models; (2) superior stability under large timestep sizes; (3) consistent trajectory generation under varying timestep sizes.