We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.