The human environment consists of a large variety of mechanical and biomechanical systems in which different types of contact can occur. In this work, we consider a monopedal jumper modelled as a three-dimensional rigid multibody system with contact and simulate its dynamics using a structure preserving method. The applied mechanical integrator is based on a constrained version of the Lagrange-d’Alembert principle. The resulting variational integrator preserves the symplecticity and momentum maps of the multibody dynamics. To ensure the structure preservation and the geometric correctness, we solve the non-smooth problem including the computation of the contact configuration, time and force instead of relying on a smooth approximation of the contact problem via a penalty potential. In addition to the formulation of non-smooth problems in forward dynamic simulations, we are interested in the optimal control of the monopedal high jump. The optimal control problem is solved using a direct transcription method transforming it into a constrained optimisation problem, see [14].