Energy and symmetry-preserving formulation of nonlinear constraints and potential forces in multibody dynamics

Practical multibody models are typically composed by a set of bodies (rigid or deformable) linked by joints, represented by constraint equations, and in many cases are subject to potential forces. Thus, a proper formulation of these constraints and forces is an essential aspect in the numerical analysis of their dynamics. On the other hand, geometric integrators are particular time-stepping schemes that have been successfully employed during the last decades in many applications, including multibody systems. One category is the so-called energy-momentum (EM) schemes that exhibit excellent stability and physical accuracy, conserving the discrete energy in conservative systems and preserving possible symmetries related with the conservation of linear and angular momenta. The use of the discrete derivative concept greatly systematizes their formulation, but its particular expression is not unique. This paper discusses the properties of several discrete derivative formulas proposed in the literature for models possessing finite-dimensional and linear configuration spaces, showing that not all of them lead to proper EM schemes. Some formulas, when applied to constraints or potentials endowed with certain symmetries (associated with the conservation of linear and angular momenta, found in many common practical systems), may produce numerical results that conserves the energy but violates the symmetries, introducing numerical instabilities and/or producing unphysical motions. This fact, surprisingly overlooked by many authors, is carefully analyzed and illustrated with several numerical experiments related with the dynamics of representative multibody systems.