The problem of 3-dimensional, convex rigid-body collision over a plane is fully investigated; this includes bodies with sharp corners that is resolved without the need for nonsmooth convex analysis of tangent and normal cones. In particular, using nonsmooth Lagrangian mechanics, the equations of motion and jump equations are derived, which are largely dependent on the collision detection function. Following the variational approach, a Lie group variational collision integrator (LGVCI) is systematically derived that is symplectic, momentum-preserving, and has excellent long-time, near energy conservation. Furthermore, systems with corner impacts are resolved adeptly using $\epsilon$-rounding on the sign distance function (SDF) of the body. Extensive numerical experiments are conducted to demonstrate the conservation properties of the LGVCI.

翻译：针对三维凸刚体在平面上的碰撞问题进行了全面研究；这包括具有尖角的刚体，无需进行切锥和法锥的非光滑凸分析即可求解。特别地，利用非光滑拉格朗日力学推导了运动方程和跳跃方程，这些方程主要依赖于碰撞检测函数。遵循变分方法，系统性地推导了李群变分碰撞积分器（LGVCI），该积分器具有辛性、动量保持性，并在长时间内具有优异的近能量守恒特性。此外，通过对刚体的符号距离函数（SDF）进行 $\epsilon$ 圆角处理，灵巧地解决了具有尖角冲击的系统。进行了大量数值实验以验证 LGVCI 的守恒性质。