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：本文全面研究了三维凸刚体在平面上的碰撞问题，包括具有尖角的刚体，且无需使用切锥和法锥的非光滑凸分析。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，这些方程在很大程度上依赖于碰撞检测函数。遵循变分方法，系统性地推导了一种李群变分碰撞积分器（LGVCI），该积分器具有辛性、动量守恒性，并展现了优异的长时间近能量守恒特性。此外，通过使用刚体的符号距离函数（SDF）的$\epsilon$-圆角化方法，巧妙地解决了带有尖角冲击的系统。进行了大量数值实验以验证LGVCI的守恒性质。