VARIATIONAL COLLISION INTEGRATORS FOR NONHOLONOMIC LAGRANGIAN SYSTEMS

The candidate reproduces the (+)-discrete Hamilton–d’Alembert–Pontryagin action with a split impact step exactly as in the paper (cf. Sd+ in Eq. (4) and Definition 3.1) and computes the first variation in the same way as the paper’s proof (the coefficient grouping matches the displayed variation formula). From these, the candidate reads off the same kinematic constraints, discrete Legendre relations, nonholonomic d’Alembert conditions, and the elasticity condition D3Ld(·)=D3Ld(·) that appear in Theorem 3.1 for the (+)-discrete case. The only nuance is the step from (πS)^*(δp̃)·(q̃−v_i) to the full matching q̃=v_i: the variation enforces the tangential matching πS(q̃−v_i)=0, while the paper lists q̃=v_i explicitly and enforces it in Algorithm 1. Aside from this mild clarification, the arguments align and are essentially the same. See the theorem statement and stated impact conditions, the action and path-space definitions, the variation computation, and the implementation algorithm in the paper: Theorem 3.1 (+)-case and conditions ; action Sd+ and Definition 3.1 ; variation formula in the proof ; path-space/variation restrictions at the impact node ; and Algorithm 1 enforcing q̃=vk at impact .