Universal Polynomial Invariants of Nonholonomic Leapfrog

Modin and Verdier observed favorable long-time energy behavior for nonholonomic leapfrog in a rotating constrained-spring example that their reversibility argument did not explain. We investigate a foundational question: which polynomial conservation laws follow from the reflected update independently of the driver? For planar rank-one constraints and 0 < h < 2, we prove that the modified quadratic energy generates all common polynomial invariants, except at h = \( \sqrt{2} \), where an independent quartic generator appears. At this step, a change of variables turns the update into an exchange of squared lengths. This mechanism extends to arbitrary orthogonal-projector sequences in any dimension. The classification identifies the conservation shared by the update family; the original passenger-energy observation remains a separate question.