Lie Group Analysis of Scaling Law for Domino Toppling Wave Velocity

This paper employs the Lie group method of invariants to re-investigate the domino toppling problem. By defining an anisotropic scaling group distinguishing horizontal propagation from vertical gravitational fall, we rigorously derive the universal scaling law \( v=\lambda \sqrt{\frac{g}{h}} f(\frac{\delta}{\lambda}) \) through both finite group transformation and infinitesimal generator approaches. Curve fitting yields the approximate power law \( v\sim \sqrt{\frac{\delta \lambda }{h}g} \). The Lie group decoupling reveals that the speed is governed by an effective dynamical length \( L_{eff} = \delta\lambda/h \) and is independent of domino width. Furthermore, we clarify that the power-law exponent \( \alpha \approx 1/2 \) corresponds to a complete scaling symmetry in the ideal frictionless limit. The introduction of friction breaks this symmetry, causing \( \alpha \) to fluctuate around \( 1/2 \), which is interpreted from the perspective of symmetry breaking.