Symmetric Reduction of Regular Controlled Lagrangian System with Momentum Map

The paper proves that RpCL-/RoCL-equivalence of unreduced RCL systems is equivalent to RCL-equivalence of the corresponding reduced systems (Theorems 4.3 and 5.3). The candidate’s argument mirrors the paper’s proof strategy: it uses the decomposition of the closed-loop field into Euler–Lagrange plus vertical-lift terms, equivariance on momentum level sets, and the induced diffeomorphism on reduced spaces; and, in the converse direction, it constructs a control on the target system by inverting the vertical-lift on the vertical difference Δ, exactly as encoded by the paper’s Eq. (3.2)/(3.3). Minor details about global smoothness and membership in the control subset are handled at the level of the paper by diagram-chasing and surjectivity/injectivity arguments; the candidate’s proof implicitly uses the same facts. Overall, both are correct and essentially the same proof at different levels of explicitness.