31 Lectures in Geometric Mechanics

The paper states and proves the Kelvin–Noether Circulation Theorem for continua: I(t) = ∮_{γ_t} (1/D) δℓ/δu and dI/dt = ∮_{γ_t} (1/D) (δℓ/δa ⋄ a), by (i) changing variables to a fixed reference loop via pull-back, (ii) using the transport identity d/dt(η_t^*α_t) = η_t^*(∂_tα_t + £_uα_t) for a one-form density α_t, and (iii) substituting the Euler–Poincaré equation (∂_t + £_u)(δℓ/δu) = (δℓ/δa) ⋄ a; see Section 25.6 with equations (25.21)–(25.23) and the displayed proof steps . The candidate solution follows the same structure: pull-back to the reference loop, differentiate via the same transport identity, use advection of mass density to handle the 1/D factor, and apply the Euler–Poincaré equation to conclude the diamond term. The only minor nuance is that the paper removes 1/D by pull-back using D_t = η_t^*D_0, whereas the candidate writes (∂_t + £_u)(1/D) = 0 for the scalar density, which is consistent provided D is treated as an advected density; both routes are standard and equivalent in this setting .