On the model of the compressible hyperelastic rods and Euler equations on the circle

Considered herein is a geometric investigation on the one-parameter γ-equations modeled in the cylindrical compressible hyperelastic rods. It is shown that the family of equations can only be realized as an Euler equation on the Lie group Diff(S1) of all smooth and orientation preserving diffeomorphisms on the circle if the material parameter γ=1, which corresponds to the Camassa–Holm equation. In contrast, the Benjamin–Bona–Mahony (BBM) equation with the parameter γ=0 in this family of equations is not an Euler equation on Diff(S1) for any inertia operator.