  Smooth periodic travelling waves in the Camassa--Holm (CH) equation are
revisited. We show that these periodic waves can be characterized in two
different ways by using two different Hamiltonian structures. The standard
formulation, common to the Korteweg--de Vries (KdV) equation, has several
disadvantages, e.g., the period function is not monotone and the quadratic
energy form may have two rather than one negative eigenvalues. We explore the
nonstandard formulation and prove that the period function is monotone and the
quadratic energy form has only one simple negative eigenvalue. We deduce a
precise condition for the spectral and orbital stability of the smooth periodic
travelling waves and show numerically that this condition is satisfied in the
open region where the smooth periodic waves exist.
