  For a given pseudo-Anosov homeomorphism $\varphi$ of a closed surface $S$,
the action of $\varphi$ on the Teichm\"uller space $\mathcal T(S)$ preserves
the Weil-Petersson symplectic form. We give explicit formulae for two invariant
functions $\mathcal T(S)\to \mathbb R$ whose symplectic gradients generate
autonomous Hamiltonian flows that coincide with the action of $\varphi$ at time
one. We compute the Poisson bracket between these two functions. This amounts
to computing the variation of length of a H\"older cocyle on one lamination
along a shear vector field defined by another. For a measurably generic set of
laminations, we prove that the variation of length is expressed as the cosine
of the angle between the two laminations integrated against the product
H\"older distribution, generalizing a result of Kerckhoff. We also obtain rates
of convergence for the supports of germs of differentiable paths of measured
laminations in the Hausdorff metric on a hyperbolic surface, which may be of
independent interest.
