Hamiltonian properties of earthquakes flows on surfaces with closed geodesic boundary

The Teichm\"uller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to $\mathcal{T}_S(\mathbf{b})$ is Hamiltonian, by providing a Hamiltonian $\mathbb{L}$. Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove that any non-Fuchsian affine representation of $\pi_1(S)$ into $\mathbb{R}^{2,1}\rtimes SO_0(2,1)$ with cocompact discrete linear part is determined by the singularities of the two invariant regular domains in $\mathbb{R}^{2,1}$ pointed out by Barbot, once the boundary lengths are fixed.