MPEJ Volume 3, No.4, 40pp Received: July 28, 1997, Revised: Sep 10, 1997, Accepted: Sep 30, 1997 Amadeu Delshams, Tere M. Seara Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom ABSTRACT: The splitting of separatrices for Hamiltonians with $1{1\over 2}$ degrees of freedom $$h(x,t/\varepsilon)=h^0(x)+\mu\varepsilon^p h^1(x,t/\varepsilon)$$ is measured. We assume that $h^0(x)=h^0(x_1,x_2)=x_2^2/2+V(x_1)$ has a separatrix $x^0(t)$, $h^1(x,\theta)$ is $2\pi$-periodic in $\theta$, $\mu$ and $\varepsilon>0$ are independent small parameters, and $p\ge 0$. Under suitable conditions of meromorphicity for $x_2^0(u)$ and the perturbation $h^1(x^0(u),\theta)$, the order $\ell$ of the perturbation on the separatrix is introduced, and it is proved that, for $p\ge\ell$, the splitting is exponentially small in $\varepsilon$, and is given in first order by the Melnikov function.