Integrable Hamiltonian for Classical Strings on AdS_5 x S^5

We find the Hamiltonian for physical excitations of the classical bosonic string propagating in the AdS_5 x S^5 space-time. The Hamiltonian is obtained in a so-called uniform gauge which is related to the static gauge by a 2d duality transformation. The Hamiltonian is of the Nambu type and depends on two parameters: a single S^5 angular momentum J and the string tension \lambda. In the general case both parameters can be finite. The space of string states consists of short and long strings. In the sector of short strings the large J expansion with \lambda'=\lambda/J^2 fixed recovers the plane-wave Hamiltonian and higher-order corrections recently studied in the literature. In the strong coupling limit \lambda\to \infty, J fixed, the energy of short strings scales as \sqrt[4]{\lambda} while the energy of long strings scales as \sqrt{\lambda}. We further show that the gauge-fixed Hamiltonian is integrable by constructing the corresponding Lax representation. We discuss some general properties of the monodromy matrix, and verify that the asymptotic behavior of the quasi-momentum perfectly agrees with the one obtained earlier for some specific cases.