Summary: The sine-Gordon equation is considered in the Hamiltonian framework provided by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space \({\mathfrak g}^*\) of a loop algebra \({\mathfrak g}\), is parameterized by a finite dimensional symplectic vector space \(W\) embedded into \({\mathfrak g}^*\) by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.