The Toda chain as a reduced system

It is proved that the phase space of a finite Toda chain can be obtained by means of the reduction of the phase space of a geodesic flow on the uniform space of symmetric positive-definite matrices. The reduction is performed with the aid of the group of triangular matrices and makes it possible to obtain explicit formulas for solutions of the equations of motion. The construction is extended to the generalised Toda chains corresponding to an arbitrary system of roots.