Complete Integrability of Cohomogeneity-one strings in ${\mathbb{R}}^{n,1}$ and Canonical Form of Killing Vector Algebra

Abstract

The equation of motion for comohogeneity-one Nambu-Goto strings in flat space ${\mathbb{R}}^{n,1}$ has been investigated. We first classify possible forms of the Killing vector fields in ${\mathbb{R}}^{n,1}$ after appropriate action of the Poincaré group. Then, all possible forms of the Hamiltonian for the cohomogeneity-one Nambu-Goto strings are determined. It has been shown that the system always has the maximum number of functionally independent, pair-wise commuting conserved quantities, i.e. it is completely integrable. We have also determined all the possible coordinate forms of the Killing vector basis for the 2-dimensional non-commutative Lie algebra.

26 pages, no figures