Hamiltonian for a string in a Newton-Cartan background

This paper is devoted to the construction of the Hamiltonian for nonrelativistic string in the Newton-Cartan background. We start with the Hamiltonian for relativistic string in general background. Then we perform a limiting procedure on the metric that leads to Newton-Cartan background. We determine the constraint structure for nonrelativistic string and show that these constraints are the first class constraints. Then we determine the corresponding Lagrangian and discuss its properties.