Can non-relativistic strings propagate without geometric baggage? A minimal Hamiltonian formulation in Newton-Cartan geometry

A bstract We present a minimal and dynamically consistent formulation of non-relativistic bosonic string theory in a Newton-Cartan background. Starting from a reparametrization-invariant Nambu-Goto action, we develop the Hamiltonian framework and perform a complete Dirac constraint analysis. The resulting structure exhibits first-class constraints that generate worldsheet diffeomorphisms, confirming the internal gauge consistency of the model. Using an interpolating Lagrangian, we derive a Polyakov-type action that enables a direct comparison with symmetry-based constructions known as gauging-the-algebra (GTA) approaches, which promote non-relativistic symmetry algebras to local symmetries. In contrast to GTA formulations which require additional background fields to achieve algebraic closure — our model derives all necessary geometric data dynamically from the string evolution itself. This establishes that standard geometry is sufficient to support consistent non-relativistic string dynamics. Our results provide a conceptually transparent and technically robust foundation for future studies of non-relativistic string theory in curved backgrounds.