Can Non-Relativistic Strings Propagate Without Geometric Baggage?

We present a minimal and dynamically consistent formulation of non-relativistic bosonic string theory in a Newton-Cartan (NC) 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 Newton-Cartan 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.