  We define three fundamental solvable bilinear deformations for any massive
non-relativistic 2d quantum field theory (QFT). They include the
$\mathrm{T}\overline{\mathrm{T}}$ deformation and the recently introduced hard
rod deformation. We show that all three deformations can be interpreted as
coupling the non-relativistic QFT to a specific Newton-Cartan geometry, similar
to the Jackiw-Teitelboim-like gravity in the relativistic case. Using the
gravity formulations, we derive closed-form deformed classical Lagrangians of
the Schr\"odinger model with a generic potential. We also extend the dynamical
change of coordinate interpretation to the non-relativistic case for all three
deformations. The dynamical coordinates are then used to derive the deformed
classical Lagrangians and deformed quantum S-matrices.
