A variational principle for Newton-Cartan theory (Q5896399)

In the framework of affine geometry with a one-form field (time metric) and a rank 3 tensor field (space metric) Newton's equations can be formulated in a way analogous to Einstein's equations (Newton-Cartan theory). So far, it has not been possible to derive these field equations from a variational principle. By setting up an action consisting solely of constraints we succeed in deriving part of the field equations, particularly Newton's equations with an unspecified force. For non- rotating fluids we do get a restriction on the shear tensor along the streamlines, however. In the variational principle, the 4-velocity of matter is non-minimally coupled to the gravitational field.