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We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2\phi=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and space). Also, the associated EOM for point particle takes $$\ddot{x}+\nabla\phi=0.$$ One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2\phi=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and space). Also, the associated equation of motion for point particle takes $$\ddot{x}+\nabla\phi=0.$$ One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and space). Also, the associated EOM for point particle takes $$\ddot{x}+\nabla\phi=0.$$ One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2\phi=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and space). Also, the associated EOM for point particle takes $$\ddot{x}+\nabla\phi=0.$$ One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $\nabla^2=4\pi \rho$, where $\rho$ is mass density (which, can depend on time and space). Also, the associated EOM for point particle takes $\ddot{x}+\nabla\phi=0$. One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?

Many thanks! :-)

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and space). Also, the associated EOM for point particle takes $$\ddot{x}+\nabla\phi=0.$$ One of the basic requirement for a classical theory is that it should not depend on the inertial reference frame we are choosing. In particular, for a non-relativist theory such as the one described above, I would expect the theory to keep its form under Galilean transformations. I am, however, not sure how to do this rigorously with a general $\phi$. Any ideas?