Gravitational Potential and Nonrelativistic Lagrangian in Modified
  Gravity with Varying G
  We have recently shown that the baryonic Tully-Fisher (BTF) and Faber-Jackson
(BFJ) relations imply that the gravitational "constant" $G$ in the force law
vary with acceleration $a$ as $1/a$. Here we derive the converse from first
principles. First we obtain the gravitational potential for all accelerations
and we formulate the Lagrangian for the central-force problem. Then action
minimization implies the BTF/BFJ relations in the deep MOND limit as well as
weak-field Weyl gravity in the Newtonian limit. The results show how we can
properly formulate a nonrelativistic conformal theory of modified dynamics that
reduces to MOND in its low-acceleration limit and to Weyl gravity in the
opposite limit. An unavoidable conclusion is that $a_0$, the transitional
acceleration in modified dynamics, does not have a cosmological origin and it
may not even be constant among galaxies and galaxy clusters.
