AQUAL is a theory of gravity based on Modified Newtonian Dynamics (MOND), but using a Lagrangian. It was developed by Jacob Bekenstein and Mordehai Milgrom in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian gravity?". "AQUAL" stands for "A QUAdratic Lagrangian".
The gravitational force law obtained from MOND,
has a serious defect: it violates Newton's third law of motion, and therefore fails to conserve momentum and energy. To see this, consider two objects with
but the third law gives
This problem can be rectified by deriving the force law from a Lagrangian, at the cost of possibly modifying the general form of the force law. Then conservation laws could then be derived from the Lagrangian by the usual means.
The AQUAL Lagrangian is:
this leads to a modified Poisson equation:
Here,