An exact Lagrangian integral for the Newtonian gravitational field strength

An exact expression for the gravitational field strength in a self-gravitating dust continuum is derived within the Lagrangian picture of continuum mechanics. From the Euler–Newton system a transport equation for the gravitational field strength is formulated and then integrated along trajectories of continuum elements. The resulting integral solves one of the Lagrangian equations of the corresponding Lagrange–Newton system in general. Relations to known exact solutions without symmetry in Newtonian gravity are discussed. The presented integral may be employed to access the nonperturbative regime of structure formation in Newtonian cosmology, and to apply iterative Lagrangian schemes to solve the Lagrange–Newton system.