An exact Lagrangian integral for the Newtonian gravitational field strength

TL;DR

This paper derives an exact Lagrangian integral expression for gravitational field strength in a self-gravitating dust continuum, enabling non-perturbative analysis of structure formation in Newtonian cosmology.

Contribution

It introduces a novel integral solution for gravitational field strength within the Lagrangian framework, extending the analytical tools for Newtonian gravity without symmetry assumptions.

Findings

Provides an exact integral expression for gravitational field strength.

Connects the integral to known solutions in Newtonian gravity.

Facilitates non-perturbative analysis of cosmic structure formation.

Abstract

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 non-perturbative regime of structure formation in Newtonian cosmology, and to apply iterative Lagrangian schemes to solve the Lagrange-Newton system.