# Free gravitational field in the metric gravity as a Hamilton system

## TL;DR

This paper develops a Hamiltonian framework for the free gravitational field in metric General Relativity, deriving Hamilton equations and the Hamilton-Jacobi equation to facilitate analysis of gravitational dynamics.

## Contribution

It introduces a Hamiltonian formulation for the free gravitational field in metric GR, including Hamilton and Hamilton-Jacobi equations, enhancing analytical tools for gravitational studies.

## Key findings

- Derived Hamilton equations for all tensor components of the gravitational field.
- Formulated the Hamilton-Jacobi equation for the free gravitational field.
- Demonstrated the effectiveness of Hamiltonian methods in metric GR applications.

## Abstract

The closed system of Hamilton equations is derived for all tensor components of the free gravitational field $g_{\alpha\beta}$ and corresponding momenta $\pi^{\gamma\delta}$ in the metric General Relativity. The Hamilton-Jacobi equation for the free gravitational field $g_{\alpha\beta}$ is also derived and discussed. In general, all methods and procedures based on the Hamilton and Hamilton-Jacobi approaches are very effective in actual applications to many problems known in the metric GR.