Reduction of the canonical Hamiltonian of the metric GR to its natural form

TL;DR

This paper transforms the canonical Hamiltonian of metric General Relativity into its natural form, enhancing analytical and numerical analysis, and establishing analogies with particle systems and methods for Poisson brackets.

Contribution

It introduces a method to reduce the Hamiltonian to its natural form and develops techniques for calculating Poisson brackets in metric GR.

Findings

Hamiltonian reduced to natural form facilitates applications.

Analogy established between gravitational fields and particle systems.

Effective method for Poisson brackets developed.

Abstract

The canonical Hamiltonian $H_C$ of the metric General Relativity is reduced to its natural form. The natural form of canonical Hamiltonian provides numerous advantages in actual applications to the metric GR, since the general theory of dynamical systems with such Hamiltonians is well developed. Furthermore, many analytical and numerically exact solutions for dynamical systems with natural Hamiltonians have been found and described in detail. In particular, based on this theory we can discuss an obvious analogy between gravitational field(s) and few-particle systems where particles are connected to each other by the Coulomb, or harmonic potentials. We also developed an effective method which is used to determine various Poisson brackets between analytical functions of the dynamical variables. Furthermore, such variables can be chosen either from the straight, or dual sets of symplectic dynamical variables which always arise in any Hamiltonian formulation developed for the metric gravity. PACS number(s): 04.20.Fy and 11.10.Ef