Energy for N-Body Motion in Two Dimensional Gravity

TL;DR

This paper defines and analyzes the concept of energy in the N-body problem within (1+1) dimensional gravity, providing a generalized Hamiltonian framework that extends previous methods.

Contribution

It introduces a general energy definition using Noether's theorem and extends Hamiltonian formulations to a broader class of solutions in 2D gravity.

Findings

Derived a consistent energy density via superpotential.

Reproduced the known Hamiltonian for N-body motion in curved spacetime.

Extended energy definitions beyond ADM-like prescriptions.

Abstract

A general definition of energy is given, via the N\"other theorem, for the N-body problem in (1+1) dimensional gravity. Within a first-order Lagrangian framework, the density of energy of a solution relative to a background is identified with the superpotential of the theory. For specific applications we reproduce the expected Hamiltonian for the motion of N particles in a curved spacetime. This Hamiltonian agrees with that found through an ADM-like prescription for the energy when the latter is applicable but it also extends to a wider class of solutions provided a suitable background is chosen.