Constants of the Motion in a Gravitational Field and the Hamilton-Jacobi Function

TL;DR

This paper introduces a method to derive constants of motion and trajectories in gravitational fields directly from spacetime metrics using the Hamilton-Jacobi function, offering new insights into classical and cosmological motion.

Contribution

It presents a novel approach starting from general relativity metrics to construct constants of motion and trajectories via the Hamilton-Jacobi function, differing from traditional Newtonian methods.

Findings

Derived Hamilton-Jacobi function from metrics

Applied method to Kepler's First Law

Formulated galaxy trajectories in Robertson-Walker space

Abstract

In most text books of mechanics, Newton's laws or Hamilton's equations of motion are first written down and then solved based on initial conditions to determine the constants of the motions and to describe the trajectories of the particles. In this essay, we take a different starting point. We begin with the metrics of general relativity and show how they can be used to construct by inspection constants of motion, which can then be used to write down the equations of the trajectories. This will be achieved by deriving a Hamiltonian-Jacobi function from the metric and showing that its existence requires all of the above mentioned properties. The article concludes with four applications, which includes a derivation of Kepler's First Law of Motion for planets, and a formula for describing the trajectories of galaxies moving in a space defined by the Robertson-Walker metric.