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

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