Newton's graphical method as a canonical transformation

TL;DR

This paper demonstrates that Newton's graphical method in the Principia is an exact representation of a canonical transformation, revealing its role as a symplectic integrator that preserves phase-space volume and areal velocity.

Contribution

It establishes a novel connection between Newton's graphical method and modern symplectic integrators, showing its exactness and conservation properties in Hamiltonian mechanics.

Findings

Newton's Proposition 1 is an exact graphical canonical transformation.

The method conserves phase-volume and areal velocity.

Error analysis explains orbital differences for different forces.

Abstract

This work shows that, Newton's Proposition 1 in the {\it Principia}, is an {\it exact} graphical representation of a canonical transformation, a first-order symplectic integrator generated at a finite time-step by the Hamiltonian. A fundamental characteristic of this canonical transformation is to update the position and velocity vectors {\it sequentially}, thereby automatically conserving the phase-volume and the areal velocity due to a central force. As a consequence, the continuous force is naturally replaced by a series of impulses. The convergence of Newton's Proposition 1 in the limit of $\Delta t\rightarrow 0$ can be proved easily and the resulting error term for the linear and the inverse square force can explain why Hooke was able to the obtain an elliptical orbit for the former but not the latter.