Geometry — Analysis — Numerics
This is the group webpage of researchers in applied dynamics at
Massey University in Palmerston North.
This diagram to Theorem I from Newton's Principia Mathematica of 1687 (The areas which revolving bodies describe by radii drawn to an immovable centre of force... are proportional to the times in which they are described) illustrates many of our themes:
Dynamical systems: a planet orbiting the sun (S).
Geometry: Newton famously presented many of his theorems with geometrical proofs, although in some cases there is controversy as to whether he originally discovered them this way. Theorem I is an example of Noether's theorem (here, of conservation of angular momentum) connecting symmetries and conservation laws, the classic instance of geometry in differential equations.
Analysis: To get the theorem on continuous motion, Newton lets the time steps (A,B,C...) "be diminished ad infinitum".
Numerics: The diagram illustrates the leapfrog method, introduced to computational chemistry in the modern era by Loup Verlet in 1967 and today the workhorse of geometric integrators. The smooth motion of the planet is discretized into discrete steps (A,B,C...) and further decomposed into a drift phase (A -> B) and a kick phase (c -> C) that updates the velocity. Unintentionally, the diagram also shows that the leapfrog method itself (and not just the continuous motion) conserves angular momentum.