Geometric integration is a new approach to simulating the motion of large systems. The new methods, inspired by chaos theory but driven by the demands of modern applications are faster, more reliable, and often simpler than traditional approaches. They are being used in areas as diverse as a possible celestial origin of the ice ages, the structure of liquids, polymers, and biomolecules, quantum mechanics and nanodevices, biological models, chemical reaction-diffusion systems, the dynamics of flexible structures, and weather forecasting.

Although diverse, these systems have certain things in common that makes them amenable to the new approach. They all preserve some underlying geometric structure which influences the qualitative nature of the phenomena they produce. In geometric integration these properties are built into the numerical method, which gives the method markedly superior performance, especially during long simulations. In our research we are exploring all possible geometric or structural features that systems can have, the implications for their long-time dynamics, and how to design efficient numerical integrators that preserve these geometric properties.

Here is a selection of pictures, some new, some historical, mostly culled from the Web, showing some of the applications of geometric integrators.    -- Robert McLachlan

(From S. Edvardsson, K. G. Karlsson, and M. Engholm, Accurate spin axes and solar system dynamics: Climatic variations for the Earth and Mars, Astronomy & Astrophysics 384, 689-701 (2002)). In this study a one million year simulation of the whole solar system including the moons and obliquity of Earth and Mars found strong correlation between the ice ages and the earth's orbit. The motion is calculated with a symplectic integrator.

A frame from a nice movie made by Sverker Edvardsson of a simulation of the asteroid belt and the inner solar system.

The protein molecule kinesin walks along a pre-prepared track inside a cell. This is a single frame of a movie from the Vale Lab, UCSF, of a simulation of this walk. It is not calculated using true molecular dynamics; although all the states shown are local energy minima, they are not linked together dynamically.

(Left) Sketch of the flow on a family of invariant tori. From V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk (Russ. Math. Surv.) 18 (1963) no. 6 (114) 91--192. (Right) Three computed tori in an actual Hamiltonian system, arising in celestial mechanics. P. Atela and R. I. McLachlan, Global behaviour of the charged isosceles three-body problem, Int. J. Bifurcations Chaos 4(4) (1994), 865-884.

Also from the 1963 Arnold book, a sketch showing a cross-section of the tori in the figure above after perturbation. Some are destroyed and replaced by chaos, some persist. Poincare famously knew of this structure but said he would not attempt to draw it! A lot of this structure is typically preserved by a symplectic integrator.

This familiar image of the phase portrait of the free rigid body (such as a hammer tossed in the air), from Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, illustrates a variety of features of a dynamical system which one may want to preserve in a simulation: (i) it is a noncanonical (Poisson) Hamiltonian system; (ii) the motion lies on a sphere, i.e. a non-Euclidean phase space; (iii) the sphere is a level set of a Casimir (conserved quantity) of the system's Poisson structure; (iv) the curves are the level sets of the energy; (v) the system has homoclinic orbits. (vi) the system is reversible under reflection through the centre of the sphere.

These early pictures showing the results of a symplectic integration, from P. J. Channell and J. C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), 231--259, prompted a rush of interest in symplectic integrators. In the first picture, the symplectic integrator has captured a slice through an invariant torus of the Henon-Heiles system. On the right, a (nonsymplectic) integrator has destroyed the torus. In the section picture, 10⁵ time steps for 10 different initial conditions are shown. Smooth curves (``KAM tori'') correspond to regular, quasiperiodic motion; clouds correspond to chaotic motion.

Energy error of leapfrog applied to the whole solar system over 10⁸ years, from J. Wisdom and M. Holman, Symplectic maps for the N-body problem, Astron. J. 102 (1991), 1528-1538. The way the energy error oscillates but does not grow in time was an early indication of the good properties of symplectic integrators.

Evidence of chaos in the solar system: also from the preceding paper, this shows the inclination of Pluto over 10⁹ years. Even after one billion years the inclination has reached a new maximum.

 

Steve Bond has a page of nice molecular dynamics movies. Here are two frames from a movie of two C60 buckyballs (left), colliding (centre) to form a more stable C120 molecule (right).