This set of dynamical equations was found by Shūichi Nosé in 1984 when researching about molecular dynamics and thermal equilibrium distribution¹:

\[\dot{q} = \frac{p}{s^2}\] \[\dot{p} = -q\] \[\dot{s} = \zeta\] \[\dot{\zeta} = \frac{p^2}{s^3} - \frac{T}{s}\]

Further research by William Graham Hoover² a year later simplified them to³:

\[\dot{q} = p\] \[\dot{p} = -q - \zeta\, p\] \[\dot{\zeta} = p^2 - T\]

Renders

With \(T = 1.5\):

Framing It

I liked the third render so much, that I printed it on 45×35 cm fine-art paper and framed it. Read more in my blog post.

Stereographic Animation

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1. S. Nosé, 1984. "A unified formulation of the constant temperature molecular dynamics methods". J. Chem. Phys. 81(1). doi:10.1063/1.447334. ↩

2. Professor Doctor Hoover has a great homepage where you can find all of his research, books and lectures. ↩

3. W.G. Hoover, 1985. "Canonical dynamics: Equilibrium phase-space distribution". Phys. Rev. A. 31(3). doi:10.1103/PhysRevA.31.1695. ↩