Liouville Flow Volume and Density

Consider the "Triangular expansion" flow field. Its map is given by (e^tx₀, e^ty₀ + x₀²(e^2t - e^t)). Let us verify the properties for Initial x = 1, Initial y = 0, and Time t = 1.

Experiment

Use the interactive module at https://riano.app/module/liouville-flow-volume-density to observe the local area change.

1. Set the Flow to Linear sink. Observe how the divergence is -3.
2. Set Initial x to 1 and Initial y to 0.
3. Change the Time to 1. Notice the parallelogram shrinks, matching the Jacobian determinant e^-3 and a negative log volume ratio. The material density increases to compensate.
4. Set the Flow to Hamiltonian shear or Hyperbolic. Notice the divergence is 0, the Jacobian determinant is 1, and the volume ratio is 0. The area is preserved but the lengths are not necessarily preserved.