Consider a system of first-order ODEs:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]x_1^{'}=f_1(x_1, \cdots ,x_n)[/tex]

[tex]x_2^{'}=f_2(x_1, \cdots ,x_n)[/tex]

[tex]\cdots[/tex]

[tex]x_n^{'}=f_n(x_1, \cdots ,x_n)[/tex]

or written more compactly just as:

[tex]X^{'}=F,\;X\in\mathbb{R}^n[/tex]

If I start with a set of initial conditions compromising some volume in phase space and evolve them according to the equations of motion above (run the equations for a whole bunch of initial conditions all contained in some region of phase space and keep track of where the final values of each x_i land after some time t), what must F be in order that the volume containing all the final points be the same as the volume containing all the initial points?

Well, I worked on this a long time ago and didn't quite understand it but someone in here recently brought up symplectic algorithms which are related and it caused me to revisit it. See, that's how science works you know.

We can calculate this volume as an n-folded integral over the n-dimensions of phase space. That is, the initial volume would be:

[tex]\int\cdots\int_{\Omega(0)} dx_1\cdots dx_n[/tex]

with [itex]\Omega(0)[/itex] being the initial domain containing the set of vectors each being a single initial condition for the system. For example, if the system only involved x(t) and y(t), that integral would simply be the area of some selected part of the x-y plane chosen to contain all the initial points [itex](x_0,y_0)[/itex] under study. It's a bit messy for n-dimensions. The final volume in that case is:

[tex]\int\cdots\int_{\Omega(t)} dx_1\cdots dx_n[/tex]

with [itex]\Omega(t)[/itex] being the region containing all the final points of the simulation after time t.

Thus, what must F be in order for:

[tex]\int\cdots\int_{\Omega(0)} dx_1\cdots dx_n=\int\cdots\int_{\Omega(t)} dx_1\cdots dx_n[/tex]

That's an interesting problem don't you guys think? This is in essence Liouville's Theorem as it pertains to Poincare's Integral Invariants. That is, the integral above is an Integral Invariant (French you know) if it doesn't change with time.

Anyway, I've just learned the proof and would like to follow-up with the details unless someone else takes the initiative.

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# Measure-preserving flow

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