From fd1038ab3a6bdec1299eb45a775c7f1bbe1db5b5 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 18 Nov 2024 13:18:54 -0500 Subject: [PATCH] vault backup: 2024-11-18 13:18:54 --- .../2024-11-18 Volume Contraction.md | 15 +++++++++++++++ .../2024-11-18.md | 0 2 files changed, 15 insertions(+) create mode 100644 300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md delete mode 100644 300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18.md diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md new file mode 100644 index 00000000..a791910d --- /dev/null +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18 Volume Contraction.md @@ -0,0 +1,15 @@ +Lorenz system is dissapative. This means: +- Volume in phase space contracts with flow? +This introduces some questions... How do volumes evolve? + +Suppose a surface $S(t)$ encloses volume $V(t)$, with normal vectors pointing away from the surface ($\vec{n}$). + +A trajectory starts on S. let them evolve for $dt$. With a flux vector $\vec{f}$, we have +- $\vec f \cdot \vec n$ - normal, outward component of velocity +In $dt$ time, $dA$ sweeps out a volume. + +Volume: $(\vec f \cdot \vec n dt)dA$ +$$V(t+dt) = V(t) + \int_S (\vec f \cdot \vec n dt)dA $$ +$$\dot{V} = \int_S (\vec f \cdot \vec n)dA $$ +Now we can apply the divergence theorem: +$$\dot{V} = \int_V (\nabla \cdot \vec f )dV $$ diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-18.md deleted file mode 100644 index e69de29b..00000000