Arnold 9: Canonical Formalism#
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import mmf_setup
mmf_setup.nbinit()
import logging
logging.getLogger("matplotlib").setLevel(logging.CRITICAL)
%pylab inline --no-import-all
import manim.utils.ipython_magic
!manim --version
This cell adds /home/docs/checkouts/readthedocs.org/user_builds/physics-521-classical-mechanics-i/checkouts/latest/src to your path, and contains some definitions for equations and some CSS for styling the notebook. If things look a bit strange, please try the following:
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Populating the interactive namespace from numpy and matplotlib
Manim Community v0.18.0
Manim Community v0.18.0
This document contains some random notes and discussions about Chapter 9 of [Arnol'd, 1989].
§44 The integral invariant of Pointcaré-Cartan#
In 3D, every vector field has a curl \(\vect{r} = \vect{\nabla}\times \vect{v}\), and one can find the integral curves of these, which Arnold calls vortex lines. Since \(\vect{\nabla}\cdot \vect{r} = 0\), these vortex lines have no “source”, hence flux is conserved. They need not form closed loops.
Here we plot some
def v(x, y, z):
"""Return a 3D vector field."""
return (y, z, x)