Hide code cell content 
import mmf_setup;mmf_setup.nbinit()
from pathlib import Path
FIG_DIR = Path(mmf_setup.ROOT) / '../Docs/_build/figures/'
os.makedirs(FIG_DIR, exist_ok=True)
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt
try: from myst_nb import glue
except: glue = None

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:

  • Choose "Trust Notebook" from the "File" menu.
  • Re-execute this cell.
  • Reload the notebook.

 
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[1], line 4
      2 from pathlib import Path
      3 FIG_DIR = Path(mmf_setup.ROOT) / '../Docs/_build/figures/'
----> 4 os.makedirs(FIG_DIR, exist_ok=True)
      5 import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
      6 get_ipython().run_line_magic('matplotlib', 'inline')

NameError: name 'os' is not defined

Floquet Theory#

This analysis falls under the broader term of Floquet theory which derives from the following theorem due to Floquet concerning the solutions to a linear first-order differential equation of the form

\[\begin{gather*} \dot{\vect{x}} = \mat{A}(t)\vect{x}, \qquad \mat{A}(t+T) = \mat{A}(t), \end{gather*}\]

where the time-dependence is periodic with period \(T\). If \(\mat{\phi}(t)\) is a fundamental matrix solution – i.e. all columns are linearly independent solutions – then

\[\begin{gather*} \mat{\phi}(t + T) = \mat{\phi}(t)\underbrace{\mat{\phi}^{-1}(0)\mat{\phi}(T)}_{e^{T\mat{B}}}. \end{gather*}\]

The matrix \(\mat{\phi}^{-1}(0)\mat{\phi}(T) = e^{T\mat{B}}\) is called the monodromy matrix, and for any \(\mat{B}\) that satisfies this (there may be several), the solutions can be expressed in terms of a periodic function matrix-valued function \(\mat{P}(t)\):

\[\begin{gather*} \mat{\phi}(t) = \mat{P}(t)e^{t\mat{B}}, \qquad \mat{P}(t+T) = \mat{P}(t). \end{gather*}\]

Note: this is a generalization of Bloch’s theorem which states that eigenfunctions for a periodic potential expressed as periodic solutions times a phase factor \(e^{\I \vect{k}\cdot\vect{x}}\).

For the previous system, we note that, about the unstable equilibrium point, the system can be expressed as

\[\begin{gather*} \underbrace{\diff{}{t} \begin{pmatrix} \theta\\ \dot{\theta} \end{pmatrix} }_{\dot{\vect{x}}} = \underbrace{ \begin{pmatrix} 0 & 1\\ \omega^2(t) \end{pmatrix} }_{\mat{A}(t)} \underbrace{ \begin{pmatrix} \theta\\ \dot{\theta} \end{pmatrix} }_{\vect{x}} \end{gather*}\]

sec:FloquetTheory.