--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.13.6 kernelspec: display_name: Python 3 (pytimeode) language: python name: pytimeode --- ```{code-cell} :tags: [hide-cell] import mmf_setup;mmf_setup.nbinit() import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL) %matplotlib inline import numpy as np, matplotlib.pyplot as plt ``` (sec:demonstration)= JupyterBook Demonstration ========================= This document demonstrates some of the features provided by the documentation. It can serve as a starting point for your own documentation. This documentation is formatted using an extension to [Markdown][] called [MyST][] which allows full interoperability with [Sphinx][], the system used to build the documentation. Here we demonstrate some features. For more details, please see: * [Jupyter Book][] * [Jupyter Book with Sphinx][] * [MyST Cheatsheet][] ## Math You can use LaTeX math with [MathJaX][]. ### Example Here we implement a simple example of a pendulum of mass $m$ hanging down distance $r$ from a pivot point in a gravitational field $g>0$ with coordinate $\theta$ so that the mass is at $(x, z) = (r\sin\theta, -r\cos\theta)$ and $\theta=0$ is the downward equilibrium position: \begin{gather*} L(\theta, \dot{\theta}, t) = \frac{m}{2}r^2\dot{\theta}^2 + mgr\cos\theta\\ p_{\theta} = \pdiff{L}{\dot{\theta}} = mr^2 \dot{\theta}, \qquad \dot{\theta}(\theta, p_{\theta}, t) = \frac{p_{\theta}}{mr^2},\\ H(\theta, p_{\theta}) = p_{\theta}\dot{\theta} - L = \frac{p_{\theta}^2}{2mr^2} - mgr\cos\theta,\\ \vect{y} = \begin{pmatrix} \theta\\ p_{\theta} \end{pmatrix},\qquad \dot{\vect{y}} = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} \cdot \begin{pmatrix} \pdiff{H}{\theta}\\ \pdiff{H}{p_{\theta}} \end{pmatrix} = \begin{pmatrix} p_{\theta}/mr^2\\ -mgr\sin\theta \end{pmatrix}. \end{gather*} ```{note} On [CoCalc][], only a subset of the math will render in the live Markdown editor. This uses [KaTeX][], which is much faster than [MathJaX][], but more limited. The final documentation will use [MathJaX][], and loads the macros defined in `Docs/_static/math_defs.tex`. ``` ### SI Units In LaTeX, I like to use the [siunitx][] package. This is not currently supported in MathJax (they [could not get funding](https://github.com/mathjax/MathJax/issues/447)), but can be mocked up with [some defines](https://github.com/KaTeX/KaTeX/issues/817). The idea is that we use the macros for HTML, but substitute the appropriate `\usepackage{siunitx}` in the LaTeX files. *(This second step is currently not implemented)* :::{admonition] To Do: Add a mechanism for different MathJaX vs LaTeX defines. ::: [math]: [KaTeX]: [siunitx]: ## Jupyter Notebooks This document is actually a [Jupyter][] notebook, synchronized with [Jupytext][]. The top of the document contains information about the kernel that should be used etc. These are parsed using [MyST-NB][] and the output of cells will be displayed. For example, here is a plot demonstrating [Liouville's Theorem][]. ```{code-cell} :tags: [hide-input, full-width] plt.rcParams['figure.dpi'] = 300 from scipy.integrate import solve_ivp alpha = 0.0 m = r = g = 1.0 w = g/r T = 2*np.pi / w # Period of small oscillations. def f(t, y): # We will simultaneously evolve N points. N = len(y)//2 theta, p_theta = np.reshape(y, (2, N)) dy = np.array([p_theta/m/r**2*np.exp(-alpha*t), -m*g*r*np.sin(theta)*np.exp(alpha*t)]) return dy.ravel() # Start with a circle of points in phase space centered here def plot_set(y, dy=0.1, T=T, phase_space=True, N=10, Nt=5, c='C0', Ncirc=1000, max_step=0.01, _alpha=0.7, fig=None, ax=None): """Plot the phase flow of a circle centered about y0. Arguments --------- y : (theta0, ptheta_0) Center of initial circle. dy : float Radius of initial circle. T : float Time to evolve to. phase_space : bool If `True`, plot in phase space $(q, p)$, otherwise plot in the "physical" phase space $(q, P)$ where $P = pe^{-\alpha t}$. N : int Number of points to show on circle and along path. Nt : int Number of images along trajectory to show. c : color Color. alpha : float Transparency of regions. max_step : float Maximum spacing for times dt. Ncirc : int Minimum number of points in circle. """ global alpha skip = int(np.ceil(Ncirc // N)) N_ = N * skip th = np.linspace(0, 2*np.pi, N_ + 1)[:-1] z = dy * np.exp(1j*th) + np.asarray(y).view(dtype=complex) y0 = np.ravel([z.real, z.imag]) skip_t = int(np.ceil(T / max_step / Nt)) Nt_ = Nt * skip_t + 1 t_eval = np.linspace(0, T, Nt_) res = solve_ivp(f, t_span=(0, T), y0=y0, t_eval=t_eval) assert Nt_ == len(res.t) thetas, p_thetas = res.y.reshape(2, N_, Nt_) ylabel = r"$p_{\theta}$" if phase_space: ylabel = r"$P_{\theta}=p_{\theta}e^{-\alpha t}$" p_thetas *= np.exp(-alpha * res.t) if ax is None: fig, ax = plt.subplots() ax.plot(thetas[::skip].T, p_thetas[::skip].T, "-k", lw=0.1) for n in range(Nt+1): tind = n*skip_t ax.plot(thetas[::skip, tind], p_thetas[::skip, tind], '.', ms=0.5, c=c) ax.fill(thetas[:, tind], p_thetas[:, tind], c=c, alpha=_alpha) ax.set(xlabel=r"$\theta$", ylabel=ylabel, aspect=1) return fig, ax fig, ax = plt.subplots(figsize=(10,5)) for n, y in enumerate(np.linspace(0.25, 1.75, 6)): plot_set(y=(y, y), c=f"C{n}", ax=ax) ``` :::{sidebar} Phase flow for a pendulum. The small colored circular region is evolved forward in time according to the Hamiltonian equations. The trajectories of 10 equally spaced points are shown at 6 different times equally spaced from $t=0$ to $T = 2\pi\sqrt{r/g}$, the period of oscillation for small amplitude modes. At small energies (blue circle), the period is almost independent of the amplitude, and the circle stays together. As the energy increases (orange, green, and red), the period starts to depend more sensitively on the amplitude, and the initial circular region starts to shear. The purple region divides the two qualitatively different regions of rotation and libration, and gets stretched as some points oscillate and others orbit. The areas remain constant, despite this stretching, as a demonstration of Liouville's theorem. ::: `````{admonition} Details We start a code-block with: ````markdown ```{code-cell} :tags: [hide-input, full-width] ... ``` ```` This indicates that it is a code cell, to be run with python 3, and has some [tags](https://myst-nb.readthedocs.io/en/latest/use/hiding.html) to make the cell and output full width, but hiding the code. (There is a link to click to show the code.) ````` If you need more control, you can [glue][] the output to a variable, then load it as a proper figure, insert it in a table, etc. ## Manim Your project was generated from a template without Manim support. To use, please regenerate the project with `use_manim = "yes"`. the [Manim Community]: [Jacobi elliptic functions]: [glue]: [MyST]: "MyST - Markedly Structured Text" [Sphinx]: [Markdown]: [MyST Cheatsheet]: [Jupyter Book]: [Jupyter Book with Sphinx]: [Jupyter]: "Jupyter" [Jupytext]: "Jupyter Notebooks as Markdown Documents, Julia, Python or R Scripts" [MySt-NB]: [Liouville's Theorem]: