Pendulum¶

In this tutorial, we demonstrate how to generate data of a pendulum, and introduce the mathematics of a pendulum.

In [1]:

import math

import plotly.express as px

from hamilflow.models.pendulum import Pendulum

import math import plotly.express as px from hamilflow.models.pendulum import Pendulum

Constants¶

In [2]:

omega0 = 2 * math.pi
theta0 = math.pi / 3

n_periods = 2**2
n_samples_per_period = 2**8

omega0 = 2 * math.pi theta0 = math.pi / 3 n_periods = 2**2 n_samples_per_period = 2**8

A pendulum¶

In [3]:

pen = Pendulum(system=omega0, initial_condition=theta0)

pen = Pendulum(system=omega0, initial_condition=theta0)

Data¶

In [4]:

df_pen = pen.generate_from(
    n_periods=n_periods,
    n_samples_per_period=n_samples_per_period,
)
df_pen.head()

df_pen = pen.generate_from( n_periods=n_periods, n_samples_per_period=n_samples_per_period, ) df_pen.head()

Out[4]:

	t	x	u
0	0.000000	1.047198	1.570796
1	0.004192	1.046897	1.593608
2	0.008384	1.045996	1.616424
3	0.012576	1.044494	1.639247
4	0.016768	1.042393	1.662082

In [5]:

df_pen.describe()

df_pen.describe()

Out[5]:

	t	x	u
count	1024.000000	1.024000e+03	1024.000000
mean	2.144268	2.320193e-17	14.124895
std	1.239809	7.453532e-01	7.261249
min	0.000000	-1.047198e+00	1.570796
25%	1.072134	-7.494689e-01	7.848279
50%	2.144268	-3.535251e-16	14.125761
75%	3.216402	7.494689e-01	20.403244
max	4.288536	1.047198e+00	26.680726

Plot¶

In [6]:

px.line(
    df_pen,
    x="t",
    y="x",
    title=rf"Simple Harmonic Oscillator ($\omega_0 = {omega0:.4f})$",
    labels={"x": r"Angle $\theta(t)$", "t": r"Time $t$"},
)

px.line( df_pen, x="t", y="x", title=rf"Simple Harmonic Oscillator ($\omega_0 = {omega0:.4f})$", labels={"x": r"Angle $\theta(t)$", "t": r"Time $t$"}, )

TODO¶

• Compare with a harmonic oscillator in terms of frequency and profile
• Animate the plot as a single pendulum
• Add references to the derivation
• Complete the derivation

Mathematical-physical description¶

Lagrangian action¶

We describe a generic pendulum system by the Lagrangian action $$ S_L[\theta] \equiv \int_{t_0}^{t_1} \mathbb{d}t\,L(\theta, \dot\theta) \eqqcolon I \int_{t_0}^{t_1} \mathbb{d}t \left\{\frac{1}{2} \dot\theta^2 + \omega_0^2 \cos\theta \right\}\,, $$ where $L$ is the Lagrangian; $\theta$ is the angle from the vertical to the pendulum as the generalised position; $I$ is the inertia parameter, $\omega_0$ the frequency parameter, and we also call $U \coloneqq I\omega_0^2$ the potential parameter.

This setup contains both the single and the physical pendula. For a single pendulum, $$ I = m l^2\,,\qquad U = mgl\,, $$ where $m$ is the mass of the pendulum, $l$ is the length of the rod or cord, and $g$ is the gravitational acceleration.

Integral of motion¶

The Lagrangian action does not contain time $t$ explicitly. As a result, the system is invariant under a variation of time, or $\mathbb{\delta}S / \mathbb{\delta}{t} = 0$. This gives an integral of motion $$ \dot\theta\frac{\partial L}{\partial \dot\theta} - L \equiv E \eqqcolon I \omega_0^2 \cos\theta_0\,, $$ where $\theta_0$ is the initial angle.

Substitution gives $$ \left(\frac{\mathbb{d}t}{\mathbb{d}\theta}\right)^2 = \frac{1}{2\omega_0^2} \frac{1}{\cos\theta - \cos\theta_0}\,. $$

Coordinate transformation¶

For convenience, introduce the coordinate $u$ and the parameter $k$ $$ \sin u \coloneqq \frac{\sin\frac{\theta}{2}}{k}\,,\qquad k \coloneqq \sin\frac{\theta_0}{2} \in [-1, 1]\,. $$ One arrives at $$ \left(\frac{\mathbb{d}t}{\mathbb{d}u}\right)^2 = \frac{1}{\omega_0^2} \frac{1}{1-k^2\sin^2 u}\,. $$ The square root of the second factor on the right-hand side makes an elliptic integral.

End of Notebook¶